WO2010001168A1 - Signal spreading system and method - Google Patents

Signal spreading system and method Download PDF

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Publication number
WO2010001168A1
WO2010001168A1 PCT/GB2009/050772 GB2009050772W WO2010001168A1 WO 2010001168 A1 WO2010001168 A1 WO 2010001168A1 GB 2009050772 W GB2009050772 W GB 2009050772W WO 2010001168 A1 WO2010001168 A1 WO 2010001168A1
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Prior art keywords
sequence
spreading
signal
spread
sequences
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PCT/GB2009/050772
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French (fr)
Inventor
Riqing Chen
David J. Edwards
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Isis Innovation Limited
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04J—MULTIPLEX COMMUNICATION
    • H04J13/00—Code division multiplex systems
    • H04J13/10—Code generation

Abstract

There is provided a signal spreading system (10) comprising a pseudonoise signal spreader (12) arranged to spread a plurality of signals (14, 16, 18, 20). The spreader (12) is arranged to spread each different signal (14, 16,18, 20) using a different spreading sequence defined by a different value of n in a set, α, of pseudonoise spreading sequences, each spreading sequence having a sequence length, L, wherein the set, α is defined by Formula (I). or Formula (II) or Formula (III) or Formula (IV) where, f(κ) satisfies the following properties: f(κ+τ)=h(τ).f(κ), while,|f(κ)|=1, |h(κ) |=1, |g(n)|=1 where | • | is the norm value of the complex number •, where 0 ≤n, κ ≤ L-1, and where λ ≥ -1, and wherein n, L, λ and ĸ are integers.

Description

SIGNAL SPREADING SYSTEM AND METHOD
The present invention relates to a signal spreading system, method and associated apparatus for spreading signals using pseudorandom noise (also known as pseudonoise) sequences. The invention applies preferably, but not exclusively, to communication systems, for example, telecommunication systems.
Spread spectrum techniques are known for spreading a signal of a first bandwidth centred at a given frequency over a larger bandwidth. In a communication system, this can be achieved by spreading the signal over different communication channels centred at different frequencies. The increase in bandwidth use is tolerated in communication systems since the spread signal appears similar to noise and is thus relatively secure since it is difficult to detect or intercept, and is also more resistant to natural interference.
A technique for providing a spread signal is to take an original signal to be spread and combine it with a pseudorandom noise, or pseudonoise,
PN, or PN code signal to obtain the spread signal. The spread signal is then suitable for transmission over a communication network with the advantages mentioned above and others that are apparent to the skilled person. When it is received at a receiver, the spread signal is again operated upon by the same PN code in order to obtain the original signal.
In general, different PN codes are used to spread different signals (e.g. for different users on a telecommunications network) . It is a known problem that interference can occur between different spread signals. To this end, it is desirable to have orthogonal PN codes (also known as spreading sequences) . Such known sequences include Walsh-Hadamard sequences. Within a particular family or set of Walsh-Hadamard sequences, each sequence is orthogonal to each other sequence under ideally synchronised conditions. This property can be useful in, for example, telecommunication systems in which the risk of cross-channel interference can be reduced by using such orthogonal Walsh-Hadamard spreading sequences. However, if (as in real conditions) there is a time delay (due to varying or unexpected signal paths) between transmission and receipt of two spread signals generated within the same Walsh- Hadamard set, the two spread signals will not be completely orthogonal. Interference due to the time delay occurs in such real world, asynchronous systems. Similarly, if the spread signals are Doppler shifted between transmission and receipt, the spread signals will no longer be orthogonal.
The present invention relates to a spreading code with improved orthogonality properties between its sequences when spread signals are time delayed and/or Doppler shifted between transmission and receipt.
According to an aspect of the invention there is provided a signal spreading system as claimed in claim 1.
As described in further detail below, the spreading system of this invention provides a plurality of signals having improved orthogonality relative to each other under asynchronous time and delay shift conditions.
According to another aspect of the invention there is provided a method of spreading a plurality of signals as claimed in claim 4.
According to another embodiment of the invention there is provided a transmitter for a communication system as claimed in claim 6. According to another embodiment of the invention there is provided a receiver for a communication system as claimed in claim 7.
According to another aspect of the invention there is provided a group of spread signals as claimed in claim 9.
According to another embodiment of the invention there is provided a method of generating a plurality of pseudonoise sequences as found in claim 10.
According to another embodiment in the invention there is provided a pseudonoise generator arranged to generate a plurality of pseudonoise sequences as claimed in claim 11.
According to another embodiment of the invention there is provided a spread spectrum communication system as claimed in claim 12.
In some embodiments there is provided a sequence length calculator arranged to determine a desired value of sequence length, L, by taking into account any one or more of the number of users, or channel state information in a communication system across which the spread signal is to be sent. In such embodiments, the efficiency of the system can be increased by determining a particular value for the sequence length which may be optimal for the number of users or channel state information or both - the sequence length can impact upon system performance as described in further detail below.
In some embodiments the signal comprises an information signal suitable for transmission through a communication network. Where the invention is claimed in one aspect (e.g. method, system etc. protection is also sought in other aspects, e.g. for the features of the dependant claims or of the description.
Embodiments of the invention will now be described by way of example only with reference to the accompanying drawings in which:
Figure 1 schematically shows a signal spreading system according to a first embodiment of this invention;
Figure 2 schematically shows a signal spreading system according to a further embodiment;
Figure 3 schematically represents a method of spreading a plurality of signals according to an embodiment of this invention;
Figure 4 schematically shows a transmitter and receiver for a communication system arranged to transmit and receive spread signals according to an embodiment of this invention;
Figure 5 compares an analytical result of phase shift of periodic autocorrelation function compared to a simulation result for a KR-I sequence index n = 4, L = 16, 64, respectively;
Figure 6 shows an analytical result of phase shift of periodic autocorrelation function compared to a simulation result for KR-2 sequence index n = 4, L = 16, 64 respectively;
Figure 7 shows the periodic correlation of a KR-I sequence, a KR-2 sequence, and Walsh-Hadamard sequence under a periodic delay shift from 0 to L-I , with L = 16, 64 respectively; Figure 8 shows periodic autocorrelation and cross-correlation amplitude of KR sequences under different normalized Doppler shift fd from 0 to
15%;
Figure 9 illustrates the correlation function of KR sequence by the different sequence index distance, id = m -n , under normalized Doppler shift, fd ;
Figure 10 shows the variation of the peak sidelobe amplitude to different sequence index distance id = m - n under KR sequence set length 16, 32, 64, 128, respectively;
Figure 11 shows periodic correlation of KR-I sequence, KR-2 sequence, Walsh-Hadamard sequence with sequence length L = 64 under a normalised Doppler shift, fd = 0.5%, 1%, respectively.
Referring to Figure 1 , there is shown a signal spreading system 10 comprising a pseudonoise signal spreader 12 arranged to spread a plurality of signals. In this embodiment, there are four signals, a first signal 14, a second signal 16, a third signal 18 and a fourth signal 20. In some embodiments, each signal comprises an information signal suitable for transmission through a communications network - such as a telecommunications network. In such embodiments, each of the first 14, second 16, third 18 and fourth 20 signals may originate from different users on the network, for example from different mobile telephone users.
In this embodiment, the PN signal spreader 12 is shown as a separate first signal spreader 22, second signal spreader 24, third signal spreader 26 and fourth signal spreader 28. In some embodiments the separate signal spreaders 22, 24, 26, 28 may be provided as a single unitary signal spreader with the same function as the separate signal spreaders each signal spreader 22, 24, 26, 28 in the embodiment of Figure 1 is arranged to spread each different respective signal 14, 16, 18, 20. In other embodiments, the PN signal spreader 12 may be a single spreader arranged to spread all of the signals 14, 16, 18, 20. In yet further embodiments, the signal spreader 12 may comprise more than one unit, each unit being arranged to spread one or more of the signals. It will be appreciated that the number of discrete physical signal spreading units is not important - it is their function as described in more detail below that is important.
Signals 14, 16, 18, 20 received at the first, second, third and fourth signal spreaders 22, 24, 26, 28 are spread to provide respectively a first spread signal 30, a second spread signal 32, a third spread signal 34 and a fourth spread signal 36.
The PN signal spreader 12 is arranged to spread each different signal 14, 16, 18, 20 using a different spreading sequence from a set, α of pseudonoise spreading sequences. Each spreading sequence has a sequence length, L. The set, α, is defined by:
Figure imgf000007_0001
where 0 ≤ n,k ≤ L -\ , and wherein n, L, and k are integers.
Each different signal is spread using a different spreading sequence as defined by a different value of n in the set, α. Referring to Figure 1 , the signal spreading units 22, 24, 26, 28 are denoted using n=0, n=l , n=2 and n=3 to symbolise the fact that different values of n are used in generating the relevant sequence for spreading the relevant incoming signal.
In this embodiment, a code generator 38 is used to generate the different spreading sequences and communicate with the PN signal spreader 12 to indicate the operations which need to be performed on each different signal. In other embodiments, the code generator 38 may be part of the PN signal spreader 12.
The spread signals 30, 32, 34, 36 have better orthogonality properties (under time-delayed and Doppler-shifted conditions) than spread signals obtained using existing (e.g. Walsh-Hadamard) spreading sequences. These properties are described in further detail below.
In other embodiments, α may be defined by
Figure imgf000008_0001
where 0 ≤ k ≤ L 1 and where λ ≥ -1 , and wherein n, L, λ and k are
Figure imgf000008_0005
integers.
Alternatively α may be defined as
Figure imgf000008_0002
where and wherein n, L, and k are integers.
Figure imgf000008_0004
Generally,
Figure imgf000008_0003
While, f(k) satisfies the following properties:
Figure imgf000008_0006
Where | • | is the norm value of the complex number • .
These alternative definitions of α also provide a family of spread signals with improved orthogonality properties as described in further detail below.
Referring to Figure 2, in another embodiment of the invention, there is provided a sequence length calculator 40 arranged to determine a desired value of the sequence length, L. In this embodiment, the signal spreading system is used in a communication system across which the spread signals 30, 32, 34, 36 are arranged to be sent. The sequence length calculator 40 determines a desired value of the sequence length L by taking into account the number of users in the communication system, channel state information in the communication system or both. The number of users and the channel state information has an impact (as discussed in further detail below) upon the optimum value for sequence length in order to provide efficient operation of the system.
Referring to Figure 3, a method 42 of spreading a plurality of signals comprises operating 44 on the signals using a set, α of pseudonoise spreading sequences, each spreading sequence having a sequence length, L, and the set, α, being defined by
Figure imgf000009_0001
or
Figure imgf000010_0001
or
Figure imgf000010_0002
or, generally where
Figure imgf000010_0003
Figure imgf000010_0004
While, f(k) satisfies the following properties:
Figure imgf000010_0005
Where | • | is the norm value of the complex number • ,
where and where , and wherein n, L, λ and k are
Figure imgf000010_0006
Figure imgf000010_0007
integers.
The step of operating 44 on the signals comprises operating on each different signal using a different spreading sequence from the set, a.
Referring to Figure 4, a transmitter 44 is arranged to transmit one or more of a group of spread signals, each different spread signal of the group having been obtained by spreading an original signal using a different spreading sequence according to the method 42 as previously described. In this embodiment this is achieved by way of the spreading system 10. Spread signals originating from the transmitter 44 are arranged to be received at a receiver 46 which is arranged to receive one or more of the group of spread signals. The transmitter 46 is in communication with a de-spreader 48 arranged to operate upon the received spread signals using the set, α, of spreading sequences in order to recover the original signals. This is a known technique for recovering information from spread signals. The de-spreader 48 may be part of the receiver setup, or in some embodiments it may be distinct from the receiver.
The generation and properties of each of the alternative sets of pseudonoise spreading sequences defined above will now be described in more detail.
In general, if we consider a polyphase sequence family a(n,k) of length L with unity modulus in the complex plane [4] , i.e. , , where,
Figure imgf000011_0006
Figure imgf000011_0001
The periodic correlation function can be written as:
Figure imgf000011_0003
Figure imgf000011_0005
[D]^ iS the complex conjugate operation of [D] , if [D] is matrix, [D]H returns the complex conjugate transpose of [D] .
KR-I Sequence Generation
One generalized family of orthogonal polyphase sequences according to this invention is called a KR-I sequence, and is defined with a unified expression as follows:
where,
Figure imgf000011_0004
Figure imgf000011_0002
According to equation (2) , the periodic correlation of can be
Figure imgf000012_0007
written as below:
Figure imgf000012_0002
Therefore, the magnitude of periodic autocorrelation function for
Figure imgf000012_0006
KR-I sequences is constant and rotates around the unit complex circle
with period where gcd(L,n) is the Greatest Common Divisor
Figure imgf000012_0004
between L and n. In Figure 5, the analytical and simulation result for the phase shifts of periodic autocorrelation function of a KR-I sequence is exactly matched with sequence index
Figure imgf000012_0005
respectively. The analytical result is obtained from equation (4) ; and the simulation result is calculated from equation (2) .
KR-2 Sequence Generation
Another generalized family of orthogonal polyphase sequences according to the invention, which the inventors call a KR-2 sequence, is defined as the following even and odd case, respectively
Figure imgf000012_0001
where
Figure imgf000012_0003
, and wherein n, L, and k are integers. According to equation (2) , the periodic correlation of a(n,k) can be written as below and the complete mathematical analysis is shown in Appendix A, equation (A.19) .
Figure imgf000013_0001
where,
Figure imgf000013_0003
Therefore, the magnitude of periodic autocorrelation function for
Figure imgf000013_0005
KR-2 sequences is maintained constant and rotates around the unit
complex circle with period , where gcd(L,n) is the Greatest
Figure imgf000013_0004
Common Divisor between L and n. In Figure 6, the analytical and simulation result for phase shift of periodic autocorrelation function of a KR-2 sequence is exactly matched with sequence index n = 4, L = 16, 64, respectively. Here the analytical result is obtained from the equation (6) ; and the simulation result is calculated from the equation (2) .
Another generalized family of orthogonal polyphase sequences, which the inventors call a KR 3 sequence, is defined as
Figure imgf000013_0002
where and where
Figure imgf000013_0007
, and wherein n, L, λ and k are
Figure imgf000013_0006
integers.
The periodic correlation of α can be written as:
Figure imgf000014_0001
Another broader generalised family of orthogonal polyphase sequences is
described by:
Figure imgf000014_0003
While, f(k) satisfies the following properties:
Figure imgf000014_0004
Where | • | is the norm value of the complex number • .
Hence, the periodic correlation function of the generalized KR orthogonal Polyphase sequences is expressed as follows:
Figure imgf000014_0002
This broader expression covers the narrower expressions (KRl , KR2, KR3) , and can be used to generate such narrower sequences. The narrower sequences can not obviously be used to generate each other, or the broader sequence generator, i.e. if KRl is known, it is not clear or obvious for a skilled person to arrive at KR2, KR3 or the generic generalised polyphase sequence generator.
Using the broader, generic expression, polyphase sequences, KR sequences can be generated and detected by conventional polyphase sequence methods. For, example by an FPGA or by a phase shift keying type architecture. Properties of KR Sequences
In Figure 7, the periodic correlation function of KR sequences, and Walsh-Hadamard sequences is displayed under a time delay shift, τ from 0 to L-I , with L = 16, 64 respectively. The advantage of using KR spread sequence codes over Walsh-Hadamard spread sequence codes of the same sequence length is clearly observed. The KR sequence demonstrates a triangular peak-value in autocorrelation and the flat zero-value in cross- correlation over any time delay shift X from 0 to L-I . This much improved cross-correlation is particularly desirable in asynchronous SSMA systems. Whereas, the Walsh-Hadamard sequence exhibits significant sidelobes over a time delay shift τ from 0 to L-I , which is extremely detrimental to asynchronous applications. In practice, these significant sidelobes would manifest as interference between signals in the communications system.
For example, as a result of these properties, in a telecommunications system, in "real world" conditions, a first spread signal and a second spread signal may be transmitted at a transmitter, and received at a receiver, and may experience some time delay in their paths between the transmitter and receiver. In this situation, upon receipt the receiver must resolve the two spread signals without interference. If the signals have been spread using one of the KR sequences of this invention, it can be seen that the level of the autocorrelation signal is high, and the level of cross-correlation (i.e. interference with the other spread signal) is uniformly very low. In contrast, if the spread signals are spread using a conventional Walsh Hadamard sequence, there is a significant cross- correlation signal which results in interference between signals, and possible non-recognition or unsuccessful receipt of signals.
Periodicity of KR Orthogonal Sequences Each KR sequence is periodic with respect to its sequence length L
From the generation equations (3) & (5) , we find that the KR sequence
Figure imgf000016_0005
is periodic with respect to the sequence length L, i.e. , where
Figure imgf000016_0003
k is integer (7)
Figure imgf000016_0004
The maximum KR sequence index n corresponds to a sequence length L.
Figure imgf000016_0001
Figure imgf000016_0002
Periodic Autocorrelation
From equation (4) and (6) , it is found that a typical feature of KR sequences is that the magnitude of the autocorrelation function remains constant with any delay shift τ . The autocorrelation value is rotated on the unity complex circle in accordance with the delay shift τ .
Cyclic Shift
This relates to the time delay shift of a polyphase sequence; if a KR sequence is cyclically shifted by p slots, the autocorrelation function remains constant.
Figure imgf000016_0006
Decimation
This relates to the sampling interval of a polyphase sequence; if a KR sequence is properly decimated by p, where /? is an integer less than L, the periodic autocorrelation function is also decimated:
Figure imgf000017_0001
Reflection
This relates to the complex conjugation of a polyphase sequence; if a KR sequence is reflected, the autocorrelation function is also reflected.
Figure imgf000017_0002
Constant Phase Shift
If a phase shift
Figure imgf000017_0004
is added to the elements of a KR sequence, the autocorrelation function maintains constant:
Figure imgf000017_0003
Periodic Cross-correlation
The KR sequence is a family of orthogonal polyphase sequences and as such it is important to investigate their Periodic Cross-correlation Properties. As we see from equation (4) and (6) , the Cross-correlation function is zero at any delay shift X , which is highly desirable for future Multi-carrier Spread Spectrum Multiple Access communication systems. Hence: in uplink transmission, the multi-users can be allowed to send data to the base station spontaneously, according to their specific requirements, which could greatly save transmission power at base station and thus multi-user interference is dramatically reduced; and
in downlink transmission, the base station is able to send data to multi- users asynchronously. Therefore, the complicated strategy of channel assignment can be deployed with energy-efficiency.
Here, assuming the KR sequence index m ≠ n :
Cyclic Shift
If a KR sequence is cyclically shifted by p slots, the Cross-correlation within the corresponding sequence set is also cyclically shifted with p, and therefore remains zero at any delay shift τ , where,
Figure imgf000018_0003
is an integer.
Figure imgf000018_0001
Decimation
If a KR sequence a(n,k) is properly decimated by p, the cross-correlation of the decimated sequence
Figure imgf000018_0004
is also decimated by p, and thus remains zero at any delay shift τ .
Figure imgf000018_0002
Meanwhile, for a KR-I sequence, the decimated sequence
Figure imgf000018_0008
becomes
Figure imgf000018_0007
mod L,k) , i.e. , g = (n - p)mod L . Therefore, the cross- correlation of
Figure imgf000018_0005
changed into the cross-correlation of
Figure imgf000018_0006
mod L,k) as well. re p is integer (16)
Figure imgf000019_0004
For a KR-2 sequence, the decimated sequence
Figure imgf000019_0010
becomes:
Provided that L is odd, where p is integer (17)
Figure imgf000019_0005
Provided that L is even, g where p is integer (18)
Figure imgf000019_0006
Reflection
If a KR sequence is reflected, the cross-correlation function of is
Figure imgf000019_0011
also reflected, and thus remains zero at any delay shift τ .
Figure imgf000019_0001
For a KR-I sequence, the reflected sequence
Figure imgf000019_0008
is also transformed into its complex conjugated sequence
Figure imgf000019_0007
Figure imgf000019_0002
For a KR-2 sequence, the reflected sequence is also changed into
Figure imgf000019_0009
the following two cases:
[ L
Provided that L is even,
Figure imgf000019_0003
Provided that L is odd,
Figure imgf000020_0001
Constant Phase Shift
If a phase shift
Figure imgf000020_0006
is added to each element of a KR sequence, the cross- correlation function is also phase shifted with
Figure imgf000020_0007
and thus maintains a zero value at any delay shift τ
Figure imgf000020_0002
Similar analysis can be carried for the KR-3 sequences as defined by:
Figure imgf000020_0003
where
Figure imgf000020_0005
and where
Figure imgf000020_0004
and wherein n, L, λ and k are integers.
Advantages of using KR spread sequences according to this invention relative to known spreading code sequences (such as Walsh-Hadamard sequences) are illustrated via the cross-correlation properties graphically represented in the previously described figures. The KR sequences according to this invention also provide spread signals which have improved orthogonality relative to each other when Doppler shifted as discussed in further detail below.
Doppler Shift Resilience
The Doppler shift resilience of a polyphase sequence relates to its linear phase shift property. The periodic cross-correlation function and autocorrelation function of KR sequences are
Figure imgf000021_0005
defined in equations (B.2, B.3, B7, B.8, B.9, B.10 from appendix B) by the following three factors:
1) Normalized Doppler shift fd
2) Sequence length L
3) The sequence index distance:
Figure imgf000021_0006
General Correlation Function with Doppler Shift
Assuming that a normalized Doppler shift fd is added to each element of a KR sequence, the general periodic correlation function in equation (2) becomes:
Figure imgf000021_0002
From the mathematical analysis of Doppler shift from Appendix B, we find:
For the KR-I sequence, the periodic cross-correlation function
Figure imgf000021_0007
can be written as:
Figure imgf000021_0001
Where
Figure imgf000021_0003
The periodic autocorrelation function
Figure imgf000021_0004
becomes:
Figure imgf000022_0001
Where, m
Figure imgf000022_0004
For the KR-2 sequence, the periodic cross-correlation function Cm n(τ ,fd) can be written as:
(27)
Figure imgf000022_0002
The periodic autocorrelation function becomes:
Figure imgf000022_0007
Figure imgf000022_0003
Where,
Figure imgf000022_0006
Magnitude of General Correlation Function with Doppler shift
From equations (25, 26, 27, 28) , we see that the magnitude of the periodic cross-correlation function and periodic
Figure imgf000022_0008
autocorrelation function under a normalized Doppler shift
Figure imgf000022_0005
Figure imgf000022_0009
is the same in both KR-I and KR-2 sequence. Therefore, for these KR sequences, the magnitude of the periodic crosscorrelation function under a normalized Doppler shift fd can
Figure imgf000023_0011
be described jointly as:
Figure imgf000023_0001
Where
Figure imgf000023_0005
Also, the magnitude of the periodic autocorrelation function
Figure imgf000023_0002
under a normalized Doppler shift fd becomes:
Figure imgf000023_0003
Where,
Figure imgf000023_0006
The effect of Doppler shift to the periodic correlation function
Figure imgf000023_0010
is on the term:
Figure imgf000023_0004
The magnitude of the correlation is determined by the
Figure imgf000023_0007
magnitude of Hence, from equation (31) , mathematically, the
Figure imgf000023_0008
periodic correlation of KR sequence length L is diminishing and fluctuating with the Doppler shift interval
Figure imgf000023_0009
The peak magnitudes are at the Doppler shift: where k is integer (32)
Figure imgf000024_0001
Whereas, the "zero" magnitudes are at the Doppler shift:
where k is integer (33)
Figure imgf000024_0002
In Figure 8 it can be seen that the peak sidelobes of sequence length L
occur at the normalized Doppler shift . With the
Figure imgf000024_0003
development of normalized Doppler shift fd , the peak sidelobes become smaller tending to a zero value. The "zero" sidelobes occur at the
normalized Doppler shift The sidelobes of KR sequence
Figure imgf000024_0004
length 64 decreases into zero faster than any other smaller sequence length 16, 24, or 32. The cross-correlation magnitudes fluctuate over the
normalized Doppler shift interval — . Therefore, a KR sequence with a
longer sequence length L has a shorter normalized Doppler shift interval, which means it is more sensitive to the variation of normalized Doppler shift fd .
In Equation (31) , we see that periodic correlation function of a KR sequence under a normalized Doppler shift fd is also affected by the sequence index distance: id = m -n . The case id= 0, it means the autocorrelation; while id ≠ 0 , represents the cross-correlation for sequence index distance: id = m -n .
In Figure 9, with the increase of normalized Doppler shift fd , the correlation magnitude becomes flatter and tends to a zero-value. The 1 highest peak sidelobe values occur at normalized Doppler shift fd = — .
In Table 1 , the peak sidelobe values of the periodic auto-correlation are k presented with the development of normalized Doppler shift fd = — ,
when fd is more than 10%, the peak sidelobe value becomes less than 0.05. The smallest value is 3.13% at fd = 25% for a KR sequence length 32.
Table 1: The peak sidelobe values of periodic auto-correlation for KR sequence length 32 decrease with the development of Doppler shift
Figure imgf000025_0001
The peak sidelobe magnitudes tend to a constant value with the development of the sequence index distance id. From Figure 10, we observe that the convergent value diminishes from 0.0628 under L = 16 to 0.0078 under L = 128. Therefore, the longer KR sequence length exhibits the flatter and the smaller values of Cross-correlation peak; and thus
better Doppler shift resilience is achieved. After id > — , the variation of
2 peak sidelobe magnitudes becomes symmetric in the case of with
Figure imgf000026_0002
symmetry axis
Figure imgf000026_0003
From Figure 11 , under a normalized Doppler shift
Figure imgf000026_0001
the KR sequence maintains the triangular correlation performance better than the Walsh-Hadamard sequence with regard to time delay shift X from 0 to L- 1. When the normalized Doppler shift fd = 1%, the Walsh-Hadamard sequence has been barely recovered from its orthogonal correlation property, the correlated component has been totally merged into the Multi-channel interference. On the other hand, the correlation magnitude of the KR sequence remains triangular with 0.8001 in auto-correlation peak and a flat average value of 0.0401 in Cross-correlation. It is clear that, the KR sequence exhibits Doppler shift resilience superior to Walsh- Hadamard sequence under a normalized Doppler shift fd ≠ 0.
Therefore four new families of generalized KR orthogonal sequences are proposed with important cross-correlation properties across any time delay shift X from 0 to L-I . The simulation results are in line with the mathematical analysis.
Under normalized Doppler shift fd = 0 , the KR sequence shows excellent correlation properties. In this case, the triangular peak-value in autocorrelation and the flat zero-value in cross-correlation are demonstrated along the time delay shift X from 0 to L-I . By contrast, the binary Walsh-Hadamard sequences exhibit significant sidelobes, which is extremely detrimental to asynchronous communication applications. The magnitude of the periodic autocorrelation function remains constant with respect to the in-phase peak value, which is not desirable from the traditional perspective of spreading sequence for Spread Spectrum Multiple Access communication systems. However, since the KR sequence is an algebraic orthogonal code sequence, their correlation functions can be derived mathematically. These analytical functions could be used in the decoding algorithms as a reference for estimation.
In the case of a normalized Doppler shift fd ≠ 0 , the KR sequence maintains triangular correlation performance along time delay shift X from 0 to L-I and exhibits Doppler shift resilience better than Walsh- Hadamard sequences. For the periodic correlation of KR sequences with sequence length L, the fluctuating sidelobes are periodic with the peak
values at the Doppler shift and the "zero" values at the
Figure imgf000027_0001
Doppler shift where k is integer. Therefore, KR sequences with
Figure imgf000027_0002
longer sequence length L have a shorter fluctuating interval of Doppler shift, and become more sensitive to the variation of Doppler shift. However, the KR sequence with a longer sequence period has the flatter and smaller magnitude of cross-correlation peak sidelobes; and thus achieves the better Doppler shift resilience.
In embodiments in which the spreading system includes a sequence length calculator arranged to determine a desired value of sequence length, L, these features can be taken into account. For example the feature that a longer sequence length, L provides a shorter fluctuating interval of Doppler shift, whereas a longer sequence length, L, provides flatter and smaller absolute values of cross-correlation sidelobes (thus achieving better Doppler shift resilience) may be considered against each other to provide a desired balance within the system depending upon its particular application.
In a communication system, at any one time the spreading system of this invention allows up to L different spreading sequences to be generated per family - none, one or more (up to L) signals may actually be spread using these sequences at any one time. For example, L might be 2, 4, 8,
16, 32, 64, 128, 256, ... Supposing L = 1024, each value of n corresponds to a different user on the network, and a different spreading sequence from a particular family - at a particular time there may only be some of the possible sequences, e.g. 20 (corresponding to 20 network users) in use. There is still the capacity for L users at such a time.
Appendix A provides sequence generation proofs for KR-I , KR-2, KR-3 and the broad generalised KR sequence generator.
Appendix B shows the effect of Doppler shift on the sequence families generated by KR-I , KR-2, KR-3 and the broad KR sequence generator.
Appendix A
KR-I Sequence Generation Proof:
From Equation (3) , the first generalized family of orthogonal Polyphase sequences is described with a unified expression as:
Figure imgf000029_0001
From Equation (2) , the periodic correlation function of a(n,k) can be written as below:
Figure imgf000029_0004
Notice that in the 2nd-term:
Figure imgf000029_0002
Therefore, both terms merge together as:
Figure imgf000029_0003
If m ≠ , thus,
Figure imgf000030_0001
Figure imgf000030_0008
m n (A.5)
Since where is an L th root of unity and α ≠ 1 .
Figure imgf000030_0009
Figure imgf000030_0007
I ,
Figure imgf000030_0002
thus,
Figure imgf000030_0003
Hence, the periodic correlation of KR-I Sequence is generalized as follows:
Figure imgf000030_0004
I. KR-2 Sequence Generation Proof: From Equation (5) , the second generalized family of orthogonal Polyphase sequence is categorized as following two cases:
Figure imgf000030_0005
where,
Figure imgf000030_0006
II.1) L is Even
According to the Equation (2) , the periodic correlation function of a(n,k) can be written as below:
Figure imgf000031_0001
Notice that in the 2nd term:
Figure imgf000031_0002
Therefore,
Figure imgf000031_0003
Figure imgf000032_0001
thus,
Figure imgf000032_0005
II.2) L is Odd
According to the Equation (2) , the periodic correlation of a(n,k) can be written as below:
Figure imgf000032_0002
Notice that in the 2nd term:
Figure imgf000032_0003
Figure imgf000032_0004
Since L is odd,
Figure imgf000033_0006
is integer, is integer as
Figure imgf000033_0005
well, hence, this term is equal to 1. After that, both terms merge together:
Figure imgf000033_0001
(A.16)
Figure imgf000033_0003
(A.17)
Figure imgf000033_0002
Figure imgf000033_0004
In conclusion, the periodic correlation of KR-2 sequence can generalized as:
Figure imgf000034_0001
KR-3 Sequence Generation Proof:
The 3rd-generalized family of orthogonal polyphase sequences, which we call KR-3 Sequence, is described united as:
Figure imgf000034_0002
(A.20)
where,
Figure imgf000034_0004
According to Eq. (2) , the periodic correlation function can be written as:
Figure imgf000034_0003
(A.21)
Figure imgf000034_0005
is the complex conjugate operation of [D] , if [D] is matrix, [D]ff returns the complex conjugate transpose of [D] . Therefore, the periodic correlation function of a(n,k) can be written as below:
Figure imgf000035_0001
(A.22)
Notice that at the end of the 2nd term:
Figure imgf000035_0002
Figure imgf000035_0005
Figure imgf000035_0003
(A.23)
Therefore,
Figure imgf000035_0004
Figure imgf000036_0001
(A.24)
If, then
Figure imgf000036_0008
Figure imgf000036_0002
(A.25)
Else if, m = n then
(A.26)
Figure imgf000036_0005
thus, ( (
Figure imgf000036_0006
(A.27)
Generalised KR sequence generation proof:
Figure imgf000036_0007
While, f(k) satisfies the following properties:
Figure imgf000036_0003
Where | • | is the norm value of the complex number • .
From Equation (2) , the periodic correlation of a(n,k) can be written as below:
Figure imgf000036_0004
Figure imgf000037_0001
Notice that in the 2nd-term:
Figure imgf000037_0002
Both terms merge together as:
Figure imgf000037_0005
Figure imgf000037_0003
Z I
Figure imgf000037_0006
thus,
Figure imgf000037_0007
Since: where α is an L th root of unity and α ≠ 1 .
Figure imgf000037_0008
Figure imgf000037_0004
Hence, the periodic correlation function of the generalized KR orthogonal Polyphase sequences is expressed as follows:
Figure imgf000038_0001
Appendix B
I. KR-1 Sequence under Doppler Shift
If a normalized Doppler shift fd is added to the corresponding element of a KR-I sequence, According to Equation (AΛ, A.2, AA) in Appendix A, the general periodic correlation function becomes:
Figure imgf000039_0008
Figure imgf000039_0001
(B. I)
In this case of KR-I sequence, can be modified as:
Figure imgf000039_0004
Figure imgf000039_0002
Therefore, the periodic Cross-correlation function can be
Figure imgf000039_0007
written as:
Figure imgf000039_0003
Where /
Figure imgf000039_0005
And, the periodic Autocorrelation function becomes as:
Figure imgf000039_0006
Figure imgf000040_0001
Where, m
Figure imgf000040_0007
Therefore, for KR-I sequence, the magnitude of the periodic Cross- correlation function under a normalized Doppler shift fd can
Figure imgf000040_0008
be described as:
Figure imgf000040_0002
Where
Figure imgf000040_0005
Meanwhile, the magnitude of the periodic Autocorrelation function
Figure imgf000040_0003
under a normalized Doppler shift fd becomes:
Figure imgf000040_0004
Where,
Figure imgf000040_0006
II. KR-2 Sequence under Doppler Shift If a normalized Doppler shift fd is added to the corresponding element of a KR-2 sequence, According to Equation (A.2, A.8, A.19) in Appendix A, the general correlation function Cm n(τ ,fd) becomes as follows: 1) L is even
In this case, the general periodic correlation function in Eq.
Figure imgf000041_0010
(A.8, A.9) can be written as:
Figure imgf000041_0001
In case of KR-2 sequence, can be modified as:
Figure imgf000041_0006
Figure imgf000041_0002
Therefore, the periodic Cross-correlation function can be
Figure imgf000041_0009
written as:
Figure imgf000041_0003
(B.9)
Where,
Figure imgf000041_0004
And, the periodic Autocorrelation function ) becomes as:
Figure imgf000041_0008
(B.10)
Figure imgf000041_0005
Where,
Figure imgf000041_0007
2) L is odd
Similarly, if L is odd, the periodic Cross-correlation function
Figure imgf000042_0006
in Eq. (A.11, A.16) can be written as:
Figure imgf000042_0001
Or
Figure imgf000042_0007
can be modified as:
Figure imgf000042_0002
(B.12) Therefore, the periodic Cross-correlation function can be
Figure imgf000042_0005
written as:
Figure imgf000042_0003
(B.13)
Where,
Figure imgf000042_0004
And the periodic Autocorrelation function becomes as:
Figure imgf000043_0005
Figure imgf000043_0001
Where,
Figure imgf000043_0006
Hence, in conclusion, for KR-2 sequence, the magnitude of the periodic Cross-correlation function under a normalized Doppler shift
Figure imgf000043_0002
can be described jointly by:
Figure imgf000043_0003
Where, m
Figure imgf000043_0007
Meanwhile, the magnitude of the Periodic Autocorrelation Function under a normalized Doppler shift fd becomes:
Figure imgf000043_0008
Figure imgf000043_0004
Where,
Figure imgf000043_0009
III. KR-3 Sequence under Doppler Shift
If a normalized Doppler shift fd is added to the corresponding element of a KR-3 sequence, According to Equation (A.20, A.21 , A.22, A.24) in Appendix A, the general correlation function becomes as
Figure imgf000044_0005
follows:
Figure imgf000044_0004
(B.17)
Or can be modified as:
Figure imgf000044_0010
Figure imgf000044_0001
Therefore, the periodic Cross-correlation function can be
Figure imgf000044_0006
written as:
Figure imgf000044_0002
Where,
Figure imgf000044_0009
And the periodic Autocorrelation function becomes as:
Figure imgf000044_0007
Figure imgf000044_0003
Where,
Figure imgf000044_0008
Hence, in conclusion, for KR-3 sequence, the magnitude of the periodic Cross-correlation function under a normalized Doppler shift
Figure imgf000045_0001
fd can be described jointly by:
Figure imgf000045_0002
Where,
Figure imgf000045_0006
Meanwhile, the magnitude of the Periodic Autocorrelation Function
Figure imgf000045_0003
under a normalized Doppler shift fd becomes:
Figure imgf000045_0004
Where,
Figure imgf000045_0007
IV Generalised KR- sequence under Doppler shift
If a normalized Doppler shift is added to the corresponding element of
Figure imgf000045_0009
a KR orthogonal Polyphase sequence, the periodic correlation
Figure imgf000045_0008
becomes as follows:
Figure imgf000045_0005
Both terms merge together as:
Figure imgf000046_0004
Figure imgf000046_0001
Figure imgf000046_0005
(B.24)
The periodic Autocorrelation becomes as:
Figure imgf000046_0010
Figure imgf000046_0002
(B.25)
Where,
Figure imgf000046_0006
Therefore, for the generalized KR orthogonal Polyphase sequences, the magnitude of the periodic Cross-correlation under a normalized Doppler shift fd can be described as:
Figure imgf000046_0007
Figure imgf000046_0003
Where,
Figure imgf000046_0008
Meanwhile, the magnitude of the periodic Autocorrelation
Figure imgf000046_0009
under a normalized Doppler shift fd becomes:
Figure imgf000047_0001
Where,
Figure imgf000047_0002

Claims

Claims
1. A signal spreading system comprising a pseudonoise signal spreader arranged to spread a plurality of signals, the spreader arranged to spread each different signal using a different spreading sequence defined by a different value of n in a set, CC , of pseudonoise spreading sequences, each spreading sequence having a sequence length, L, wherein the set, CC is defined by:
Figure imgf000048_0001
or
Figure imgf000048_0002
or
Figure imgf000048_0003
or
Figure imgf000048_0004
where,
Figure imgf000048_0005
satisfies the following properties:
Figure imgf000049_0002
where | • | is the norm value of the complex number • ,
where
Figure imgf000049_0003
, and where
Figure imgf000049_0004
and wherein n, L, λ and k are integers.
2. The system of claim 1 comprising a sequence length calculator arranged to determine a desired value of sequence length, L, by taking into account any one or more of: the number of users; or channel state information in a communication system across which the spread signal is to be sent.
3. The system of claim 1 or claim 2 wherein the signal comprises an information signal suitable for transmission through a communications network.
4. A method of spreading a plurality of signals comprising operating on the signals using a set, CC , of pseudonoise spreading sequences, each spreading sequence having a sequence length, L, the set, CC , being defined by
Figure imgf000049_0001
or
Figure imgf000050_0001
or
Figure imgf000050_0002
or
Figure imgf000050_0003
where, satisfies the following properties:
Figure imgf000050_0004
Figure imgf000050_0005
where | • | is the norm value of the complex number • ,
where and where and wherein n, L, λ and k are
Figure imgf000050_0006
Figure imgf000050_0007
integers, wherein operating on the signals comprises operating on each different signal using a different spreading sequence from the set, CC , of spreading sequences.
5. The method of claim 4 comprising calculating the sequence length, L, by taking into account any one or more of: the number of users; or channel state information in a communication system across which the spread signal is to be sent.
6. A transmitter for a communication system arranged to transmit one or more of a group of spread signals, each different spread signal of the group having been obtained by spreading an original signal using a different spreading sequence, each spreading sequence having a sequence length, L, the different spreading sequences being from a set, CC , of pseudonoise spreading sequences, the set defined by:
Figure imgf000051_0001
or
Figure imgf000051_0002
or
Figure imgf000051_0003
or
, where
Figure imgf000051_0005
Figure imgf000051_0004
where, satisfies the following properties:
Figure imgf000051_0007
Figure imgf000051_0006
where | • | is the norm value of the complex number
where
Figure imgf000052_0004
and where
Figure imgf000052_0005
and wherein n, L, λ and k are integers.
7. A receiver for a communication system arranged to receive one or more of a group of spread signals, each different spread signal of the group having been obtained by spreading an original signal using a different spreading sequence, each spreading sequence having a sequence length, L, the different spreading sequences being from a set, CC , of pseudonoise spreading sequences, the set defined by:
Figure imgf000052_0001
or
Figure imgf000052_0002
or
Figure imgf000052_0003
or
Figure imgf000052_0006
where, f(k) satisfies the following properties:
Figure imgf000053_0001
where | • | is the norm value of the complex number • ,
where
Figure imgf000053_0004
and where and wherein n, L, λ and k are
Figure imgf000053_0005
integers.
8. The receiver of claim 7 comprising a de-spreader arranged to operate upon the received spread signals using the set, α , of spreading sequences in order to recover the original signals.
9. A group of spread signals, each different spread signal of the group having been generated by spreading an original signal using a different spreading sequence, each spreading sequence having a sequence length, L, the different spreading sequences being from a set, CC , of pseudonoise spreading sequences, the set defined by:
Figure imgf000053_0002
or
Figure imgf000053_0003
or
Figure imgf000054_0001
or
Figure imgf000054_0004
where,
Figure imgf000054_0005
satisfies the following properties:
Figure imgf000054_0006
where is the norm value of the complex number • ,
where and where and wherein n, L, λ and k are
Figure imgf000054_0008
Figure imgf000054_0007
integers.
10. A method of generating a plurality of pseudonoise sequences, each sequence having a sequence length, L, comprising generating two or more pseudonoise sequences from a set, cc , defined by:
Figure imgf000054_0002
or
Figure imgf000054_0003
or
Figure imgf000055_0001
or
Figure imgf000055_0004
where, satisfies the following properties:
Figure imgf000055_0005
Figure imgf000055_0002
where | • | is the norm value of the complex number • ,
where 0 ≤ n,k ≤ L — l , and where λ ≥ -1 , and wherein n, L, λ and k are integers.
11. A pseudonoise generator arranged to generate a plurality of pseudonoise sequences, each sequence having a sequence length, L, the plurality of sequences being from a set, CC , defined by
Figure imgf000055_0003
or
Figure imgf000056_0001
or
Figure imgf000056_0002
or
Figure imgf000056_0003
where, f(k) satisfies the following properties:
Figure imgf000056_0004
where | • | is the norm value of the complex number • ,
where and where and wherein n, L, λ and k are
Figure imgf000056_0005
Figure imgf000056_0006
integers.
12. A spread spectrum communication system comprising a signal spreader for spreading signals, a transmitter for transmitting spread signals, a receiver for receiving spread signals and a de-spreader for recovering original signals, wherein the spreader and the de-spreader use a set, cc , of pseudonoise spreading sequences to operate upon a plurality of original signals in order to produce a plurality of spread signals and to operate upon the spread signals to recover the original signals respectively, wherein the set, CC , of spreading sequences is defined by:
Figure imgf000057_0001
or
Figure imgf000057_0002
or
Figure imgf000057_0003
or
Figure imgf000057_0005
where, /(^) satisfies the following properties:
Figure imgf000057_0004
where | • | is the norm value of the complex number • ,
where
Figure imgf000057_0006
and where
Figure imgf000057_0007
, and wherein n, L, λ and k are integers.
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Non-Patent Citations (5)

* Cited by examiner, † Cited by third party
Title
BOCHE H ET AL: "Estimation of deviations between the aperiodic and periodic correlation functions of polyphase sequences in vicinity of the zero shift", SPREAD SPECTRUM TECHNIQUES AND APPLICATIONS, 2000 IEEE SIXTH INTERNATI ONAL SYMPOSIUM ON 6-8 SEPTEMBER 2000, PISCATAWAY, NJ, USA,IEEE, vol. 1, 6 September 2000 (2000-09-06), pages 283 - 287, XP010517568, ISBN: 978-0-7803-6560-5 *
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