[sdiy] Non maximal-length LFSR

[Richie Burnett]

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