PRBS injection frequency.
I'm looking at methods of system identification using an injected Psuedo Random Binary Sequence. There are a few sources and I think this one is available. [ http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.2303&rep=rep1&type=pdf ] My problem comes in how the minimum injected frequency is calculated. The PRBS number of bits is given as N. The PRBS length is given as 2^n-1. There is an injection frequency of f.
It makes sense that the maximum frequency would be equal to your injection frequency when you have a sequence something like 010. Intuitively, I would think the minimum frequency would occur when the sequence was at maximum length N. Fmin = F/N But all of the literature says the minimum frequency is Fmin/M
Example to clear up my question.
F=10kHz
N=5
M=31
Fmax=10kHz
I would think Fmin=10kHz/5=2kHz
But it is actually 322Hz.
I would really appreciate it if someone could clear up this discrepancy. Thanks.
Posted Mar 25, 2013, 3:20 AM UTC
Many psuedo-random sequences actually emit energy all the way to DC. The typical LFSR as shown in the linked paper, for example, will always have an imbalance that guarantees a slight DC bias.
The question isn't so much "What's the minimum frequency?" as "What's the minimum frequency resolution?" Any repeating sequence, by definition, has a discrete spectrum. (Any sequence that repeats has spectral energy concentrated at discrete frequencies, spaced at F/M, where M = 2^(n)-1 is the repetition period.) For the linear system being identified, there is no information whatsoever about the frequency response in between those discrete frequencies. This is a reasonable approximation only if the response can reasonably be approximated by sampling in this fashion. F_min refers not to the minimum frequency of interest, but the minimum sampling resolution of the frequency response that will guarantee an adequate representation.
Ok let me see if I understand. (Forgive me DSP is not my thing) When you say a DC bias, you mean that there will always be some more number of 1s than 0s (like on differential pairs). Ideally this number approaches, but never reaches zero. Is it this slight DC bias that gives rise to the low end frequencies? I still can't quite visualize why the resolution breaks down to F/M. Thanks for the help so far.
You are correct. As an example: A maximum-length sequence with 10 bits repeats every 1023 bits, an odd number. Therefore there will always be 512 1's and 511 0's or vice-versa.
This imbalance means that there will always be some energy precisely at DC.
It may help to contrast a repeating psuedo-random sequence true-random, nonrepeating bit sequence. In the nonrepeating case, we know that the average over a long enough interval will be a random variable centered on zero. Even a completely fair coin isn't likely to get exactly the same outcomes after 1000 flips, or even 1000000 flips, but it'll approach equality. In another way of viewing the same phenomenon, you can imagine averaging every 1000 bits. This forms a sort of low pass filter with a cutoff around F/1000, filtering the "noise" of the original bit sequence. When you average more bits, your cutoff frequency approaches zero, but there's always going to be some tiny bandwidth that lets through a tiny bit of noise.
The repeating sequence doesn't have this property. If it repeats every 1000, averaging 1000 bits gets you the same result as averaging 10000 or 1000000000 bits. In other words, there's a sliver of energy exactly at DC, but no additional energy in the spectrum anywhere up to F/1000 (the cutoff of the original filter).
If you run the repeating sequence through a Fourier transform, you can prove that there's a discrete frequency component at 0F/1000, 1F/1000, 2*F/2000, and so on. This is somewhat analogous to an average that multiplies the samples by a sinusoid or a square wave of increasing frequencies. For a period of 24, it looks like this:
... and so on.