[music-dsp] Throw Moi from the Pulse Train

[Nigel Redmon]

I need to turn in--I'll have to read your post in its entirety 
tomorrow, but thanks for your reply. A couple of quick comments:

On Mar 6, 2005, at 9:48 PM, robert bristow-johnson wrote:
>> Nigel Redmon wrote:
>>
>>> First, there is nothing between the samples. The samples (the values 
>>> in
>>> memory) are all we have. No sinc, no zeros, nothing. There doesn't 
>>> have
>>> to be anything.
>
> maybe i agree here with Nigel.  but saying that there is "No sinc, no 
> zeros,
> nothing" is not the same as saying there are zeros in between.

I think you just misread this. When I said there are "no zeros" between 
the samples, indeed it "is not the same as saying there are zeros in 
between"--explicity. If what you're getting at is that I later talk of 
zeros, note that I justify insertion of zeros not because they exist 
between the samples, but between the pulses in the pulse amplitude 
modulated signal that the samples represent. (I'm not saying that the 
samples must represent a PAM signal, but they can, and I'm saying that 
they do in fact in my model. I also said that the value doesn't have to 
be zero--there can be any DC offset in the PAM signal--but we choose 
zero for practical reasons.)

> when Nigel says he's putting zeros in between the samples x[n] to
> reconstruct a continuous-time domain function x(t), i think at some 
> point
> down the road he's running that into a low-pass filter.  now if i were 
> doing
> this, i wouldn't bother to insert zeros and run that zero-inserted 
> upsampled
> sequence through an LPF, i would recognize that those zeros will 
> contribute
> zero times the impulse response they multiply in the convolution, and 
> what's
> left is a sinc-like function times all the other original non-zero 
> samples.

Right. I explained that once you add the zeros to double the sample 
rate, you've just double the bandwidth but not changed the frequency 
spectrum. That means that mirror image that would normally lie between 
Nyquist and the sample rate (pi to 2pi) is now between half Nyquist and 
Nyquist--the upper half of the widened audio band. We run a low pass 
filter at this point (usually, a windowed sinc FIR, especially if we 
want phase linearity). I also noted that we don't actually mess with 
inserting the zeros and multiplying--but that's one of the big reasons 
we chose zero and not some other constant in the first place.

> i'll bet it's clear as mud now.  i think i regret getting into this.

I'm sure we all do ;-) I think it will be a lot easier if we stick to 
the pulse modulation view of sampling is valid or not, because the 
upsampling and zero stuffing business doesn't mean squat if Bob doesn't 
buy the view of sampling.