[sdiy] Digital filtering question

[cheater cheater]

Hi Rich,
thanks for a cool post.

On 23/08/2010, Richie Burnett <rburnett at richieburnett.co.uk> wrote:
> Hi Tom,
>
>> All the textbooks tell me that the ideal digital lowpass filter has a
>> filter kernel that is the Sinc function, Sin(x)/x.
>> This function needs windowing for practical use since it continues to
>> infinity in both directions, but that's just an implementation detail!
>
> This is a bit misleading.  The ideal brickwall lowpass filter in the
> frequency domain is equivalent to a sinc function that extends to plus and
> minus infinity in the time domain.  This is of academic significance only
> since practical systems need to give results in finite time.

Only finite-precision systems. Arbitrary-precision systems (especially
for offline editing, which is VERY neglected in digital audio for no
apparent reason) can extend their precision down to the LSb, beyond
this precision any further input data in a monotonically convergent
system is inexistent for all intents and purposes.

> Once you
> truncate and window the sinc function it is no longer the optimal solution
> to get close to an ideal filter with a finite number of coefficients.  There
> are better FIR filter kernels with the same number of sample points.
>
> Firstly, it is more helpful to think of the the sinc function as:
> sin(pi*x*n) / (pi*x*n)
>
> Where n is the sample number and x is the "design parameter" that is used to
> dilate the sinc function in the time domain and control the cutoff frequency
> in the frequency domain.
>
>> My question is: What's the cutoff frequency?
>
> The cutoff frequency is equal to the Nyquist frequency of the system (Fs/2)
> divided by the number of samples between zero crossings in the sinc
> function.   The number of samples between the zero crossings in the sinc
> function is equal to 1/x in the above equation.

Between which zero crossings? The sin(x)/x function has infinitely many.

> Or to put it another way, if you want your filter to cutoff at a tenth of
> the system's Nyquist frequency then you need to set x to 0.1 in the above
> equation, and it will result in a discrete sinc function with ten sample
> points between each zero crossing.
>
> All this assumes that the sinc function exists for infinite time in both
> directions.  Truncation and windowing of the sinc function in the time
> domain both move the frequency domain behaviour away from the ideal
> brickwall lowpass response.  Passband droop, ripples in the stopband and a
> wide gradual transition region are the main symptoms of truncation and
> windowing.
>
> If you know the filter design requirements and the number of taps then
> iterative techniques like Remez exchange (Parks McClellan) will give a more
> optimal FIR filter design.

Very interesting. Have you got any good papers or book recommendations on that?

Thanks,
D.