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Complex Psudo-Noise Sequences

Started by porterboy December 3, 2004
I am trying to design a training sequence for a communications system.
I originally had a real signal and I was using a periodic M-Sequence
as the training signal (FYI an m-sequence is a special series of
{+1,-1} which has an autocorrelation function which is almost a
perfect spike: very like white noise).

However, since I am doing bandpass communication, a real signal is
double sideband, and is thus wasting half the bandwidth. In order to
double the bandwidth efficiency I want to transmit a complex signal,
in which there is no redundant information on any frequency. Basically
I want a periodic complex sequence which looks like noise. (A spike
for an autocorrelation). Does such a sequence exist?

ADVANCED QUESTION
I dont really expect an answer to this part, but sure I'll give it a
try. I normally produce a SSB signal by using a Hilbert filter to
generate an analytic signal. But using a Hilbert filter screws up the
nice autocorrelation properties of my m-sequence. Therefore, it would
be better to use a complex sequence originally and avoid the Hilbert
(hence the original question). However, the analytic signal has all
its energy in the positive frequencies, whereas a complex "m-sequence"
would have its energy distributed evenly between positive and negative
frequencies. So the question is: Is there a complex sequence which
looks like noise, AND which has all of its energy in the positive
frequencies. Answers on a postcard... Muchos Gracias
porterboy wrote:

> I am trying to design a training sequence for a communications system. > I originally had a real signal and I was using a periodic M-Sequence > as the training signal (FYI an m-sequence is a special series of > {+1,-1} which has an autocorrelation function which is almost a > perfect spike: very like white noise). > > However, since I am doing bandpass communication, a real signal is > double sideband, and is thus wasting half the bandwidth. In order to > double the bandwidth efficiency I want to transmit a complex signal, > in which there is no redundant information on any frequency. Basically > I want a periodic complex sequence which looks like noise. (A spike > for an autocorrelation). Does such a sequence exist?
Take your m-sequence, I'll assume that you can represent it as m(0), m(1), m(2) ... If it's autocorrelation is truly a spike, then both its odd samples and its even samples should be statistically independent, and each should have an autocorrelation should be a spike. So you should be able to make your complex sequence like this: mc(n) = m(2*n) + j m(2*n + 1)
> > ADVANCED QUESTION > I dont really expect an answer to this part, but sure I'll give it a > try. I normally produce a SSB signal by using a Hilbert filter to > generate an analytic signal. But using a Hilbert filter screws up the > nice autocorrelation properties of my m-sequence. Therefore, it would > be better to use a complex sequence originally and avoid the Hilbert > (hence the original question). However, the analytic signal has all > its energy in the positive frequencies, whereas a complex "m-sequence" > would have its energy distributed evenly between positive and negative > frequencies. So the question is: Is there a complex sequence which > looks like noise, AND which has all of its energy in the positive > frequencies. Answers on a postcard... Muchos Gracias
You can't put all of the energy into the positive frequencies without placing constraints on the time sequence, which in turn places constraints on the autocorrelation function. So if you consider "like noise" to be "white noise" then no, you can't. You _can_ make a pseudo-noise sequence, with a non-impulsive autocorrelation function that's flat in the positive frequencies. Once you run this through your SSB filters, you will see the same results as you would with your "purely random" noise. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
On Fri, 03 Dec 2004 07:54:40 -0800, Tim Wescott
<tim@wescottnospamdesign.com> wrote:

>porterboy wrote: > >> I am trying to design a training sequence for a communications system. >> I originally had a real signal and I was using a periodic M-Sequence >> as the training signal (FYI an m-sequence is a special series of >> {+1,-1} which has an autocorrelation function which is almost a >> perfect spike: very like white noise). >> >> However, since I am doing bandpass communication, a real signal is >> double sideband, and is thus wasting half the bandwidth. In order to >> double the bandwidth efficiency I want to transmit a complex signal, >> in which there is no redundant information on any frequency. Basically >> I want a periodic complex sequence which looks like noise. (A spike >> for an autocorrelation). Does such a sequence exist? > >Take your m-sequence, I'll assume that you can represent it as m(0), >m(1), m(2) ... If it's autocorrelation is truly a spike, then both its >odd samples and its even samples should be statistically independent, >and each should have an autocorrelation should be a spike. > >So you should be able to make your complex sequence like this: > >mc(n) = m(2*n) + j m(2*n + 1) >> >> ADVANCED QUESTION >> I dont really expect an answer to this part, but sure I'll give it a >> try. I normally produce a SSB signal by using a Hilbert filter to >> generate an analytic signal. But using a Hilbert filter screws up the >> nice autocorrelation properties of my m-sequence. Therefore, it would >> be better to use a complex sequence originally and avoid the Hilbert >> (hence the original question). However, the analytic signal has all >> its energy in the positive frequencies, whereas a complex "m-sequence" >> would have its energy distributed evenly between positive and negative >> frequencies. So the question is: Is there a complex sequence which >> looks like noise, AND which has all of its energy in the positive >> frequencies. Answers on a postcard... Muchos Gracias > >You can't put all of the energy into the positive frequencies without >placing constraints on the time sequence, which in turn places >constraints on the autocorrelation function. So if you consider "like >noise" to be "white noise" then no, you can't. > >You _can_ make a pseudo-noise sequence, with a non-impulsive >autocorrelation function that's flat in the positive frequencies. Once >you run this through your SSB filters, you will see the same results as >you would with your "purely random" noise.
I'll second Tim's ideas, plus add some. First, why do you want energy in only the positive frequencies? Did you mean when it's bandpass or at baseband? Preamble design for just this sort of thing has been around for a long time. Depending on the constraints of your system there may be some suitable preambles around that already exist, and they generally are pretty good with the properties that you're looking for. Another way to generate the sequence would be to take a PN sequence or an m sequence with good autocorrelation and modulate it as a QPSK signal. This puts energy in both I and Q channels and will still have good autocorrelation properties. It's essentially the same thing that Tim suggested, only you can play with the mappings, make it differential, etc. Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions. http://www.ericjacobsen.org
"porterboy" <porterboy76@yahoo.com> wrote in message 
news:c4b57fd0.0412030249.4ddf0e13@posting.google.com...
> However, since I am doing bandpass communication, a real signal is > double sideband, and is thus wasting half the bandwidth. In order to > double the bandwidth efficiency I want to transmit a complex signal, > in which there is no redundant information on any frequency. Basically > I want a periodic complex sequence which looks like noise. (A spike > for an autocorrelation). Does such a sequence exist?
Yes, there are many constructions that have been discovered in the past 20 years. Some of the papers are not easy to follow. You might find what you need in an earlier paper: S. M. Krone et al. "Quadriphase sequences for spread-spectrum multiple-access communication," IEEE Transactions on Information Theory, vol. 30, May 1984.
> ADVANCED QUESTION > Is there a complex sequence which > looks like noise, AND which has all of its energy in the positive > frequencies.
If the autocorrelation function is a "spike", then the power spectral density (a.k.a. Fourier transform of autocorrelation) is "flat" and broadband .
"Tim Wescott" <tim@wescottnospamdesign.com> wrote in message 
news:10r1320kdck6772@corp.supernews.com...
> Take your m-sequence, I'll assume that you can represent it as m(0), m(1), > m(2) ... If it's autocorrelation is truly a spike, then both its odd > samples and its even samples should be statistically independent, and each > should have an autocorrelation should be a spike. > > So you should be able to make your complex sequence like this: > > mc(n) = m(2*n) + j m(2*n + 1)
The sequence whose n-th element is m(2*n) is just a cyclic shift of the sequence m(n). For example, here are two periods of an m-sequence of period 7. 11101001110100 Let us number the bits as 0 through 13 from left to right. Then the even-numbered bits are 1 1 1 0 1 0 0 while the odd numbered bits are 1 0 0 1 1 1 0 which is the same m-sequence shifted over. The real part of the complex autocorrelation is the sum of the autocorrelations of the I and Q sequences while the imaginary part is the sum of the crosscorrelation functions of the I and Q sequences. If the I and Q sequences are shifted versions of one another, then this crosscorrelation is actually a shifted autocorrelation, and thus there is a spike in the imaginary part of the complex autocorrelation function at the appropriate (nonzero) time delay. Unless one resolutely ignores the imaginary part, one cannot pretend that the complex autocorrelation function is a spike.