Re: Effect of Nonlinear Group Delay on Signals
- From: robert bristow-johnson <rbj@xxxxxxxxxxxxxxxxxxxx>
- Date: Thu, 3 Sep 2009 21:43:41 -0700 (PDT)
On Sep 3, 10:54 pm, "JM1970" <ra...@xxxxxxxxxxxxx> wrote:
I am trying to model the effect of group delay of an analog filter with
complex transfer function H(f) on a phase modulated signal s(t), with a
complex spectrum S(f). The analog filter I would like employ is a Chebyshev
n=4, with 0.1dB ripple, but the exact filter is not important right now..
In continuous time, the answer seems simple: The filtered signal is just
the inverse fourier transform of the product of S(f) and H(f).
The concept for my discrete time model is the same, but I am unsure if it
volates any laws of DSP. If I take N samples of s(t)at a rate fs, then
perform an FFT of length Nfft=N,it is sampled in the frequency domain at
intervals nfs/N, where n=0:N-1. If I sample the complex filter transfer
function at these intervals, I get phasors H(f) that I can directly
multiply the spectrum S(f)by to get the filtered spectrum SF(f), where
f=n*fs/N. To get the filtered singal samples sf(nTs), perform the IFFT of
SF(f).
all this is fine, except that you will be performing circular
convolution with your FFT, multiply, IFFT process. you probably need
to use either "overlap-add" or "overlap-save".
If there is a way to create a digital Chebyshev filter with the same
magnigude and phase characteristics as an analog Chebyshev filter (or
approximately), then it would seem possible to perform a fast convolution
of the sampled signal, and the impulse response of the filter. Is there a
way to design a digital Chebyshev filter with phase equivalent to an analog
counterpart?
here is the deal with digital emulations of analog counterparts:
analog filters are made up of adders (or subtractors) of signals,
scalers (multiplication by constant) of signals, and integrators
(w.r.t. time) of signals. integration is the same as multiplying by 1/
s.
digital filters are made up of adders (or subtractors) of signals,
scalers (multiplication by constant) of signals, and delay elements
(shifting w.r.t. time). delay is the same as multiplying by 1/z = e^(-
sT).
the problem is that we cannot represent 1/s exactly. it would be 1/s
= -T/log(1/z). (T = 1/Fs) so wherever you see an integrator, just
replace it with -T/log(1/z). but we don't have anything that performs
log(1/z), so we approximate
1/s = T/2 * (1 + 1/z)/(1 - 1/z)
so, wherever we see an integrator in our analog filter, we replace it
with the expression above which is in terms of delays. the magnitude
and phase characteristics are the same as the analog, except they
happen at slightly different frequencies.
r b-j
.
- References:
- Effect of Nonlinear Group Delay on Signals
- From: JM1970
- Effect of Nonlinear Group Delay on Signals
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