IE20040150U1 - Method and device for designing and communicating with orthogonal codes with good autocorrelation properties - Google Patents

Abstract

ABSTRACT The invention describes a method and device for designig a set of orthogonal codewords, such a set is known as an orthogonal code, which have good properties for use in communication systems. These codes are useful because they provide advantages to a communication system which uses them in that they result in lower error rates in the receiver. The method for designing the codes is to take a good orthogonal code, i.e. a good set of orthogonal codewords, and manipulate it so as to transform it into a better code, e. g. a code with higher mean Golay merit factor.

Description

METHOD AND DEVICE FOR DESIGNING AND COMMUNICATING WITH ORTHOGONAL CODES WITH GOOD AUTO CORRELATION PROPERTIES.

In the following description, certain specific details of the disclosed embodiment such as architecture, example codes, interfaces and techniques, etc, are set forth for purposes of explanation rather than limitation, so as to provide a clear and thorough understanding of the present invention. However, it should be understood readily by those skilled in this art, that the present invention may be practiced in other embodiments which do not conform exactly to the details set forth herein, without departing significantly from the spirit and scope of this disclosure. Further, in this context, and for the proposes of brevity and clarity, detailed descriptions of well-known apparatus, circuits and methodology have been omitted so as to avoid unnecessary detail and possible confusion.

BACKGROUND OF THE INVENTION Orthogonal codes have many uses. Various types of electronic communications schemes use orthogonal codes e. g. M—ary bi-orthogonal keying is just one of many that rely on orthogonal or nearly orthogonal codes.

What is an orthogonal code‘? The following text provides a definition: Say b is a vector ofn numbers. (bo,b1,b2, . . b,,.1) Say c is a vector of n numbers. (co,c1,c2. . . c,..,) b and c are said to be orthogonal if and only if the inner product of b and c is zero i.e. b and c are orthogonal ilf b'c = b0c0+b1c.+....b,,.,c,,_1 = 0 Say H is an in x n matrix such that each row of the matrix is a vector of length n. Each of these vectors is known as a codeword. The matrix H of codewords is known as a code.

H is an orthogonal code if and only if H - HT: ml Where ' denotes matrix multiplication (inner product), m is a scalar and I is the identity matrix.

In the example ofM—ary orthogonal signalling a transmitter sends information by sending one of M orthogonal signals. In M-ary biorthogonal signalling the transmitter sends one of M signals drawn from an alphabet of M/2 orthogonal signals and the negatives of each of these signals. The receiver then has to distinguish which signal was sent. In many cases the receiver does this by correlating the received signal with each of the M/?. orthogonal signals and selecting the one which gave the highest correlation.

If the signals have not been distorted by the channel, the value of this correlation will be positive for the correct signal. For the negative of the correct signal, the correlation will be negative and for all the other signals, the correlation will be zero. This latter is because all the other signals are orthogonal to the correct one and hence have zero correlation with them. Because of these correlation properties the correct signal can be detected. An orthogonal code may be used as the orthogonal signals. Signals with amplitudes corresponding to the vectors elements are used as the orthogonal signals. This is one reason that orthogonal codes are useful. it is advantageous if the autocorrelation properties of each of the orthogonal signals are good. A merit factor which is often used to measure how good the auto correlation function is, is the Golay Merit Factor.

This measures how strong the correlation is for a correctly synchronised code compared to how strong the correlation is for an unsynchronised codeword. In other words, how strong the peak in its autocorrelation function is, compared to the side lobes.

The Golay merit factor is defined as the square of the synchronised correlation of the code with itself to the sum of the squares of all the non—synchronised correlations. The synchronised correlation of the code is the inner product of the code with itself. The non-synchronised correlation with delay n is the inner product of following two vectors. The first vector is the code with n, a positive integer number, zeros concatenated to one end of the code. The second vector is the code with the same number of zeros concatenated to the other end.

The length 13 Barker sequence has the highest known Golay merit factor of any binary sequence This factor is 14.083. By comparison, probably the most well known orthogonal codes are formed by Walsh- Hadamard matrix. The mean Golay merit factor of the length 32 Walsh-Hadamard matrix is 0. I94.

The invention here describes a new technique which can generate orthogonal codes with relatively large Golay Merit Factors. [1] "Sieves for Low Autocorrelation Binary Sequences", IEEE Trans. Info. Th., vol. IT -23, pp. 43-51, Jan. l977, M. J. E. Golay.

DETAILED DESCRIPTION OF THE INVENTION One aspect of the invention is to take a Hadamard matrix and produce an orthogonal code with a better mean Go lay Merit factor than the code formed by the original Hadarnard matrix.

It is well known that a code constructed with a lower GMF ifor a communication system will give better receiver performance. Communication systems which use codes constructed in this way have better error perfomiance and are more immune to noise and to Inter Symbol Interference.

The following procedure may be used to produce to this: Step 1: Take any Hadamard Matrix H with m codewords of length n, this means that H is composed +1 and —l and also that H ‘ HT = nil ggnlantioc Let row i of H be denoted H; Then it is easy to show that (Hi+HJ-) is orthogonal to (H,+H,,,) if i,_j,l and m are all difi"erent. Also, (Hi+HJ-) is a ternary sequence, i.e. it is composed of only three levels. It is also easy to show that (Hi—H,-) is orthogonal to both (H5+Hj) and (H.+H,,,) and that (Hi-H,-) is a ternary sequence.

Step 2: Measure the Golay Merit Factor(GMF) of each codeword (i.e row) in H and calculate the mean of these. This is saved as the best GMF sofar Step 3: Now take the codewords in H and randomly group them into m/2 pairs.

Step 4: Measure the Golay Merit F actor(GMF) of each codeword (i.e row) in K and calculate the mean of these.

Step 5: If this new mean GMF is better than the original best, use it as the best GMF so far and continue to Step 6, otherwise go back to Step 3.

Step 6: Remember this grouping of pairs.

Step 7: Randomly switch a couple of codeword pairings, in other words select two of the m codewords at random and switch places.

Step 8: Form a new matrix, K, the rows of which are the sums and dilferences of each of the pairs. All the rows of the matrix K are orthogonal to each other and all are ternary rows i.e. composed of 3 values. +2, -2 and 0. So a ternary code with m elements of length n may be formed from K in this way.

Step 9: Measure the Golay Merit factor of each codeword in K. and calculate the mean GMF.

Step 10: If this mean is better than the best GMF sofar, remember the value and the groupings of pairs of codewords otherwise revert to the pairings and best GMF that were in force at Step 6.

Step 1 1: Increment a counter and if less than a predetermined number of loops have been made, go back to step 4.

Stop: The matrix K should now have a better mean GMF than the matrix H had.

As might be expected, it has been found that if the techniques starts ofi‘ with a good GMF in H, then a good GMF results for K. Quite a few variations on this technique are possible to change the outcome, for example, the number of random pair switches made in step 7 can be varied. A limit on the number of trials can be imposed in step 5.

In order to start with a good H a number of approaches are possible. One possibility is to take a known Hadamard matrix and find the inner product of it and some other matrix Q. If Q is zero everywhere except along the diagonal and the diagonal consists only of +1 or -1 , then if H is a Hadamard matrix, H’ = H - Q is also a Hadamard matrix. Its mean GMF is not the same as H. It is possible then to search all possible Q to find the best possible Q for a given H. There are only 2n possible values ofQ. e.g. Using one example value of Q as the diagonal matrix with the following as the diagonal transfoims the mean golay merit factor of the length 32 Walsh Hadamard matrix from 0.194 to 3.

Note: in this case where + denotes a value of +1 and ~ denotes a value of -1 This diagonal results in the same improvement of these codewords have a Golay merit factor of 3.56 and the other have a Golay merit factor of 2.46.

Using one of these as a starting point and using the steps explained above, the following sets of 32 x 32 ternary codewords or codes were obtained using the steps above First set For all of these codes (Le, sets of codewords), the average Golay merit factor is 4.26 and each set consists of 16 codewords with the Golay merit factor of 5.333 and 16 with a Golay merit factor of 3.2.

The codewords have slightly different characteristics, for example the first and third set have no more than 2 consecutive zeros, whereas the second set has better cross correlation properties.

One characteristic of the method is that it always produces ternary codes where half of all the codewords are zero valued. This can be seen to be the case in the above examples. This is advantageous in the receiver because when the codeword value is +1, an addition of the corresponding receiver sample is usually required to do the correlation. When the codeword value is -1, a subtraction of the corresponding receiver sample is usually required to do the correlation, but when the codeword value is zero, no operation is required for the corresponding receive sample.

A communications system using these codes or codes designed by this method e. g. a transmitter—receiver pair communicating using m-ary biorthogonal coding and using these codes to transmit the information will perform better in a practical channel with a non-ideal impulse response than one using codes with a lower Golay merit factor.

An example helps to illustrate why this is the case. Take the case where one of two signals cwl (t) or cw2(t) are sent through a channel with an impulse response tit).

Assuming the delay through the channel is known, it is well known by those skilled in the art, that a good way of deciding in the receiver, which of the two possible signals was sent, is to correlate the received signal with the impulse response of the channel and then to correlate the result with each of the two possible transmit signals, to sample the result at zero phase given the known delay through the channel and the correlators, and then to choose the highest valued result.

If the noise injected by the channel or transmission medium is ignored, then even if correlation operations and convolution of the channel are done in a different order , the same result is obtained. This means that the |E040150 result is the same as if the transmit signal was correlated with cwl (t) (and cw2(t)), the result was passed through a channel consisting of the autocorrelation function of the actual channel, the central sample of the result was taken as the final result and the larger of cwl or cw2 was chosen.

The charts in Fig.1 to Fig 8 below shown some examples of this.

Claims (16)

1.CLAIMS 202 1. A method of communication wherein an orthogonal or nearly orthogonal code is used to transmit data from the transmitter to the receiver and where the code is designed with an algorithm which takes a starting code, measures some desirable quality of the code and produces a new code from the linear combination of the codewords in the starting code, uses this as a new starting code and continues to improve the code in this way. 203 2. A method of communication of claim 1 where the desirable quality being improved is the mean Golay

2.Merit Factor. 204

3. A method of communication of claim 1 where the desirable quality being improved is the mean cross- correlation of the codewords. 205

4. A method of communication of claim 1 where the desirable quality being improved is the maximum cross- correlation of the codewords. 206

5. A method of communication of claim 1 where the desirable quality being improved is the maximum peak to mean ratio of the spectrum of all of the codewords. 207

6. A method of communication of claim 1 where the code used is the first set of ternary codewords given in the description above 208

7. A method of communication of claim 1 where the code used is the second set of ternary codewords given in the description above 209

8. A method of communication of claim 1 where the code used is the third set of ternary codewords given in the description above 210

9. A device for communication wherein an orthogonal or nearly orthogonal code is used to transmit data from the transmitter to the receiver and where the code is designed with an algorithm which takes a starting code, measures some desirable quality of the code and produces a new code from the linear combination of the codewords in the starting code, uses this as a new starting code and continues to improve the code in this way. 21]

10. A device for communication of claim 9 where the desirable quality being improved is the mean Gclay Merit Factor. 212

11. A device for communication of claim 9 where the desirable quality being improved is the mean cross- correlation of the codewords. 213

12. A device for communication of claim 9 where the desirable quality being improved is the maximum cross- correlation of the codewords. 214

13. A device for communication of claim 9 where the desirable quality being improved is the maximum peak to mean ratio of the spectrum of all of the codewords. 215

14. A device for communication of claim 9 where the code used is the first set oftemary codewords given in the description above IE040150 216

15. A device for communication of claim 9 where the code used is the second set of ternary codewords given in the description above 217

16. A device for communication of claim 9 where the code used is the third set of ternary codewords given in the description above