US20070110229A1  Ternary and MultiValue Digital Signal Scramblers, Descramblers and Sequence of Generators  Google Patents
Ternary and MultiValue Digital Signal Scramblers, Descramblers and Sequence of Generators Download PDFInfo
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 US20070110229A1 US20070110229A1 US11/618,986 US61898607A US2007110229A1 US 20070110229 A1 US20070110229 A1 US 20070110229A1 US 61898607 A US61898607 A US 61898607A US 2007110229 A1 US2007110229 A1 US 2007110229A1
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 H03K19/00—Logic circuits, i.e. having at least two inputs acting on one output; Inverting circuits
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Abstract
Reversible and self reversing multivalue scrambling functions created by applying multivalue inverters are disclosed. The generation of possible multivalue inverters is also presented. Corresponding multivalue descrambling functions are also disclosed. The multivalue functions are used in circuits that scramble and descramble multivalue signals. The multivalue functions can also be used in signal generators. Such signal generators do not require the use of multipliers. The autocorrelation of the signals generated by the signal generators is also presented. Electronic circuits that implement the multivalue functions are also described.
Description
 This application is a continuationinpart of U.S. patent application Ser. No. 10/935,960 filed on Sep. 8, 2004 which claims the benefit of U.S. provisional patent application Ser. No. 60/547,683 filed on Feb. 25, 2004, both of which are incorporated herein by reference.
 This invention relates to the scrambling of multivalue digital (nonbinary) signals and the generation of multivalue (nonbinary) digital signals consisting of sequences of multivalue (nonbinary) digital elements such as nvalue pseudonoise sequences. Multivalue signals, also referred to as xvalue signals, can assume one of x states, wherein x is greater than or equal to three.
 Its potential application is in telecommunication systems, control systems and other applications. Specific examples of utility where the invention can be used include spreadspectrum technologies, signal scrambling, CDMA, linecoding and scrambling application in video and other signal distribution.
 Digital scramblers are used to change the appearance of a digital signal in such a way that during transmission the signal is different from the original signal. The original signal can be recovered from the scrambled signal at the receiving end by a descrambler. Most commonly in today's telecommunications, the scramblers relate to binary signals.
 Scrambling of a binary signal can be achieved by combining the binary signal to be scrambled with a second known binary signal through a digital circuit that has the characteristics of a reversible function. The inventor has recognized that the reversible function has the property that its two inputs generate an output. When applying this output as the input to another, but similar, circuit with as the other input a signal similar to the known input of the first digital circuit, the output of the second circuit is identical to the to be scrambled data signal input of the first circuit. This other circuit is a descrambler.
 There are two known binary functions that can perform this reversible function: the Exclusive Or (XOR) and the Equal function. The XOR function is also known as the modulo2 adding function.
 Telecommunication markets such as wireless communications and internet communications demonstrate an ongoing increase in demand for higher information transmission rates. This demand in increased information transmission rates in wireless communications is addressed by increasing bandwidth of communication channels, by compression of the information and by moving into much higher radio spectra (such as Ultra Wide Band in the 5 GHz area). Eventually, new technology has to be applied to obtain better performance from existing bandwidth, starting with highly congested spectrum areas. Current transmission technology predominantly uses digital binary signals. One possible technology to provide better bandwidth usage is the application of multivalue digital signals. Scrambling, descrambling and signal sequence generation is an important element of signal processing technology, especially in wireless communications. However, currently very little technology exists in prior art that can perform multivalue digital scrambling, descrambling and sequence generation. Most of prior art in scrambling, descrambling and sequence generation only performs binary functions, as previously discussed.
 Consequently, new and improved methods and apparatus to perform scrambling, descrambling and sequence generation on nonbinary, multivalue digital signals are required.
 The existence of multivalue reversible logic functions has been previously mentioned. However, no mention has been found of multivalue reversible logic in scrambling/descrambling applications, nor in the generation of multivalue signal sequences.
 In view of the more limited possibilities of the prior art in binary and multivalue scrambling (or reversible) logic functions, the current invention offers an easier design as well as a greater variety in multivalue scramblers and in LFSR based sequence generators. It also provides improved performance.
 The general purpose of the present invention, which will be described subsequently in greater detail, is to provide new scrambler/descrambler systems and LFSR based generators of multivalue signal sequences. It also provides a new electronic circuit and gates to process multivalue signals, and provides sequence generators with reduced component and count.
 Before explaining at least one embodiment of the invention in detail it is to be understood that the invention is not limited in its application to the details of construction and to the arrangements of the components set forth in the following description or illustrated in the drawings. The invention is capable of other embodiments and of being practiced and carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein are for the purpose of the description and should not be regarded as limiting.
 Multivalue and nvalue in the context of this application mean a number n, with n being a positive integer greater than two.
 A primary object of the present invention is to provide new multivalue scrambler/descrambler systems that will overcome the shortcomings of the prior art devices.
 An object of the present invention is to provide a maximum set of multivalue digital, single function, scrambling/descrambling devices.
 Another object is to provide new LFSR based multivalue digital scramblers and self synchronizing LFSR based descrambler devices.
 Another object is to provide a method for applying multivalue logic scrambling/descrambling functions in a LFSR architecture to generate multivalue digital signal sequences.
 Another object of the invention is to provide nValue LFSR based scramblers/descramblers as nvalue signal spreaders/despreaders with no synchronization requirements. As is known, these are easier to realize, but provide less security.
 Another object of the invention is to provide nValue LFSR based scramblers/descramblers as nvalue signal spreaders/despreaders with synchronization requirements. As is known, these are harder to realized, but provide higher security.
 Another object is to provide multivalue digital inverters that will change the value of a multivalue signal without upsetting the distinguishing characteristics of the individual elements of a digital signal.
 Another object is to provide composite systems comprising two or more individual systems that are connected through components that execute multivalue logic digital scrambling functions.
 Another object is to provide a method called “design by inverters” that allows creating any multivalue truth table for scrambling and descrambling functions from a set of individual multivalue digital inverters.
 Another object of the invention is to provide a method to translate prior art modulon adder functions combined with nvalue multipliers at the inputs into nvalue scrambling functions of this invention.
 Another object of the invention is to apply multivalue logic reversible inverters to received signals in order to bring desired autocorrelation and crosscorrelation levels of the signals to zero.
 Another object of the invention is to provide a method for creating multivalue logic devices of which the scrambling functions are a subset, by applying components which only consist of the following types: a gate and a universal multivalue inverter.
 Prior art in scrambling/descrambling and signal sequence generation mainly applies binary technology. Even in cases where multivalue digital signals are generated the scrambling, descrambling and sequence generation is based on modulo2 theory and embodiment. In some cases modulon addition theory and embodiment is proposed. However modulon addition is a nonself reversing technology. As such modulon addition requires a reversing function to create a nvalue scrambler/descrambler set. Also modulon addition solution in creating LFSR based scramblers, descramblers and sequence generators require multipliers to achieve statistically optimal performance.
 Consequently the present invention offers a much broader, more diverse and easier method to create multivalue scramblers, descramblers and signal sequence generators. Combined with prior art in realizing multivalue logic electronic embodiments this invention can prevent the need for “switching back to binary logic” in multivalue logic signal processing applications. Prior art such as realizing adequate 4B3T transmission signals demonstrates the overhead from binary logic to manage ternary (3value) signal generation and recovery with binary logic circuitry.
 One aspect of the present invention provides multivalue functions, including ternary and higher multivalue functions, that are useful in many applications, including those requiring scrambling, descrambling, spreading, dispreading and sequence generation.
 In accordance with one aspect of the present invention, a method of scrambling a ternary signal with a scrambler is provided. The ternary signal is able to assume one of three states and the scrambler has a first scrambling ternary logic device that implements a ternary logic function, sc, and a scrambling logic circuit. The method involves inputting the ternary signal and an output from the scrambling logic circuit to the first scrambling ternary logic device and inputting an output from the first scrambling ternary logic device to the scrambling logic circuit. The ternary logic function, sc, can be specified by a scrambling ternary truth table having three columns, and each of the columns being defined by a ternary inverter. When the ternary signal is input, the output from the first scrambling ternary logic device is a scrambled version of the ternary signal.
 In accordance with another aspect of the present invention, each column is defined by one of the following ternary inverters i_{1 }to i_{6}:
i_{1} i_{2} i_{3} i_{4} i_{5} i_{6} 0 0 0 1 1 2 2 1 1 2 0 2 0 1 2 2 1 2 0 1 0  It is noted, however, that every column cannot be defined by the identity inverter i_{1}. In that case, there would be no scrambling performed.
 In accordance with a further aspect of the present invention, at least one ternary inverter can be used to define a scrambling ternary truth table with the exception of the identity inverter, as described in the previous paragraph. Also, more than one ternary inverter can be used to define a scrambling ternary truth table.
 In one embodiment of the present invention, the scrambling logic circuit includes a scrambling nlength shift register having n elements and a first and a second scrambling ternary logic device, the scrambling nlength shift register having outputs from two of the n elements that are provided to two inputs of the second scrambling ternary logic device, and wherein the output from the first scrambling ternary logic device is provided to an input of the nlength shift register and an output of the second scrambling ternary logic device is provided to an input of the first scrambling ternary logic device. One of the inputs to the second scrambling ternary logic device is usually from the last element in the nlength shift register. Further, additional ternary logic devices and taps from additional or different elements in the shift register can be used. The ternary logic devices implement a ternary logic function that is defined by ternary inventors.
 For example, the scrambling logic circuit could include a scrambling nlength shift register having n elements and at least one second scrambling xvalue logic device (where x is greater than or equal to three). There are, at most, n−1 second scrambling xvalue logic devices. The n−1 xvalue second logic devices can be functionally equivalent or can be different, but each are specified by a xvalue scrambling truth table having x columns, each of the x columns being defined by one of x! reversible xvalue inverters and wherein not all of the columns are defined by the identity inverter. Each of the n−1 xvalue second logic devices has two inputs and one output. One of the inputs of each of the n−1 xvalue second logic devices is uniquely connected with an output of an element of the nlength shift register. If there are two or more second logic devices in the scrambling logic circuit then the second logic devices have as their second input the output of the next second logic device, with exception of the last second logic device which has the output of the nlength shift register as is second input. The output of the first (and possibly only) second logic device is the output of the scrambling logic circuit.
 In accordance with a further aspect of the present invention, the scrambled version of the ternary signal is descrambled with a descrambling device. The descrambling device is related and similar to the scrambler. The descrambling device has a first descrambling ternary logic device and a descrambling logic circuit. The first descrambling ternary logic device implements a ternary logic function, dc. The method of descrambling includes inputting the scrambled version of the ternary signal to the first descrambling ternary logic device and to the descrambling logic circuit, inputting an output from the descrambling logic circuit to the first descrambling ternary logic device, and outputting a descrambled ternary signal on an output of the first descrambling ternary logic device. The ternary logic function, dc, can be specified by a descrambling ternary truth table having three columns, and each of the columns is defined by one of the ternary inverters i_{1 }to i_{6}.
 In accordance with another aspect of the present invention, a column in the descrambling ternary truth table is defined by: (1) the same inverter that defines the corresponding column in the scrambling ternary truth table if the inverter is i_{1}, i_{2}, i_{3 }or i_{6}; (2) the inverter is if the corresponding column in the scrambling ternary truth table is defined by the inverter i_{4}; and (3) the inverter i_{4 }if the corresponding column in the scrambling ternary truth table is defined by the inverter i_{5}.
 In accordance with another aspect of the present invention, a column in the descrambling ternary truth table is defined by: (1) the same inverter that defines the corresponding column in the scrambling ternary truth table if the inverter is self reversing and (2) a different inverter if the corresponding column in the scrambling ternary truth table is defined by a reversible inverter that is not selfreversing. This also applies to the multivalue case where a signal represents one of four or more values.
 In accordance with one embodiment of the present invention, the scrambling logic circuit includes a scrambling nlength shift register having n elements and a second scrambling ternary logic device, the scrambling nlength shift register having outputs from two of the n elements that are provided to two inputs of the second scrambling ternary logic device, and wherein the output from the first scrambling ternary logic device is provided to an input of the nlength shift register and an output of the second scrambling ternary logic device is provided to an input of the first scrambling ternary logic device. The corresponding descrambling logic circuit includes a descrambling nlength shift register having n elements and a second descrambling ternary logic device, the scrambling nlength shift register having outputs from two of the n elements that are provided to two inputs of the second descrambling ternary logic device, and wherein the output from the second descrambling ternary logic device is provided to an input of the first descrambling ternary logic device and the scrambled version of the ternary signal is input to the descrambling nlength shift register. The descrambling unit is similar and related to the scrambling unit, having the same length of shift register, output taps at the same positions and applying identical ternary functions.
 In accordance with another aspect of the present invention, apparatus for scrambling a ternary signal that can assume one of three states is provided. The apparatus includes a first scrambling ternary logic device that implements a ternary logic function, sc, the first scrambling ternary logic device having a first and second input and an output. It also includes a scrambling logic circuit having an input and an output. The ternary signal is input to the first input of the first scrambling ternary logic device, the output of the scrambling logic circuit is input to the second input of the first scrambling ternary logic device and the output of the first scrambling ternary logic device is provided to the input of the scrambling logic circuit. The ternary logic function, sc, can be specified by a scrambling ternary truth table having three columns, each of the columns being defined by a ternary inverter. A scrambled ternary signal is provided on the output of the first scrambling ternary logic device.
 Apparatus for descrambling ternary signals is also provided in accordance with the present invention. The apparatus for scrambling and descrambling ternary signals corresponds, for the most part, to the previously described method for scrambling and descrambling ternary signals.
 The present invention also provides method and apparatus for scrambling and descrambling multivalue signals that can assume one of x states, wherein x is greater than or equal to 4. The scrambler has a first scrambling multivalue logic device that implements a multivalue logic function, fsc, and a scrambling logic circuit. The method involves inputting the multivalue signal and an output from the scrambling logic circuit to the first scrambling multivalue logic device and inputting an output from the first scrambling multivalue logic device to an input to the scrambling logic circuit. The multivalue logic function, fsc, can be specified by a multivalue scrambling truth table having x columns, each of the columns being defined by a multivalue inverter. The output from the first scrambling multivalue logic device is a scrambled version of the multivalue signal.
 In the case where x equals 4, each column is defined by one of the following multivalue inverters i_{1 }to i_{24}:
i_{1} i_{2} i_{3} i_{4} i_{5} i_{6} i_{7} i_{8} i_{9} i_{10} i_{11} i_{12} i_{13} i_{14} i_{15} i_{16} i_{17} i_{18} i_{19} i_{20} i_{21} i_{22} i_{23} i_{24} 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 1 1 1 2 2 3 3 0 0 2 2 3 3 0 0 1 1 3 3 0 0 1 1 2 2 2 2 3 1 3 1 2 2 3 0 3 0 2 1 3 0 3 0 1 1 2 0 2 0 1 3 3 2 3 1 2 1 3 2 3 0 2 0 3 1 3 0 1 0 2 1 2 0 1 0  In accordance with one aspect of the present invention, at least one different multivalue inverter defines a multivalue truth table, as long as every column is not defined by the identity inverter for scrambling applications. Alternatively, a greater number of different multivalue inverters can define the multivalue truth table.
 Further, the invention is applicable where x is greater than four. There are x! reversible inverters in any xvalue logic system.
 The scrambling logic circuit for processing the multivalue signal is similar to the scrambling logic circuit for processing the ternary logic signal. It includes a scrambling nlength shift register having n elements and one or more second scrambling multivalue logic device. The scrambling nlength shift register has outputs from two of the n elements that are provided to two inputs of one of the second scrambling multivalue logic device. The output from the first scrambling multivalue logic device is provided to an input of the nlength shift register and an output of the second scrambling multivalue logic device is provided to an input of the first scrambling multivalue logic device. As before, additional logic devices and additional taps can be used.
 In accordance with another aspect of the present invention, the scrambled version of the multivalue signal is descrambled with a descrambling device having a first descrambling multivalue logic device and a descrambling logic circuit. The first descrambling multivalue logic device implementing a multivalue logic function, fdc. The method involves inputting the scrambled version of the multivalue signal to the first descrambling multivalue logic device and to the descrambling logic circuit, inputting an output from the descrambling logic circuit to the first descrambling multivalue logic device and outputting a descrambled multivalue signal on an output of the first descrambling multivalue logic device. The multivalue logic function, fdc, can be specified by a descrambling multivalue truth table having x columns, and each of the columns is defined by one of the multivalue inverters i_{1 }to i_{24}, if x is equal to 4. Again, the descrambling unit is related and similar to the scrambling unit, as previously described.
 In accordance with another aspect of the present invention, a column in the descrambling multivalue truth table is defined by: (1) the same inverter that defines the corresponding column in the scrambling multivalue truth table if the inverter is self reversing; and (2) a different inverter if the corresponding column in the scrambling multivalue truth table is defined by a reversible inverter that is not selfreversing.
 In accordance with another aspect of the present invention, a multivalue sequence generator is provided. The sequence generator includes a shift register having a plurality of n elements and one or more (but not more than n−1) multivalue logic devices, each having a first input, a second input and an output. The output from one (but generally not the last) of the plurality of elements is connected to the first input of the first multivalue logic device. If the first multivalue logic device is the only logic device, the output of the last of the plurality of elements is connected to the second input of the multivalue logic device and the output from the first or only multivalue logic device is connected to the input of the first of the plurality of elements. If there are two or more multivalue logic devices in the sequence generator, the second input of the first multivalue logic device is connected with the output of the next multivalue logic device. The next multivalue logic device has two inputs. The first input of the next multivalue logic device is connected with a unique output of the plurality of elements of the shift register (but not with the last one). The second input of the next multivalue logic device is connected with the output of the following multivalue logic device, unless this next multivalue logic device is the final one. In that case its second input is connected with the output of the last element of the plurality of elements of the shift register.
 In accordance with another aspect of the present invention an electronic circuit for implementing a ternary logic function is provided. The ternary logic function can be defined by a truth table that has a first column defined by a first ternary inverter, a second column defined by a second ternary inverter and a third column defined by a third ternary inverter. The electronic circuit processes the ternary logic signal and a second input signal, both signals being able to assume a first state, a second state and a third state, such that it implements the ternary logic function defined by the truth table.
 The electronic device has a first input port for the ternary logic signal, a second input port for the second input signal and an output. The electronic device also has a first channel including a series connection of the first ternary inverter and first switching means for connecting the first input to the output when the second input signal assumes a first of the three states. This first channel implements the first column of the truth table defining the ternary logic function. It also has a second channel including a series connection of the second ternary inverter and second switching means for connecting the first input to the output when the second input signal assumes a second of the three states. This second channel implements the second column of the truth table defining the ternary logic function. It also has a third channel including a series connection of the third ternary inverter and third switching means for connecting the first input to the output when the second input assumes a third of the three states. The third channel implements the third column of the truth table defining the ternary logic function.
 Various other objects, features and attendant advantages of the present invention will become fully appreciated as the same becomes better understood when considered in conjunction with the accompanying drawings, and wherein:

FIG. 1 is a diagram showing a five stage Linear Feedback Shift Register binary scrambler with feedback taps at the output of shift register stage 2, 4 and 5 and a matching binary 5stage LFSR based descrambler; 
FIG. 2 is a diagram showing a binary 5stage Linear Feedback Shift Register (LFSR) based binary sequence generator; 
FIG. 3 shows the truth table of a 2 input single output binary function that passes on the value of one input regardless of the value of the other input; 
FIG. 4 shows a diagram of a binary 5stage LFSR based scrambler applying the function described inFIG. 3 to form a binary sequence generator; 
FIG. 5 shows the canonical form of a 2 input single output logic device, in this case, depicting a binary XOR or modulo2 addition device; 
FIG. 6 shows the truth table belonging to the 2 input single output XOR device; 
FIG. 7 shows a two input (‘a’ and ‘b’) single output (‘c’) ternary logic device scr2; 
FIG. 8 shows the ternary logic truth table for device scr2 ofFIG. 7 ; 
FIG. 9 shows the truth tables for three self reversing ternary logic functions; 
FIG. 10 shows the canonical truth table of the self reversing nvalue logic function, in accordance with one aspect of the present invention; 
FIG. 11 shows a table of six ternary reversible inverters in accordance with one aspect of the present invention; 
FIG. 12 shows a table of a symbolic representation of four (4) self reversing ternary inverters including the identity inverter and also shows a truth table of a self reversing scrambling function; 
FIG. 13 shows the truth tables belonging to a pair of ternary reversible functions that are each others reverse but are not self reversing; 
FIG. 14 shows a diagram of two devices that embody the two ternary functions ofFIG. 13 that demonstrate the scrambling and descrambling capabilities; 
FIG. 15 shows a truth table of a weak ternary scrambling function; 
FIG. 16 shows a table of 24 reversible 4value logic inverters including the identity function; 
FIG. 17 shows 4 self reversing 4value logic truth tables that are not equal to the 4value logic canonical self reversing functions; 
FIG. 18 shows a diagram of a prior art binary Linear Feedback Shift Register based binary scrambler and its matching descrambler; 
FIG. 19 shows a multivalue LFSR based scrambler; 
FIG. 20 shows a multivalue LFSR based descrambler; 
FIG. 21 shows a 4value 5stage LFSR based scrambler comprising only one type of selfreversing commutative function and its matching descrambler; 
FIG. 22 shows a 4value 5stage LFSR based scrambler comprising 3 different types of selfreversing commutative function and its matching descrambler; 
FIG. 23 shows a 4value 5stage LFSR based scrambler comprising 2 different types of selfreversing commutative function and one type of noncommutative selfreversing function and its matching descrambler; 
FIG. 24 shows the truth table of a 4value selfreversing but noncommutative function; 
FIG. 25 shows a reversible commutative 4value function and its reverse, a noncommutative reversible function; 
FIG. 26 shows a 4value 5stage LFSR based scrambler comprising 4 different types of reversible functions and its matching descrambler, the different functions having two types of 4value self reversing functions, one commutative reversible function and its reverse, a 4value noncommutative reversible function; 
FIG. 27 shows a binary LFSR based scrambler with constant input and an equivalent circuit; 
FIG. 28 shows two binary revertible inverters; 
FIG. 29 shows a binary LFSR based scrambler with a nonconstant input into a connecting identity function and an equivalent circuit; 
FIG. 30 shows the truth table of the binary identity; 
FIG. 31 shows a generic 3stage 2tap nvalue modulon nvalue sequence generator for which the inputs to the modulon adder are multiplied with factors p and q (modulon); 
FIG. 32 shows a table with the possible inputs to a ternary device multiplied by the possible values of p and q being 1 and 2 modulo3 (zero is considered a trivial case); 
FIG. 33 shows a table with the possible inputs to a 5value device multiplied by the possible values of p and q being 1, 2, 3 and 4 modulo5 (zero is considered a trivial case); 
FIG. 34 shows a diagram of a 2 input single output ternary device with one of its inputs multiplied with a factor 2; 
FIG. 35 shows the truth table of a modulo3 adder; 
FIG. 36 shows an equivalent depiction of the ternary device ofFIG. 34 and its corresponding truth table; 
FIG. 37 shows a diagram of a modulo3 adder with the bottom input multiplied by 2 as well as the truth table for this circuit; 
FIG. 38 shows a diagram of a modulo3 adder with both inputs multiplied by 2 as well as the truth table for this circuit; 
FIG. 39 shows a table of transformation for an nvalue reversible inverter; 
FIG. 40 shows a diagram of a one ternary function LFSR based sequence generator; 
FIG. 41 shows the truth table of ternary reversible function ‘ter1’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 42 shows the truth table of ternary reversible function ‘ter2’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 43 shows the truth table of ternary reversible function ‘ter5’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 44 shows the truth table of ternary reversible function ‘ter6’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 45 shows the truth table of ternary reversible function ‘ter9’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 46 shows the truth table of ternary reversible function ‘ter10’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 47 shows the truth table of ternary reversible function ‘ter3’ applied in the layout of the ternary sequence generator as shown inFIG. 40 , as well as a graph showing the autocorrelation for the generated sequences; 
FIG. 48 shows a graph of the autocorrelation of a sequence generated by an LFSR based ternary sequence generator with the layout ofFIG. 40 applying the ternary function ter10 and applying ternary inverter i3 as shown in the table of reversible ternary inverters inFIG. 11 ; 
FIG. 49 shows a diagram of a conducting gate with the control input equal to ‘0’; 
FIG. 50 shows a diagram of a nonconducting gate with the control input not equal to ‘0’; 
FIG. 51 shows a table applying ternary inverter i4 once and twice; 
FIG. 52 shows a table applying ternary inverter i5 once and twice; 
FIG. 53 is the truth table of a ternary function that will be realized in diagram with the gate/universal inverter method; 
FIG. 54 shows a diagram of a ternary device realizing the truth table ofFIG. 53 with the gate/universal ternary gate method; 
FIG. 55 shows a diagram of a ternary LFSR based signal spreader with its matching selfsynchronizing LFSR based despreader wherein the applied scrambling function is the ternary function ‘sc2’; 
FIG. 56 shows the truth table of ternary logic function ‘sc2’; 
FIG. 57 shows a table of incoming ternary data signals and the resulting spread signals as generated by the system as shown inFIG. 56 ; 
FIG. 58 shows the autocorrelation graph for the spread signal created by spreading ternary input signal 0 with the spreading system as shown inFIG. 56 ; 
FIG. 59 shows the autocorrelation graph for the spread signal created by spreading ternary input signal 1 with the spreading system as shown inFIG. 56 ; 
FIG. 60 shows the adjusted autocorrelation graph for the spread signal created by spreading ternary input signal 1 with the spreading system as shown inFIG. 56 ; 
FIG. 61 shows the adjusted autocorrelation graph for the spread signal created by spreading ternary input signal 2 with the spreading system as shown inFIG. 56 ; 
FIG. 62 shows a diagram of a ternary spreading/dispreading system applying an independent sequence generator; 
FIG. 63 shows the truth table of ternary logic function ‘sc4’; 
FIG. 64 shows the truth table of ternary logic function ‘sc9’; 
FIG. 65 shows the truth table for ternary logic function ‘ds9’; 
FIG. 66 shows the cross correlation graph for the possible ternary spread signals generated by the spreader as shown in the diagram ofFIG. 62 and the ternary spreading sequence generated by the ternary LFSR based sequence generator wherein the ternary inverter i6 as shown inFIG. 11 is applied to the incoming spreaded signal before determining the cross correlation; and  FIGS. 67 to 69 illustrate further embodiments of the present invention.
 The inventor has filed a provisional patent application that has been assigned Ser. No. 60/501,335 that relates to commutative multivalue scrambling/descrambling logic functions. This provisional patent application contains a description of how these functions can be applied in signal scramblers and in LFSR based multivalue scramblers and descramblers. The disclosure, including the entire specification and drawings, in provisional patent application Ser. No. 60/501,335, filed on Sep. 9, 2003, is hereby incorporated by reference in its entirety into this document. Also, the disclosure, including the entire specification and drawings, in nonprovisional patent application Ser. No. 10/912,954, filed on Aug. 6, 2004, that claims the benefit of provisional patent application Ser. No. 60/501,335 and is entitled TERNARY AND HIGHER MULTIVALUE DIGITAL SCRAMBLERS/DESCRAMBLERS, is hereby incorporated by reference in its entirety into this document. Further, the entire disclosure, including the specification and drawings in the provisional patent application Ser. No. 60/547,683, filed Feb. 25, 2004, is hereby incorporated by reference into this document.
 While powerful in their own right, the application of the functions described in the aforementioned provisional patent application form a subset of a more elaborate collection of possible functions that are specified in this specification. The increased number of possible functions can provide a greater diversity and thus, among other benefits, a greater security.
 Furthermore, in accordance with another aspect of the present invention, a new method referred to as “inverter based design” allows for designing, realizing and enabling commutative as well noncommutative scrambling/descrambling functions. With the “inverter based” scrambling functions, multivalue pseudonoise signal sequence generators can be created with a multivalue shift register, and these inverter based sequence generators do not require a multiplier. Thus, fewer components are required than previous designs.
 Furthermore, in accordance with another aspect of the present invention, the “method of inverters” allows for creating electronic circuits executing multivalue scramblers, descramblers and digital sequence generators by using a limited functional set of components including: a gate and a universal inverter and, if needed, A/D and D/A converters.
 In accordance with one aspect of the present invention, the invention can contribute in obtaining better performance from existing bandwidth. For example, ternary and quaternary signals are used in linecoding to obtain a better usage from spectrum bandwidth. Signals are scrambled to avoid long series of constant values of a signal (such as long series of zeros). Usually this involves scrambling of a binary signal followed by coding from binary to multivalue. In accordance with another aspect of the present invention, linecoding is made much less complex.
 Spreadspectrum signals can be created by scrambling a slow data signal with a much faster “chip” sequence. The signal can be despread, so that the original sequence is recreated, by descrambling the received signal with a copy of the faster “chip” sequence. In accordance with one aspect of this invention, direct multivalue spreading and despreading is enabled. Since an important element in wireless technology, such as CDMA, is the availability of spreading sequences, this invention by enabling the creation of multivalue sequences for spreading and despreading, enables a much better usage of bandwidth by wireless and CDMA applications.
 As described in provisional patent application No. 60/501,335, filed on Sep. 9, 2003, and in the nonprovisional patent application Ser. No. 10/912,954 filed on Aug. 6, 2004, that claims the benefit of provisional patent application Ser. No. 60/501,335, this invention can be embodied in binary logic with A/D and D/A converters in such a way that it performs multivalue digital functions.
 In one scrambling technique, scrambling a binary signal requires a second known binary signal. In order to descramble or recover the original signal, the second known binary signal should be available as an input to the descrambler. Application of a Linear Feedback Shift Register (LFSR) connected with the applied XOR function allows for the selfgeneration of this second known binary signal. A descrambler consisting of the mirror image of the LFSR based scrambler creates, in a similar fashion, the appropriate second known binary signal which can be used to descramble the signal.
 It is known that LFSRs with XOR functions can create binary digital signals consisting of a sequence of binary elements that may have attractive statistical characteristics know as Pseudo Noise or Pseudo Random Noise. It can be shown that LFSR based binary sequence generators are in fact LFSR based scramblers with a connecting device to the input that makes the input not relevant.

FIG. 1 shows the canonical form of a LFSR based binary scrambler 300 and its corresponding descrambler 302. The binary nstage LFSR scrambler 300 comprises a binary shift register 304 with n elements 305 to 309. Thus, the LFSR scrambler 300 inFIG. 1 is a 5stage LFSR scrambler. The output of each element in the shift register (except the last element) is the input to the next element. Based on preferred characteristics, a feedback tap can be installed from an element into a binary scrambling/descrambling function or device. In binary technology, the binary scrambling/descrambling function or device 310, 311 and 312 of choice is predominantly the Exclusive OR or XOR function, also known as Modulo2 addition. One can vary the number of stages and the number of feedback taps to determine the characteristics of the scrambler. The theory for this is widely known.  The LFSR based binary descrambler 302 to recover the original signal is the mirror image of the LFSR scrambler 300. Thus, the descrambler 302 includes a shift register 314 having n elements 316 to 320 and devices 322, 323 and 324. The devices 322, 323 and 324 implement the XOR function.
 LFSR based binary pseudorandom sequence generators are also prior art.
FIG. 2 shows a canonical form of a binary LFSRbased sequence generator 326. In the sequence generator, the scrambling/descrambling device or function between a data input and the LFSR feedback has been removed (shortcircuited), and no input signal is needed. The only requirement is that the initial content of the shift register is not all zero. As soon as a clocksignal is provided to shift content from one element to the next, the output of this device will be a sequence of binary symbols. The characteristics of the sequence are determined by the length of the shift register, the number of feedback taps and the initial content of the shift register. As is well known, the output from the sequence generator 326 can be taken from any point in the circuit. The binary sequence generator ofFIG. 2 is prior art.  All LFSR based scramblers, descramblers and sequence generators require a clock signal. On every occurrence of the clock signal, the content of the elements of the shift registers is moved one position. In all figures describing systems applying an LFSR, the availability of a clock signal is assumed (though not always drawn). Furthermore in the figures of this application, all content of elements of a shift register is assumed to move from left to right one position on the occurrence of a clock signal.
 A novel approach to create a sequence generator is to apply the scrambler configuration of
FIG. 1 , where the scrambler has an input data signal. This is shown inFIG. 4 . The device or function S that connects the input and the feedback from the LFSR inFIG. 4 is different from the XOR function.FIG. 3 shows the truth table of function or device S.  The device or function S will pass on the value of one input no matter what the other input is, as indicated by the truth table in
FIG. 3 . A binary LFSR based scrambler or signal generator applying this function or device can be constructed, as is shown inFIG. 4 . The circuit ofFIG. 4 includes a five element shift register 328, XOR devices 330 and 332 and the device S. The order of the applied inputs of function or device S inFIG. 4 is important. If the order is switched, the scrambler will pass on the input signal input Sigin without change. In the appropriate configuration, the scrambler will not depend on signal Sigin, but will act as a binary LFSR based sequence generator. Thus, the input, Sigin, should be passed through the device S  The binary sequence generators and binary LFSR scramblers are closely related and apply the similar structures, functions (XOR) and circuits. The scrambling performance of binary scramblers and LFSR scramblers changes with the number of applied number of XOR functions and in LFSR devices also with the length of the shift register.
 Presently digital signal scramblers mainly are applied for binary signals and use components that will execute the binary XOR function. The current description, in accordance with one aspect of the present invention, limits itself (for making the explanation easier) to function devices that have two multivalue inputs (say ‘a’ and ‘b’) and one multivalue output (say ‘c’). The binary XOR function is also represented by ≠.
FIG. 5 shows a 2 input single output logic device, representing the binary ≠ function. The functional behavior of this component (the binary XOR) is well known and is described by its truth table shown inFIG. 6 .  The Ternary Scrambler and Descrambler

FIG. 7 shows a diagram for a 2 input single output ternary logic device scr2. This device scr2 processes a ternary signal that can assume one of three different states.FIG. 8 shows the corresponding ternary logic truth table in accordance with one aspect of the present invention. As described in provisional patent application 60/501,335 and in nonprovisional patent application Ser. No. 10/912,954, filed Aug. 6, 2004, which claims the benefit of provisional patent application 60/501,335, the inventor has created a method and apparatus for digital ternary signal scrambling/descrambling by applying one or more of three truth tables that have the property of being selfreversing. Selfreversing functions have the property that if applied twice in sequence to an original signal, the original signal will be recreated. Thus, a signal scrambled with a selfreversing function can be descrambled with the same function. Functions that are self reversing satisfy the three equations: A sc B=C; C sc B=A and C sc A=B. 
FIG. 9 shows the truth tables of three commutative ternary logic self reversing functions: scr2, scr3 and scr5. Commutative functions are mirrored about the diagonal axis of their truth tables. Note that there are more noncommutative selfreversing ternary logic functions. The top row in the depicted ternary truth tables represents the possible values that an input ‘a’ to a twoinput ternary device (with inputs ‘a’ and ‘b’) can assume. It is noted that ‘0’, ‘1’and ‘2’ represent a way to distinguish between three different states. The physical value of these states can be different from ‘0’, ‘1’ and ‘2’. The left column in a truth table represents the possible values that the input ‘b’ to a two inputs ternary device (with inputs ‘a’ and ‘b’) can assume. The three by three matrix under the horizontal line and to the right of the vertical line in a truth table represents the possible values (or ‘states’) that an output ‘c’ of a ternary device can assume, depending on the input values of ‘a’ and ‘b’. One can use a truth table as representing all possible combinations of the equation: c→(a scr b). This equation can be interpreted as: the value of ‘c’ generated by a device with describing truth table ‘scr’ with inputs ‘a’ and ‘b’. One can also take the values of ‘a’ and ‘b’ as the coordinates of the generated result. ‘a’ represents in this case the column of the ‘scr’ matrix and ‘b’ the row. Of course, the term row can mean column and the term column can mean row. The truth tables of functions or devices ‘scr5’, ‘scr3’ and ‘scr2’ shown inFIG. 9 are commutative and self reversing. This property of reversibility provides the invention with the unique capability for a scrambling device to be its own descrambler.  NValue Reversing Functions
 An nvalue digital logic device comprised of 2inputs and a single output and with n a positive integer greater than 2, has at least a number of n selfreversing commutative functions that can be used as scrambler/descrambler functions. The setup for the shown ternary scramblers and descramblers applies to nvalue logic scramblers and descramblers executing self reversing commutative functions complying with the canonical form.

FIG. 10 shows the canonical form of a self reversing nvalue truth table. The value of m=(n−1). By moving all rows in the truth table down one position and moving the bottom row to the top, additional commutative self reversing (but different) nlogic functions are obtained. This process can be repeated n−1 times before the truth table returns to its original position.  NValue Inverters—Ternary Reversible Inverters

FIG. 11 illustrates ternary reversible inverters. A ternary reversible inverter is a function or a device that transforms each element from the set of symbols, states or values of a ternary or 3value logic into an element of a ternary logic. The element that is being transformed is called the parent and the result of the transformation is called the child. Each child has one parent and each parent has one child.  Consequently a ternary logic signal has six possible ternary inverters. Each of these inverters is reversible.
FIG. 11 shows the six ternary reversible inverters i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, and i_{6}. The first inverter i_{1 }is the identity as all elements in transformation return their own value or state. Inverters i_{2}, i_{3 }and i_{6 }are self reversing inverters. Applying these inverters once, followed by again applying the inverter on the result of the first inversion returns the original elements. Of course, inverter i_{1}, called the identity inventor, is also self reversing, but, by itself, has limited usefulness in scrambling technology.  A novel way to form a self reversing ternary scrambler function is to construct a 3×3 truth table by putting a self reversing inverter as one of three columns in the truth table. An example is shown in
FIG. 12 . This is repeated for each column in the truth table.  The truth table of the ternary scrambling function ‘scr’ shown in
FIG. 12 is formed by using inverter i_{3 }once and inverter i_{2 }twice. Thus, inFIG. 12 , the column under “0” is created by applying inverter i_{3}. The column under “1” is created by applying inverter i_{2}, as is the column under inverter i_{3}. It is assumed that i_{1}, i_{2}, i_{3 }and i_{4 }inFIG. 12 are self reversing inverters, including the identity. As long we do not use only identity inverters and self reversing inverters are used, the constructed function will be a self reversing function and can be used as a scrambling/descrambling function. Just a limited number of the thus constructed truth tables will be commutative (or input order independent) such that the inputs can be switched without effect on the output.  Considering commutative and noncommutative scrambling functions and by excluding self reversing functions that will generate identical images of the to be scrambled input, 4×4×4−1=63 different commutative and noncommutative self reversing ternary functions can be created from reversible ternary inverters that can be applied in ternary scrambling/descrambling devices.
 The inverters i_{4 }and i_{5 }in
FIG. 11 are reversible inverters but are not self reversing. This means that by applying another reversible inverter after applying i_{4 }or i_{5}, one can recover the original input. The reverse of i_{4 }is i_{5 }and the reverse of i_{5 }is i_{4}. Overall, one can create 6×6×61=215 reversible ternary scrambler functions (the 1 is a function consisting of all identity inverters, which is not a scrambler function). 
FIG. 13 shows, as an example, a pair of two ternary reversible functions that are each others inverse but are not self reversing.  The function scr3_a shown in
FIG. 13 is formed with ternary inverters: i_{1}, i_{4 }and i_{5 }in consecutive columns of the truth table. The ternary function scr3_b which should reverse the scrambling activity of scr3_a has the ternary inverters i_{1}, i_{5 }and i_{4 }in its consecutive columns. Thus, the ternary functions scr3_a and scr3_b are not self reversing because they use the inverters i_{4 }and i_{5 }fromFIG. 11 to construct the truth tables. 
FIG. 14 shows the two functions ofFIG. 13 embodied in the ternary devices Dscr3_a and Dscr3_b. This configuration shown inFIG. 14 demonstrates its scrambling/descrambling capabilities. Thus, a signal scrambled by a scrambler having the architecture shown inFIG. 14 , but using the ternary logic functions scr3_a and scr3_b, could be descrambled by the circuit ofFIG. 14 . Anytime a ternary logic signal is scrambled using a ternary logic function whose truth table has a column defined by a nonself reversible inverter, such as i_{4 }and i_{5 }fromFIG. 11 , a one step descrambler function must use the reverse of the inverters i_{4 }and i_{5 }to define the descrambling function for those columns. As previously mentioned, the reverse of i_{4 }is i_{5 }and the reverse of i_{5 }is i_{4}.  Some of these functions will not have very strong scrambling properties. In the case of scramblers/descramblers executing noncommutative self reversing functions, care should be taken in maintaining the order of inputs. If not, the descrambler will not correctly recover the original input. As an example of a very weak scrambler, the following truth table, in
FIG. 15 , is shown.  The function in
FIG. 15 is an example of a very weak noncommutative (only the column under the input “1” has scrambling properties) self reversing ternary function. The first column is an identity, so no scrambling takes place. The second column transforms 1 to 2 and 2 to 1, but leaves 0 a 0. The third column is again an identity. Thus, there is little scrambling activity.  Scrambler systems comprised of consecutive reversible (commutative or noncommutative) devices have matching combined descrambler systems that have the reversing functions of the scrambler system in reverse order.
 Canonical commutative selfreversing functions operate under the odd/even rule. This rule specifies that a composite nvalue digital scrambler comprising an odd number of nvalue commutative selfreversing scrambling functions is its own descrambler. The rule that applies to the canonical commutative selfreversing functions does not apply to systems applying one or more functions not belonging to the group of canonical commutative scrambling/descrambling functions.
 There are 215 reversible ternary logic function based signal scrambling devices, of which 63 are self reversing. The modulo3 addition function for ternary scrambling does not properly scramble and descramble a ternary signal.
 4Value Inverters
 The same reasoning applied to ternary logic can be applied to 4value logic and higher value logic. 4value logic has 4×3×2=24 inverters of which one is the identity. With these 24 inverters 24×24×24×24−1 different commutative and noncommutative reversible and selfreversing 4value scrambling 4 by 4 truth tables can be constructed.

FIG. 16 shows a table with the 24 possible 4value reversible inverters. A reversible inverter of a logic signal is a transformation of all possible states or values of that logic to a state that also belongs to that logic. The original state before inverting or transformation is called the parent. The result of the transformation is called the child. Each child has one parent. Each parent has one child.  There are 24×24×24×24−1 different reversible 4value digital signal scrambling devices. The modulo4 addition function does not properly perform the scrambling and descrambling functions for 4value scrambling.
 In the provisional application for patent 60/501,335 filed on Sep. 9, 2003 and in the nonprovisional patent application having Ser. No. 10/912,954, filed Aug. 6, 2004, 4 canonical commutative self inverting 4value scrambling/descrambling functions are described. In
FIG. 17 , an additional 4 commutative selfinverting 4value scrambling/descrambling functions are provided, comprising the 4value digital inverters: i_{1}, i_{8}, i_{17 }and i_{24}.  NValue Inverters
 nvalue inverters can also be created. The inverter based approach of the present invention allows creating 2×3×4× . . . x n=n! reversible nlogic inverters (with n a positive integer). The invention here described can create (n! to the power of n) minus 1 different nvalue logic scrambler/descrambler devices applying the (n! to the power n) minus one different reversible nvalue logic truth tables.
 It is believed that there are (n! to the power n)—1 different reversible nvalue digital signal scrambling devices. The one not included is the function comprising all three identity inverters.
 All the scrambler devices designed according to this invention can be used as single or combined systems as described in the ternary case. The combined nvalue scrambling systems comprising an odd number of canonical commutative self reversing individual scramblers can be used as its own descrambler.
 NValue Scrambles and Descrambles as Signal Spreaders and Despreaders
 The present invention allows for nvalue scramblers and descramblers as signal spreaders and despreaders. Spreading and despreading of binary signals as known in the prior art of spread spectrum technology is in essence scrambling of a rapidly changing signal sequence (chip sequence) with a much slower changing data signal. Despreading is the descrambling of the spreaded signal with a copy of the original rapidly changing signal sequence. nvalue scramblers, comprising one or more of the nvalue reversible logic functions, are an excellent way to create nvalue spread spectrum digital signals. The application of nvalue scramblers as nvalue digital signal spreaders is one aspect of the present invention.
 As an example,
FIG. 67 shows a 4value spreading application by applying the 4value scrambling function fs3, as shown inFIG. 68 . In order to show the signals as being symmetrical, the representation of the 4value logic states 0, 1, 2 and 3 have been adapted to −2, −1, 1 and 2, as shown inFIG. 68 .  The spreading application shows how a 4 element 4value digital signal is scrambled with a much faster 4value signal comprising 4 16 chip digital sequence. The resulting output is the spread 4value signal.
 The recovery or despreading of the spread (or scrambled) 4value signal takes place at the despreader or descrambler shown in
FIG. 69 . Because the 4value function fs3 that spreads the signal is self reversing, it can be applied in the despreader to recover the signal. The diagram ofFIG. 69 is similar toFIG. 67 . The addition is a synchronizing clock signal that makes sure that the incoming spread signal and the descrambling 4value chip sequence are in phase. If these two signals are out of phase, the resulting output of the device is not equal to the original 4value data signal. Prior art describes several ways to create a synchronizing clock and the supporting synchronization circuit. One way is to synchronize signals with a common pilot signal or derive synchronization from the properties of the signal (such as auto and cross correlation). In accordance with one aspect of the invention, nvalue descrambling functions applied as nvalue digital signal despreaders are used.  LFSR Based MultiValue Scramblers
 LFSR based multivalue scramblers are contemplated by the present invention. The binary case will be considered first.
FIG. 18 shows the basic configuration of a binary Linear Feedback Shift Register scrambler 350 and its matching descrambler 352. The scrambler 350 inFIG. 18 comprises a shift register having memory elements. A shift register has a series of at least 2 memory elements. Each memory element can contain (in the binary case) the representation of the value 0 or 1. Triggered by an event (like a clockpulse) the content of each element is shifted one position to its next neighbor, in this case to the right. As long as a memory element contains a value, this value is available as an output signal of the memory element. At least one of the outputs of the memory elements is fed back to the incoming signal. There it is combined with the incoming signal through a two input/single output digital ‘exclusive or’ device. The output of the ‘exclusive or’ is usually considered to be the scrambled signal. The devices labeled “≠” are XOR devices.  The scrambled signal is then transmitted to the descrambler 352. It is also fed to a shift register and on the specified event (like a clock pulse) will become the content of the first element of the shift register. The feedback connections from the outputs of elements of the shift register are called taps. In a binary device, taps are connected with each other through an ‘exclusive or’ (XOR) device. If there is only one tap the feedback tap is connected with the incoming (to be scrambled) binary signal. The number of taps as well as the number of elements (or the length) of a shift register determine the statistical characteristics of the scrambled signal. The initial appearance of the scrambled signal is also determined by the initial content of the shift register elements.
 The descrambler is a mirror image of the scrambler as shown in the above figure. The principles are exactly the same as in the scrambler. The binary descrambler should have a shift register of equal length as the binary scrambler and feedback taps from the elements in the same position as in the related binary scrambler. The connections between the feedback taps are also made through binary ‘exclusive or’ (XOR) devices. Complete recovery of the signal is possible if the initial content of the elements of the shift register of the descrambler at the start of receiving is the same as the content of the initial content of the shift register of the scrambler. However after a certain number of steps (having cleared the initial content of the shift register) the descrambled signal will be equal to the original signal no matter what the initial content of the descrambler shift register was.
 All LFSR based devices described in this application are governed by a clock signal that will not be drawn or identified in all figures. Such a clock signal is well known.
 The Ternary LFSR Scrambler/Descrambler Case
 The canonical layout of ternary LFSR scramblers (or nvalue LFSR scramblers) can be the same as the binary form. However, as recognized by the present invention, the applied scrambling functions are different.
FIG. 19 illustrates one possible (multitap) layout of a multivalue LFSR based scrambler 354 based on one aspect of the present invention.  In the ternary case, all memory elements 355 to 359 of the shift register 360 are able to contain ‘0’, ‘1’ or ‘2’ value or state representations. An nstage LFSR ternary scrambler can have a maximum of n feedback taps and a maximum of (n−1) devices to connect the taps and one final device (named Opscr) to connect the incoming ternary signal with the feedback (or the result of the feedbacks). The devices that connect the output of a shift register element with a previous result (or in the case of the last tap with another element output) are named ‘Lfsr’ functions with a number added to indicate different functions.
 The device that combines the incoming digital data signal with the feedback from the LFSR and has as output the input to the first stage of the shift register is called Opscr.
 The multivalue LFSR based descrambler 362, illustrated in
FIG. 20 , has a layout (but not always the functions) that is the mirror image of the scrambler 354. This is shown inFIG. 20 which illustrates a descrambler that can descramble a signal created by the scrambler ofFIG. 19 .  We will maintain a similar naming convention in the multivalue LFSR based descrambler in
FIG. 20 . The device that combines the incoming (scrambled) signal with the feedback from the LFSR and has as output the recovered original data signal is called Opdes. All other functions are called ‘Lfsr’ with a number added to indicate different functions.  For an LFSR system to form a ternary LFSR based scrambling/descrambling system it has to comply with the following requirements and design rules:
 First, the shift register in scrambler and descrambler are of equal length. The elements or stages of a shift register are memory elements. In a nvalue logic they can contain values that range from 0 to n−1.
 Second, scrambler and descrambler have identical tap configurations.
 Third, the functions Opscr and Opdes have to be a reversing pair (one is a scrambling and the other one is a descrambling function. The placement in the scrambler or the descrambler does not matter. But if one is in the scrambler the other has to be in the descrambler). The functions Opscr and Opdes can also be equal if selfreversing.
 Fourth, the functions Lfsr can be any of the reversible ternary functions formed from the ternary reversible inverters.
 Fifth, a selected Lfsr function at a certain tap in the scrambler should correspond with the same function at the corresponding tap in the descrambler. In case of noncommutative functions, the order of inputs to the devices in scrambler and descrambler configurations should be the same.
 Sixth, the initial content of the scrambler and descrambler shift register should be identical.
 One form for an LFSR based scrambler/descrambler set would be a scrambler that only applies one commutative selfreversing function for Lfsr and Op functions. In that case, the ternary scrambler and descrambler would contain the same elements. The modulon addition function does not meet the criterion of being selfreversing for n being an integer greater than 2.
 Another ternary LFSR scrambler/descrambler set can apply a mix of commutative ternary selfreversing functions for Lfsr and Opfunctions. Another ternary LFSR scrambler/descrambler set can apply a mix of commutative and noncommutative functions for Lfsr and Opfunctions. The statistical performance of the ternary LFSR scrambler is determined by the length of the shift register and the number of taps as well as where the taps are placed and what functions are used.
 Proper scrambling/descrambling performance is determined by complying with the above stated design rules.
 The NValue LFSR Scrambler/Descrambler
 The design rules can be expanded to the more general nvalue case for LFSR scrambler/descrambler systems.
 First, both nvalue LFSR scramblers and descramblers should have shift registers of equal length. Each element is able to preserve as its content the representation of the values 0, 1, 2 . . . . , n−1.
 Second, scramblers and descramblers should have identical number of feedback taps at corresponding elements of the shift register.
 Third, connecting devices should all be reversible nvalue functions constructed from nvalue reversible inverters.
 Fourth, the functions Opscr and Opdes have to be a reversing pair (one is a scrambling and the other one is a descrambling function. The placing in either scrambler or descrambler does not matter. But if one is in the scrambler the other has to be in the descrambler) Or the functions Opscr and Opdes are equal if selfreversing.
 Fifth, the functions Lfsr can be any of the reversible multivalue functions formed from the multivalue reversible inverters.
 Sixth, a selected Lfsr function at a certain tap in the scrambler should correspond with the same function at the corresponding tap in the descrambler. In case of noncommutative functions the order of inputs to the devices in scrambler and descrambler should be the same.
 Seventh the initial contents of the shift register in scrambler and descrambler should be identical for complete recovery of the original signal.

FIG. 21 illustrates one example of an nvalue LFSR based scrambling system in accordance with one aspect of the present invention.FIG. 21 shows a diagram of a 4value 5stage LFSR based scrambler and its matching descrambler applying only devices with the self reversing functional truth table scr4_d, as shown inFIG. 17 . 
FIG. 21 illustrates another example of an nvalue LFSR based scrambling system in accordance with another aspect of the present invention.FIG. 22 shows a diagram of a 4value 5stage LFSR based scrambler and its matching descrambler applying devices with the self reversing functional truth tables scr4_a, scr4_c and scr4_d as shown inFIG. 17 . Specifically, functions S and D implement scr4_d, function L1 implements scr4_a and function L2 implements scr4_c, as shown in the figure.  Not all self reversing multivalue functions are commutative. Another example of an nvalue LFSR based scrambling system is shown in
FIG. 23 . Here, one of the functions applied as the connecting device between incoming input and feedback from the shift register is a noncommutative selfreversing 4value function, scr4_g. Because function scr4_g is self reversing it is also applied in the position corresponding with Opdes in the descrambler. The truth table of the 4value selfreversing function scr4_g is shown inFIG. 24 . Note that the order of the inputs to the table of scr4_g is important to determine the output (the inputs cannot be switched) so that the function is noncommutative.  Not every commutative reversible nvalue function is self reversing.
FIG. 26 shows the application of the reversible and commutative 4value function scr4_e in a 4value 5stage LFSR based scrambler. This function is applied in the position of Opscr. Consequently the matching function Opdes in the descrambler should be the reverse of Opscr. This reverse is the noncommutative, reversible function scr4_f. The truth tables of scr4_e and scr4_f are shown inFIG. 25 .  Prior art mentions multivalue LFSR based scrambling and descrambling but often in a cursory way, indicating a belief that multilevel LFSR based scrambling and descrambling is an extension of the binary form and displaying a limited understanding of the rules for nvalue LFSR based scrambling and descrambling. As illustrated herein, multilevel LFSR based scrambling and descrambling is not a simple extension of the binary rules.
 One reference the inventor found to a related approach was in the Help Pages of Mathworks Simulink Communication Blockset on Feb. 12, 2004, found on the internet. These pages describe an algorithm for nvalue integer signals for LFSR based scramblers and descramblers. The help file on the web site described a set of LFSR based scramblers/descramblers that strictly apply modulon adders. This is appropriate for binary application. However if all applied functions (in the scrambler as well as the descrambler) as described by Mathworks are modulon addition the descrambler can in the described form not recover the original signal. This approach does not work to make ternary and multivalue LFSR scramblers/descramblers, and fails to recognize that modulon addition (with n greater than 2) is not a self reversing function.
 Another example in prior art mentions modulon subtraction as the descrambling function for scrambling by modulon addition. Such an observation is not specific enough. Modulon addition can be implemented with an nvalue commutative function. However subtraction is noncommutative. That means that if the order of inputs to an nvalue subtraction is switched at the descrambler, the recovered signal in case of nvalue (n greater than 2) is not the original signal. Conversely modulon subtraction is a set of two nvalue logic functions.
 Another observation is that modulon subtraction is a self reversing but noncommutative function in nvalue logic. This has an interesting consequence for nvalue scramblers (be it with or without LFSR). If an nvalue scrambler, such as in
FIG. 19 , has nvalue subtraction as nvalue function Opscr, it can have the same nvalue subtraction function as Opdes in the descrambling function as shown inFIG. 20 , because the subtraction function is self reversing. However the descrambler as shown inFIG. 20 can also have the function modulon addition as Opdes function.  Consequently the rules as described in this application provide rules that state a set of conditions that will ensure that scramblers and descramblers that comply with the rules will demonstrate complete recovery of scrambled multivalue signals. The set of rules may not be exclusive. There are other combinations of nvalue functions possible that also show scrambling/descrambling capabilities.
 The inventor believes he is the first to have invented methods for creating LSFR ternary and multivalue scramblers and descramblers. He also believes he is the first to be able to articulate and design different matching sets of multivalue LFSR based scramblers and descramblers.
 It should be clear that because many reversible possible functions can be applied there is a large range of possible nvalue logic LFSR scrambler/descrambler combinations.
 NValue LFSR Based Scramblers/Descramblers as NValue Signal Spreaders/Despreaders
 The nvalue LFSR based scramblers and descramblers can be used as single channel or single user nvalue spreader and despreader. The present invention contemplates at least two different spreader/despreader realizations. These include an Nvalue LFSR based spreader/despreader with no synchronization and an Nvalue LFSR based spreader/despreader with synchronization.
 NValue LFSR Based Spreader/Despreader with No Synchronization
 In this configuration an nvalue LFSR based scrambler is applied to scramble an incoming nvalue data signal. The scrambler operates on a clock signal. The clock signal is significantly faster than the maximum rate of change of the nvalue data signal. Each element of the nvalue digital data input signal to the scrambler is assumed to have a constant duration. The number of clock pulses during the presence at the input of the scrambler of one nvalue data signal element is N_{s}. One nvalue digital input data signal element will thus be scrambled with N_{s }chips. The result is a scrambled signal with N_{s }elements. N_{s }is called the spreading factor.
 The configuration of this setup uses the LFSR based nvalue scrambler to generate the spreading chipsequence as well as to create the scrambled sequence.
 The despreading part of the invention is the matching nvalue LFSR based descrambler to the previous scrambler operated at the same clock frequency. Because the descrambler is selfsynchronizing there is no requirement for determination of the starting position of the received scrambled sequence. The descrambler may become temporarily desynchronized. As soon as the shift register is ‘flushed’ the descrambler is synchronized again.
 It may be required to synchronize the clock signal of the receiving descrambler with the incoming scrambled signal in order to ensure optimal signal level detection.
 A diagram of the Nvalue LFSR based spreader/despreader in accordance with one aspect of the present invention is shown in
FIG. 55 .FIG. 56 shows the truth table of applied self reversing ternary function ‘sc2’. The representation of ternary data input signal and the resulting spreaded signals by the invention are shown inFIG. 57 . The spreading factor (26 in the example) is determined by the frequency of the clocksignal. The statistical characteristics of the spreaded signal are determined by the configuration of the LFSR based scrambler.  The configuration of the scrambler as shown in
FIG. 55 generates a ternary msequence for input data signal ‘0’. The autocorrelation for the spreaded signal is shown inFIG. 58 . The autocorrelation of the spreaded ternary signal ‘1’ is shown inFIG. 59 . The autocorrelation adjusted by applying one of the reversible ternary inverters on the received signal is shown inFIG. 60 . The adjusted autocorrelation with a different ternary inverter for spreaded digital data signal ‘2’ is shown inFIG. 61 . The autocorrelation has 3 equal peaks within period 2N_{s }−1 and is not well suited for signal synchronization. By application of the selfsynchronizing descrambler no further synchronization should be required.  NValue LFSR Based Spreader/Despreader with Synchronization

FIG. 62 shows a spreader/despreader system that applies the same nvalue spreading sequence generated by an nvalue LFSR based sequence generator. In the figure as an example a 4stage ternary LFSR, using ternary function ‘sc4’ is applied.FIG. 63 shows the truth table for ternary function ‘sc4’.  The LFSR in
FIG. 62 works independent of the incoming ternary data signal ‘Sigin’. The output of the LFSR based sequence generator (the spreading sequence) is scrambled with the incoming ternary datasignal ‘Sigin’ by a ternary scrambling/descrambling device with truth table ‘sc9’. The truth table of ‘sc9’ is shown inFIG. 64 . Because the rate of change in the spreading sequence is greater than the input data signal, the data signal will be spread.  The despreader applies almost the same configuration as the spreader. The only difference is the descrambling function, which is ‘ds9’ or the inverse (or reverse) of ‘sc9’. The truth table of ‘ds9’ is shown in
FIG. 65 .  The sequence generator of the despreader again works independently of the incoming signal. The incoming signal to the despreader function ‘ds9’ is the spread signal. If the spread signal and the spreading sequence are descrambled while in the right synchronization, the result of the descrambling will be the original data input signal.
 The advantage of this arrangement is that just one spreading sequence is required. Each of the three different ternary input signals will create a different spreaded sequence.
 There are several basic ways to synchronize the spreading sequence at the despreader with the incoming spreaded signal. The best known way of synchronization is to apply a spreading sequence that has a single sharply defined correlation value.
 In this example there are three different sequences, originating from the same spreading sequence. In many cases, there are sequences, such as in this example, where each ternary input signal spread by the spreading sequence creates a spread signal that has identical ideal crosscorrelation with the spreading signal. This is shown in
FIG. 66 . By applying ternary reversible inverter i_{6 }as shown inFIG. 11 to the incoming spread signal, the crosscorrelation for all three possible spread signals will be as shown inFIG. 66 . This is an advantageous correlation graph, as it has just one large peak at the same location. This allows for easy detection of synchronization of the incoming signal with the LFSR based sequence generator, without having to distinguish for original input data signal.  Inverter Use for Adjusting Cross Correlation
 Each individual reversible nvalue inverter can be used as a “simple” scrambler, inverting selectively elements from an incoming digital nvalue signal. Though this does not significantly change the statistical makeup of an nvalue digital signal, it can change the appearance. It can also be fully reversed. It is possible to put several inverters consecutively behind each other such that the input of the next one is the output of the previous one.
 Another application for using the inverters in an nvalue digital signal consisting of 2 or more elements is in comparing two nvalue digital sequences by calculating their cross correlation. In some cases the calculated cross correlation between two digital nvalue signals consisting of a sequence of more than 1 element will be a value not equal to zero. In applications it may be advantageous to be able to change signals such that their crosscorrelation is zero while being able to recover the original sequence. A simple way to achieve this is to apply one or more reversible inverters to one or both of the signals, calculate the cross correlation and recover the original signals by applying the inverters in reverse order.
 Another application of reversible nvalue inverters is that it changes the appearance of the crosscorrelation or autocorrelation in such a way that one clear peak exists that is significantly different from other values. The existence of such a distinguishing peak allows for synchronization of the signal.
 Generating NValue Digital Sequence Signals
 There are many applications in digital telecommunications where the availability of binary or multivalue digital signals consisting of a sequence of binary or multivalue digital signals is desirable or beneficial. For instance the availability of certain binary sequences with particular autocorrelation and/or crosscorrelation properties is critical to the appropriate functioning of spreadspectrum communication applications. More specifically multiuser systems require specific crosscorrelation properties in wireless applications.
 One expectation is often that the sequence signal has the appearance of being statistically random. Linear feedback shift registers are used to create this type of sequences which are known as Pseudonoise (PN) or pseudorandom noise. These sequences may have the appearance of being random, however, they are fully deterministic in how they are created and what their characteristics are.
 The terms digital multivalue sequence and signal can be interchanged in this application.
 The theory of creating pseudorandom sequences is well established and there are well defined rules how to create sequences (binary as well as multivalue). As far as the disclosed prior art is concerned: n value sequences appear to be strictly generated by LFSRs with feedback taps comprising modulon adders and modulon multipliers.
 One aspect of the present invention is focused on generating nvalue (such as ternary) sequence signals by applying the previously defined ‘reversible’ nvalue functions, constructed from the set of nvalue reversible inverters without use of multipliers. As such, this is a new invention that has not been disclosed in prior art. It allows for replacing the theoretically combination of adders and multipliers with single multivalue digital function devices. It also can generate additional sequences with attractive properties such as crosscorrelation and autocorrelation that cannot be generated by application of modulon addition and modulon multiplication.
 In prior art pseudorandom number (PN) generators (be it binary or multivalue (nvalue)) are Linear Feedback Shift Registers with feedback taps connected to Modulon adders. Further more for optimal statistical performance, some taps will contain modulon multipliers in the nvalue case (n a positive integer greater than 2) to comply with the coefficients of calculated irreducible polynomials over the Galois Field GF(n^{P}).
 In accordance with one aspect of the present invention, nvalue sequences are generated by new inverterbased, reversible nvalue functions (with exception of the modulon adder). This aspect of the present invention offers a new and more efficient way to generate nvalue sequences.
 PN generators are in fact scramblers with any input from the outside combined with the feedback from the LFSR through a function that is constructed from strictly identity inverters. This is demonstrated in its binary form in
FIG. 27 .  The first diagram in
FIG. 27 shows a binary 5stage, 2tap scrambler with inputs all ‘0’. This scrambler can be replaced with a configuration as shown in the second diagram ofFIG. 27 . The device labeled as “+” is an XOR device.  There exist two reversible binary inverters as shown in
FIG. 28 . The first inverter inFIG. 28 , i_{1}, is the identity, the second inverter i_{2 }is what is generally recognized as an inverter in that ‘0’ is transformed to ‘1’ and ‘1’ is transformed to ‘0’.  By replacing the XOR device, +, that connects the incoming signal with the feedback from the LFSR with the identity device, i_{1}, a binary scrambler can be turned into a PN generator. The input signal does not influence the outgoing signal ‘out’. This new circuit is illustrated in
FIG. 29 .FIG. 30 shows the truth table for the binary identity.  Generalized NValue Sequence Generators
 All multivalue or nvalue digital LFSR pseudonoise generators can be treated as nvalue LFSR scramblers with the operational connecting device between the incoming signal and the feedback signal being the noncommutative nvalue identity function.
 It is the object of this invention to create devices and methods that can generate nvalue (with n an integer greater than 2) sequences, applying nvalue scrambling functions, nvalue inverters and nvalue shift registers, without regard for the statistical properties of the sequences. Prior art describes ways to determine LFSR based nvalue pseudo noise sequence generators by applying irreducible polynomials over the Galois Field GF(n) and realizing these with modulon adders, modulon multipliers and nvalue shift registers.
 It is an object of this invention to provide a method to realize prior art nvalue LFSR based noise generators and others with functions and devices of this invention.
 Prior art LFSR based nvalue pseudo noise generators pertain to values of n which are positive integer and prime (e.g. 2, 3, 5, 7, . . . .). The realized nvalue LFSR generators comprise taps containing a multiplier p at one input of the modulon adder and multiplier q at the other (with p and q an integer smaller than n).
FIG. 31 shows an example of such a generic nvalue LFSR based sequence generator.  The entities p and q are modulon numbers that multiply the input to the modulon adder with. The value of p and q can be determined from the irreducible polynomials over GF(n). If n is a prime integer then the multiplying factors (in this case p and q) are reversible inverters.

FIG. 32 shows the possible inputs to a modulo3 adder, multiplied with 1 and 2 modulo3. Multiplying a ternary input with 1 in ternary logic is the same as inverting the input with the identity inverter. Multiplying a ternary input with a factor 2 in ternary logic is the same as subjecting the input to the selfreversing inverter i_{2}. 
FIG. 33 shows the possible inputs to a modulo5 adder, multiplied with factors 1, 2, 3, and 4 modulo5. In 5value logic, multiplying the input with an integer modulo5 is the same as subjecting the input to a reversible inverter (including the 5value identity). In general: multiplying the input to an nvalue digital device is similar to subjecting that input to an nvalue digital reversible inverter.  Modifying the Truth Table by Modifying the Input to an NValue Digital Device
 Modifying the input to an nvalue digital logic device, characterized by an nvalue logic truth table, is the same as not modifying the input, but modifying the characterizing truth table (and thus the function) of the device. This is illustrated in
FIG. 34 with a diagram of a ternary (or modulo3) adder with inputs In_1 and In_2 and output Out. The input In_1 is multiplied by a factor 2. The inputs to the modulo3 adder are In_1mod and In_2. In_1mod is created by multiplying the input In_1 with a factor 2. 
FIG. 35 shows the truth table of the modulo3 adder. The top row of the ternary truth table represents the possible values for input In_1mod. The left column of the ternary truth table represents the possible values for input In_2.  If the input In_1 is 0 the modulo3 adder also sees a zero, as 2×0 is 0. If the input In_1 is 1 the modulo3 adder sees 2, which is 2×1. As a consequence we can say that instead of the column below input In_1 is 1 (adder without In_1 multiplier) it “sees” the column below In_1mod is 2.
 If the input In_1 is 2 the modulo3 adder “sees” the column in the truth table below In_1mod is 1. The truth table of the modulo3 adder with the top input multiplied by a factor 2 is the same as the original modulo3 truth table modified such that the column under In_1mod is 1 is replaced by the column under In_1mod is 2 and the column under In_1mod is 2 is replaced by the column under In_1mod is 1.
 This is illustrated in
FIG. 36 . The original device ofFIG. 34 is now replaced with the equivalent device depicted inFIG. 36 with inputs In_1 and In_2 and output Out. The corresponding truth table is also shown inFIG. 36 .FIG. 36 shows the equivalent ternary logic circuit for the modulo3 adder with the top input multiplied by a factor two and the truth table of the equivalent circuit. 
FIG. 37 shows the equivalent circuit and its truth table for a modulo3 adder with the bottom input multiplied by a factor 2. The effect of the multiplication of the bottom input by a factor 2 is the switching of the rows of In_2 is 1 and In_2 is 2. 
FIG. 38 shows the equivalent circuit and its truth table for the modulo3 adder with both inputs multiplied by a factor 2. The effect of the multiplication of both inputs with a factor 2 is the switching of the rows In_2 is 1 and In_{—}2 is 2 as well as the switching of the columns for In_1 is 1 and In_1 is 2.  General NValue Inverter Method for Finding Equivalent Truth Tables and Circuits
 An nvalue (reversible) inverter has n different elements. An nvalue inverter transforms the possible input values 0, 1, 2, 3, . . . , n−2, n−1 (with n a positive integer) into the possible output values: p, q, r, s . . . t, u with all possible output values also being integers smaller than n, in such a way no output value is equal to another output value. The transformation is shown in the
FIG. 39 .  The effect of a reversible inversion on an nvalue truth table is:
 if the inversion is applied to the top input the column under input is 0 is replaced with the column under input is p, the column under input is 1 is replaced by the column under input q, the column under input is 3 is replaced by the column under input is s. Up to the column under input is (n−1) is replaced by the column under input is u.
 If the inversion is applied to the bottom input, the same rules are applied to the rows of the truth tables. If an inversion is applied to both inputs, the above rules apply to columns and rows, and are not dependent on the order of execution.
 This method fulfills the object of the invention to create nvalue digital logic functions and circuits that can generate not only n value signal sequences, but also all sequences that can be created with prior art but with fewer component functions applied, as this makes the multipliers obsolete.
 In ternary logic, 5 different reversible inverters exist that are different from identity. But there is just one modulo3 multiplication not equal to 0 or 1. Consequently the inverter based method can generate more different functions than modulo3 arithmetic and provide a much greater diversity in functions and devices than prior art applying modulo3 arithmetic.
 The same reasoning and diversity advantage applies to higher value generation of sequences.
 The Ternary LFSR Sequence Generators
 The literature describes a 3value digital PN generator to be a LFSR able to contain 3valued numbers with feedback taps into modulo3 adders. The feedback taps contain modulo3 multipliers. The behavior of the ternary pseudo noise generator is determined by the number of taps and the values in the multipliers. Literature and prior art describe how to apply irreducible polynomials to design nvalue noise generators.
 One aspect of the present invention comprises ternary or 3value sequence generators formed from a 3valued shift register with feedback taps leading into connecting devices which have describing ternary truth tables comprised of columns or rows which are selected from the set of 6 reversible ternary inverters. A ternary sequence generator can be any of the possible ternary LFSR scramblers with no input and the connecting device between LFSR signal and input signal being replaced by a short circuit.
FIG. 40 shows an example of such a ternary PN generator.  Several terminals can be selected for collecting the ‘output’ signal. In fact any of the inputs or outputs of the elements of the shift register will do. The ‘output’ on these terminals will differ on the first output signals as the shift register is being flushed. In general the initial state should be such that transitions will be generated.
 The two irreducible polynomials for a 4stage LFSR based ternary PN generator with two feedback taps as shown in
FIG. 40 are: (x^{4}+x+2) and (x^{4}+2x+2). Literature says that these two (modulo3 adder based) solutions are able to generate what is known in prior art as ternary msequences.  The performance or desirability of a sequence is frequently measured by correlation and autocorrelation. Autocorrelation and crosscorrelation are important indicators of certain qualities of digital sequences. The subject of autocorrelation is well documented. Correlation is usually calculated between two sequences. In autocorrelation the second sequence is a copy of the original one.
 In principle, correlation of two sequences of equal lengths (assume length is n elements) is calculated by multiplying the assigned value of the k^{th }element of the first sequence with the assigned value of the k^{th }element of the second sequence. This multiplication is done for all n elements of both sequences. The result of these multiplications is added into one sum. This sum is then called the correlation value. In general, the correlation between two sequences is calculated for all time shifted instances of one of the sequences. If the length of the sequences is n elements, there will be 2n−1 different matches between two sequences of length n.
 The graph that displays the 2n−1 correlation values between the two sequences is in general called the autocorrelation graph (if the two sequences are equal) or the crosscorrelation graph if the correlation is determined between two different sequences of equal length.
 Msequences or maximum length sequences in the binary case are sequences that have an autocorrelation value A in all time shifted calculations and a value B when a sequence is correlated with a nonshifted copy of itself. The value of A and B depends on the assigned value to the binary elements. In case of a balanced assigned value (say 0 is assigned the value −1 and 1 is assigned the value +1) the value of A can be 0. It is well known that sequences with certain auto and crosscorrelation have attractive properties. One aspect of this invention determines configurations to generate ternary and nvalue sequences with msequence like properties. It is possible to normalize correlation calculations for nvalue sequences with n an integer greater than 2. In general, for calculating correlation properties of an n value digital signal will be considered having the values: 0, 1, 2, 3, . . . , n−1. For calculating the correlation, these values will be used or these values will be substituted so they are balanced around 0, (for instance by subtracting (n−1)/2 of all values if n is odd; and by substituting the first n/2 states with—(n/2)+i for i=0, 1, . . . (n/2−1) and the next n/2 states with i−(n/2−1) for i=n/2, . . . . . , n−1 if n is even). The above type of substitution does affect the individual correlation values but not the shape of the correlation graph. However, the substitution can also reflect use of different reversible inverters. This can change the overall shape of the correlation graph, mainly the orientation of some of the peak values in the correlation graph as being above or below the average line.
 One measure for the performance or desirability of a sequence is its autocorrelation. The modulo3 adder with multipliers provides 2 configurations for generating msequences based on the irreducible polynomial in the layout of
FIG. 40 . The application of reversible ternary functions can provide at least 6 different configurations under the layout ofFIG. 40 that will generate msequences. This invention in general provides greater diversity with fewer components to generate attractive nvalue digital sequences. 
FIGS. 41, 42 , 43, 44, 45 and 46 show the truth tables of the applied different ternary devices in the layout of the sequence generator as inFIG. 40 for the LFSR based ternary msequence generator and a graph of the autocorrelation of the generated sequences.  The autocorrelation graphs show that applying the functions ter1, ter2, ter5, ter6, ter9 and ter10 in the ternary 4 stage LFSR with taps on the output of the 3^{rd }and 4^{th }element creates sequences with attractive properties. One reason they are attractive is because they will make it easier to determine synchronization. Analysis of the sequences will show that the shifted sequences are not equal to each other. This indicates that the thus generated individual shifted sequences also have very attractive orthogonal properties.
 For illustration purposes,
FIG. 47 showing the autocorrelation and truth table for the above sequence generator ter3 has been included. The application of this function demonstrates much less attractive properties. This is because it will be difficult to detect a peak value where synchronization occurs.  The NValue Digital Sequence Generators
 The above reasoning can be extended to all multivalue or nvalue (with n an integer greater than two) linear feedback shift register based digital sequence generators.
 Thus, one aspect of the present invention provides a nvalue (with n an integer greater than 2) sequence generators formed from nvalue Linear Feedback Shift Registers with a maximum number of n feedback taps connected by nvalue logic function devices of which the truth tables are constructed from nvalue reversible inverters. These systems have no input signal. The resulting feedback is directly guided to the first stage of the shift register.
 The inventor has tested the sequences generated by the different configurations depicted in
FIGS. 41, 42 , 43, 44, 45 and 46. While the sequences generated by each configuration are time shifted, the sequences of each configuration are totally different from the sequences generated by the other configurations. Consequently this invention can generate a greater set of msequences than prior art by nature of its greater set of connecting multivalue functions or devices than in prior art.  Exceptions to this invention are any LFSR system that contains only modulon adding devices as twoinput connectors in LFSR based sequence generators. These modulon adding devices based systems are prior art.
 Adjusting AutoCorrelation and CrossCorrelation of Sequences
 The autocorrelation of an nvalue digital sequence can be considered to be crosscorrelation between this sequence and a time shifted version of this sequence. It is useful to have a crosscorrelation or autocorrelation that is zero over as large a period as possible.
 The ternary LFSR based sequence generator as shown in
FIG. 46 applying ternary function ‘ter10’ has an autocorrelation value of −1 over a large area, with low peaks with value 26 at n=40 and n=120 and a large peak of value 53 at n=80.  It would be useful to be able to change the autocorrelation such that is has just one positive peak for synchronization. It would also be very beneficial if the crosscorrelation between time shifted sequences could be made 0 (which means that these sequences are orthogonal).
 By applying ternary inverter i_{3 }as shown in the table of reversible ternary inverters in
FIG. 11 to the sequence after generating the sequence but before determining the auto or cross correlation, the correlation is adjusted to 0 over the majority of the sequence period. Further, the peak value is adjusted to −54 at n=40 and at n=120 and to +54 at n=80. 
FIG. 48 shows a graph of the adjusted autocorrelation for the LFSR based ternary sequence generator with layout as inFIG. 40 with device ‘ter10’ and applying ternary reversible inverter i_{3 }to the sequence before determining the correlation.  In accordance with another aspect of the present invention, the level of the crosscorrelation of two nvalue digital sequences can be adjusted by applying one or more similar or different nvalue inverters to each sequence before determining the crosscorrelation.
 In accordance with a further aspect of the present invention, the level of the autocorrelation of an nvalue digital sequence can be adjusted by applying one or more similar or different inverters to the sequence before determining the autocorrelation. The autocorrelation is determined by shifting the original nvalue sequence in time (or phase) and calculating the correlation between the original and shifted nvalue sequence. Appropriate adjustment can be achieved by applying similar or different inverters to the original and its shifted sequences.
 Consequently the invention does not only provide a greater diversity in creating sequence generators it also provides better ways to make use of them.
 As many apparently different embodiments of this invention may be made without departing from the spirit and scope thereof, it is to be understood that the invention is not limited to the specific embodiments thereof.
 One embodiment of the execution of the multivalue scrambling, descrambling and sequence generators is by way of programmable Digital Signal Processors (DSPs). In such an embodiment the incoming multivalue digital signal (in case of a scrambler or descrambler) can be translated by an A/D converter into binary representation and read into the DSP. The DSP can be programmed to execute the scrambling, descrambling and signal generating functionality in binary form and apply a D/A converter to create the multivalue digital output signal.
 Another embodiment of the execution of the multivalue scrambling, descrambling and signal generating functionality is by way of binary logic and binary or hybrid memory elements. Multivalue digital signal inputs can be translated through an A/D converter into binary representation. The ternary and multivalue logic functions can be contained in electronic memory devices in such a way that the binary representation of the input signals form the addresses of memory locations. The content of the memory devices at these locations then form the result of the truth table representing the ternary and multivalue logic functions, also in binary form. The content of the specific memory address location can be used for further processing or can be translated from binary form into a ternary or multivalue signal by using a D/A converter. Consequently multivalue shift registers in LFSR applications can be either clocked binary memory devices. They can also be analog or hybrid devices where the content of a memory location is a physical entity (such as electrical charge) representative of the value of a ternary or multivalue signal.
 Another embodiment of the execution of the multivalue scrambling, descrambling and signal generating functionality is by way of complete discrete binary logic.
 A new embodiment of ternary and multivalue scrambling, descrambling and signal sequence generating execution is the application of existing and emerging ternary and multivalue electronic logic circuits either in CMOS or otherwise.
 A method to design the embodiment in ternary and multivalue electronic form is part of this invention. It is shown for ternary logic but can easily be visualized in any higher value logic.
 The object is to realize any ternary two inputs/single output truth table in ternary logic. The assumption is that the following two electronic elements are available: The first element is an electronic gate that conducts a current if a control input has a value equivalent to a state ‘0’. The gate is nonconducting for all other control values. This gate is shown in conducting state in
FIG. 49 and in nonconducting state inFIG. 50 . It is assumed that the gate is direction sensitive and should be read from left to right.  The second element is a universal ternary inverter equivalent to ternary inverter i_{4 }or i_{5}. Applying inverter i_{4 }twice acts like applying i_{5}. Applying ternary inverter i_{5 }twice acts like applying ternary inverter i_{4 }once. Applying inverter i_{4 }followed by inverter i_{5 }is the same as identity. Applying inverter i_{5 }followed by inverter i_{4 }is also like applying identity. Also applying i_{4 }three times as well as applying i_{5 }three times is equal to identity.

FIG. 51 shows the result of applying ternary reversible inverter i_{4 }twice.FIG. 52 shows the effect of applying ternary reversible inverter i_{5 }twice.  The number columns between the vertical lines in
FIG. 51 andFIG. 52 represent possible states before and after inversion. It should be clear that by applying one inverter once or twice each initial ternary state can be transferred to each desirable other ternary state.  With the gate and universal ternary inverter (i_{4 }or i_{5}) as only available components (but in unlimited amounts), every possible ternary twoinput single output device can be realized. To demonstrate this, the ternary truth table shown in
FIG. 53 will be realized in diagram by applying the gate/universal inverter method. 
FIG. 54 shows a diagram of a device realizing the ternary truth table exclusively with the gate and ternary universal inverter i_{5 }as components. The electronic device 370 has two inputs 371 and 372. The signal b is input to the first input 271 and the signal a is input to the second input 372. The electronic device 370 also has three channels 373, 374 and 375.  Each channel in this case corresponds with the realization of a column of the truth table. The empty circles indicate the inverter i_{5}. The circle 385 with a line inside shows a gate controlled by an input 386. If the state of the control input to the gate 385 is 0, then the gate is conducting and signals are passed from the input 387 to the output 388. If the state of the control input to the gate 385 is not 0, then the gate 385 is not conducting and no signals are conducted from the input 387 to the output 388.
 The electronic device 370 also has an output 381.
 If a=0, then state 0 is applied to gate 376 so that gate 376 is enabled so that the first channel 371 conducts, and input 371 is connected to output 381. In this case, gates 378 and 380 have a non0 state applied from the input signal “a” at input 372. In case of gate 378, the input state 0 is provided to the input of the inverter i_{5 }in the control input of gate 378. The inverter transforms the provided input state to the state 2 which is the control input state to the gate 378, which is thus not conducting. In the case of gate 380, the input state 0 is provided to the input of the first inverter i_{5 }in the control input of gate 380. The inverter transforms the provided input state to the state 2 which is then the input state to the second inverter i_{5 }in the control input of gate 380. The second inverter transforms the provided input state 2 to the state 1 which is then the input state to the control input of gate 380, which is thus not conducting. Thus, the channels 374 and 385, respectively, do not conduct, and cannot contribute to the signal at output 381. Only signals through channel 373 contribute to the output signal at output 381.
 A signal coming from input 371 going through 378 to output 381 is transformed by the inverter i_{5 } 389 in its path. So if a=0 and b=0 then the state of the signal b=0 is transformed to state 2 and output 381 assumes the state 2. If a=0 and b=1 then the state of the signal b=1 is transformed to state 0 and output 381 assumes the state 0. If a=0 and b=2 then the state of the signal b=2 is transformed to state 1 and output 381 assumes the state 1. These states represent the first column of the truth table of the ternary logic function ‘ternary’ as shown in
FIG. 53 .  If a=1, then only gate 378 conducts. Gates 376 and 380 are nonconducting. Then one of the gates 382, 383 or 384 will conduct, depending on the value of the input b. If b=0, then gate 382 will conduct and the subchannel 390 is conducting. Subchannels 391 and 392 are nonconducting. The signal representing b 0 will be inverted by inverter i_{5 }in subchannel 390, and the output will be 2. If b 1, then gate 383 will conduct and subchannel 391 is conducting. In this case, subchannels 390 and 392 are nonconducting. The signal representing b=1 will be inverted by inverter i5 in subchannel 391 and the output will be 0. If b=2 then gate 384 will conduct and the subchannel 392 will be conducting. Subchannels 390 and 392 will be nonconducting. The signal representing b=2 will be inverted twice by the inverters i_{5 }in subchannel 392, first from 2 to 1 and then from 1 to 0 and the output will be 0. The output states represent the second column of the truth table of the ternary logic function ‘ternary’ as shown in
FIG. 53 .  If a=0 then only gate 380 conducts. Gates 376 and 378 are nonconducting since a nonzero value is applied to the control input of these gates as a result of the input a. In this case, the output of the electronic device 370 is 0 when the input b is 0, 1 when the input b is 1 and 2 when the input b is 2. These states represent the third column of the truth table of the ternary logic function ‘ternary’ as shown in
FIG. 53 .  Similar embodiments with components employing a gate that conducts if the control input is ‘0’ and a reversible nvalue inverter would be able to realize any nvalue digital logic function. Thus, any of the nvalue logic functions defined herein can be constructed.
 The reduction of combinations of nvalued inverters at inputs of nvalued logic functions have been disclosed in accordance with one aspect of the present invention. Examples have been provided and
FIGS. 3138 highlight elements of the reduction. The methods as provided apply to all nvalued 2 input/single output functions. They will also apply in essence to functions with more than 2 inputs. The method has been demonstrated for any n>2 and any GF(n). In general the reduction method will be used for LFSRs which initially may be derived from irreducible polynomials over GF(n). It has been shown as another aspect of the present invention that LFSRs can use reversible nvalued logic functions that can not readily be derived from such polynomials. The truth tables of functions may be created from nvalued reversible inverters as rows and/or columns of the truth table.  Appropriate nvalued logic functions for LFSRs cannot be created from truth tables of modulon adders (which are reversible) with modulon multipliers if n can be factorized, such as n=4, n=6, n=9, etc, or in other words if n is not prime. A known method to create a field over GF(n) if n is not prime, but n=u^{P}, wherein u is a prime number and p≧2, is to create an extended field GF(u^{P}). It is known that in order to define an extended field GF(u^{P}) one will use a primitive polynomial over GF(u^{P}) to define the extended field. Within the extended field a reversible multiplication exists and a reversible addition. According to the method explained earlier, the constant multipliers in GF(u^{P}) can be considered as nvalued reversible inverters with n=u^{P}. Accordingly the constant multipliers at inputs of an addition in GF(u^{P}) can be reduced to a single reversible nvalued logic function, wherein the truth table of the addition over GF(u^{P}) is modified according to nvalued inverters representing the multipliers over GF(u^{P}) at the inputs of the addition function. This means that an expression z=(r×x)+(q×y) wherein × is a multiplication over GF(u^{P}) and + is an addition over GF(u^{P}), r and q are nvalued constants and x and y are nvalued variables, can be replaced by an expression z=x sc4 y, wherein sc4 is a nvalued logic function. The truth table of sc4 can be created by modifying the truth table of + according to the nvalued inverter representing (r×x) and the nvalued inverter representing (q×y). Inverters (r×x) and (q×y) are commutative.
 An illustrative example will be provided for GF(2^{2}). Assume the expression z (2×x)+(3×y) over GF(2^{2}). This can be replaced by the expression z=(x sc4 y). The following tables provide the truth tables of addition + and multiplication × over GF(2^{2}), as well as the truth table of sc4.
+ 0 1 2 3 × 0 1 2 3 sc4 0 1 2 3 0 0 1 2 3 0 0 0 0 0 0 0 3 1 2 1 1 0 3 2 1 0 1 2 3 1 2 1 3 0 2 2 3 0 1 2 0 2 3 1 2 3 0 2 1 3 3 2 1 0 3 0 3 1 2 3 1 2 0 3  It should be clear that the function reduction can be performed for any expression z=(r×x)+(q×y) over any extended field GF(u^{P}) with u^{P}>2, u≧2 and p≧1. The situation u^{P}=2 is the binary case. The solution for either r=0 or q=0 is trivial. The solution for r=1 provides z=x+(q×y), which will have a function sc4 created by modifying the + truth table according to (q×y) and q=1 will provide z=(r×x)+y, which will have a function sc4 created by modifying the + truth table according to (r×x).
 In view of the above description of the present invention, it will be appreciated by those skilled in the art that many variations, modifications and changes can be made to the present invention without departing from the spirit or scope of the present invention as defined by the claims appended hereto. All such variations, modifications or changes are fully contemplated by the present invention.
 While the invention has been described with reference to an illustrative embodiment, this description is not intended to be construed in a limiting sense. For example, while the disclosed embodiments utilize discrete devices, these devices can be implemented using one or more appropriately programmed processors, specialpurpose integrated circuits, digital processors, or an analog or hybrid counterpart of any of these devices.
Claims (20)
1. An apparatus for evaluating an u^{P }valued arithmetical expression z=(r×x)+(q×y) wherein × is a multiplication defined in GF(u^{P}) and + is an addition defined in GF(u^{P}), with u≧2 and p≧1 except u^{P}=2, which can be represented by a u^{P }valued truth table, x and y are variables that can each assume one of u^{P }values and r and q are constants each of which is assigned one of (u^{P}−1) values not including 0, and z is a result of the u^{P }valued expression, comprising:
a device with a first input enabled for receiving a signal representing variable x and a second input enabled for receiving a signal representing variable y and an output;
the device implementing a switching function sc4 with an u^{P}valued truth table different from the truth table of the addition; and
the output enabled to provide a signal representing z.
2. The apparatus as claimed in claim 1 , wherein:
an u^{P}valued expression (r×x) can be represented by a first u^{P}valued inverter;
an u^{P}valued expression (q×y) can be represented by a second u^{P}valued inverter; and
the truth table of the u^{P}valued function sc4 can be created by modifying the truth table of the addition according to the first and the second inverter.
3. The apparatus as claimed in claim 2 , wherein an u^{P}valued expression (x×r) can also be represented by the first u^{P}valued inverter.
4. The apparatus as claimed in claim 1 , wherein the apparatus is part of an u^{P}valued Linear Feedback Shift Register (LFSR).
5. The apparatus as claimed in claim 2 , wherein either the first or the second u^{P}valued inverter is a unity inverter.
6. A method for physically evaluating an nvalued arithmetical expression z=(x×r)+(y×q), with n≧3, x and y being nvalued variables of which each can assume one of n values, r and q being nvalued constants each of which is assigned one of (n−1) values not including 0, with nvalued arithmetical functions × and +, with ×being a multiplication and with + being an addition defined in a Finite Field GF(n=u^{P}) with u≧2 and p≧1, and wherein the addition can be defined by a nvalued truth table comprising, executing a nstate switching expression (x sc4 y), wherein sc4 is an nstate switching function with a truth table that is different from the truth table of the addition.
7. The method as claimed in claim 6 , further comprising:
applying the method for implementing an nvalued Linear Feedback Shift Register.
8. The method as claimed in claim 6 , further comprising:
assigning r=1 to the nvalued expression, making it z=x+(y×q).
9. The method as claimed in claim 6 , further comprising:
assigning q=1 to the nvalued expression, making it z=(x×r)+y.
10. A method for implementing an Linear Feedback Shift Register (LFSR) that can be described by a nvalued irreducible polynomial over GF(n=u^{P}) with u≧2 and p≧1 except u^{P}=2 that includes executing at least one nvalued expression (r×x)+(q×y) wherein x and y are nvalued variables, r and q are nvalued constants and × is a commutative multiplication defined in GF(n=u^{P}), and + is a nvalued addition defined in GF(n=u^{P}) which can be represented by an nvalued truth table, comprising:
evaluating an nvalued expression (x sc4 y), wherein sc4 is a nvalued switching function which can be represented by a nvalued truth table that is different from the truth table of the addition.
11. The method as claimed in claim 10 , wherein the irreducible polynomial is primitive.
12. The method as claimed in claim 10 , wherein (r×x)=x.
13. The method as claimed in claim 10 , wherein (q×y)=y.
14. The method as claimed in claim 10 , further comprising using the LFSR for generating a sequence of nvalued symbols.
15. The method as claimed in claim 10 , further comprising using the LFSR for scrambling a sequence of nvalued symbols.
16. The method as claimed in claim 10 , further comprising using the LFSR for descrambling a sequence of nvalued symbols.
17. A method for physically evaluating an nvalued logic expression (x sc1 r) sc2 (y sc3 q) with n≧3, wherein x and y are nvalued variables, r and q are nvalued constants and sc1, sc2 and sc3 are nvalued switching functions which can be represented by nvalued truth tables, comprising:
creating a nvalued switching function sc4 with a truth table by modifying the truth table of function sc2 according to a first inverter representing (x sc1 r) and a second inverter representing (y sc3 q);
implementing the nvalued switching function sc4 in a device with a first input, a second input and an output;
providing signals representing nvalued variables x and y to the device; and
generating a signal representing z=(x sc4 y) on an output of the device.
18. The method as claimed in claim 17 , further comprising:
reducing the expression (x sc1 r) sc2 (y sc3 q) to an nvalued expression ((x sc1 r) sc5 y) wherein sc5 is an nvalued switching function which can be represented by an nvalued truth table which can be created by modifying the truth table of sc2 according to a second nvalued inverter representing (y sc3 q); and
reducing the nvalued expression ((x sc1 r) sc5 y) to the nvalued expression (x sc4 y) wherein sc4 can be represented by an nvalued truth table which can be created by modifying the truth table of sc5 according to a first nvalued inverter representing (x sc1 r).
19. The method as claimed in claim 18 , further comprising the steps:
selecting a first element of the second nvalued inverter representing (y sc3 q);
selecting in the truth table of sc2 a column corresponding with an input equal to the first element of the second inverter of the previous step;
placing the selected column of the truth table of sc2 in the previous step in the truth table of the nvalued switching function sc5 in the column position corresponding to the position of the element in the second inverter;
selecting a next element of the second nvalued inverter as the first element; and
repeating the previous three steps until all n elements of the second nvalued inverter have been evaluated.
20. The method as claimed in claim 19 , further comprising the steps:
selecting a first element of the first nvalued inverter representing (x sc1 p);
selecting in the truth table of sc5 a row corresponding with an input equal to the first element of the inverter of the previous step;
placing the selected row of sc5 of the previous step in the truth table of nvalued switching function sc4 in the row position corresponding to the position of the element in the first inverter;
selecting a next element of the first nvalued inverter as the first element; and
repeating the previous three steps until all n elements of the first inverter have been evaluated.
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US13/027,387 US8589466B2 (en)  20040225  20110215  Ternary and multivalue digital signal scramblers, decramblers and sequence generators 
US14/064,089 US20140055290A1 (en)  20030909  20131025  Methods and Apparatus in Alternate Finite Field Based Coders and Decoders 
US14/622,860 US9218158B2 (en)  20030909  20150214  Nvalued shift registers with inverter reduced feedback logic functions 
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