[music-dsp] Unitary power condition for FIR filter coefficients

Charles Henry czhenry at gmail.com
Tue Jan 29 12:05:41 EST 2008

```> Does a simple and general condition on the FIR coefficients exist so that
> feeding the filter with a unit variance white noise a unit variance output
> signal is obtained? Or is it necessary to evaluate the filter spectral power
> explicitly?

All-pass filters will do this, I think.

There is a property of convolution operators, in general (proofs
available on request), and their norms in L1 and L2

the L1 norm |f|.1 = integral( -inf, inf, |f|dt)
| conv(f,g) |.1  = |f|.1 * |g|.1

the L2 norm squared |f|^2 = integral( -inf, inf, f^2dt)
| conv(f,g) |^2 <= |f|^2*|g|^2

When you're talking about variance, that's an L2 norm squared (where
the signal mean is 0).  So, there's an inequality.  The total energy
remains constant, only if the filter has unitary gain for all
frequencies (i.e. an all-pass filter).

Chuck

On Jan 29, 2008 10:38 AM, Massimiliano Tonelli <mtonelli at anwida.com> wrote:
> Hello everyone,
>
> Does a simple and general condition on the FIR coefficients exist so that
> feeding the filter with a unit variance white noise a unit variance output
> signal is obtained? Or is it necessary to evaluate the filter spectral power
> explicitly?
>
> Thanks
> Massimiliano Tonelli
>
>
>
>
>
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