[sdiy] Non maximal-length LFSR

Tom Wiltshire tom at electricdruid.net
Thu Mar 3 11:54:25 CET 2016

I've done some experiments with simulations of non-maximal length sequences for the same reasons you're considering it. "Not worth it" was my conclusion.

The trouble is that for non-maximal length sequences, you break the output up into several cycles instead of one. So yes, starting value is key, because that decides which one of the non-maxmal series you'll get. Also you no longer get *all* the values output, so you can't guarantee that you'll get a unbiased result, although in practice for long LFSRs, it seems like you get something near enough for audio. If you do the experiments with short LFSRs (like 5 or 6 bits) you can find combinations which do give a serious bias, but it gets harder as the numbers go up.

I agree with Ove. Use the 47-bit register and ignore the top bit. It'll give you a better result than a degenerate 48-bit register would anyway.


On 3 Mar 2016, at 08:16, Richie Burnett <rburnett at richieburnett.co.uk> wrote:

> Hi guys,
> I'm wondering if one of the maths gurus here can help me out with something...
> I want to make a white noise generator using a Linear Feedback Shift Register.  However, rather than using one of these (see link below) polynomials to choose the feedback tap locations to achieve the maximum possible length sequence, I would like to choose a sub-optimal polynomial for reasons of computational efficiency on the chosen implementation platform.  My question is then, how long will the resulting pseudo-random bit sequence be?
> For example there are maximum-length polynomials shown for shift-register length 47 and length 49 that contain just two taps.  That's great because they're cheap to implement, but the maximum-length polynomial for shift-register length 48 contains four taps and is more expensive to implement.  In-short I want to use a shift-register of length 48 bits, but with just 2 taps, and accept the fact that I won't get a maximum-length sequence of (2^48)-1 bits out of it, but it will hopefully still be quite long and adequate for white noise synthesis purposes.
> In response to "why don't you just try it and find out"  I've coded up my noise generator, and I can't hear any repetition, but obviously my ear probably isn't going to detect the repeats if they're more than a few seconds apart.  I've also looked at the average power spectrum of 3 mins of noise, versus the average power spectrum of 6 mins, and the second spectrum is much flatter suggesting to me that the sequence isn't simply repeating over in the additional 3 mins.  My problem is getting some handle on the sequence length but I suspect it is going to get complicated, and possibly even depend on the initial conditions of the shift-register!?  I can't even get an answer in a reasonable time using simulation or autocorrelation because even though the polynomial doesn't result in a sequence that's maximum-length, it's probably still very long?
> Any help or pointers would be much appreciated.
> Many thanks,
> -Richie,
> * http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf 
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