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I want to design a filter with a custom phase delay related to frequency. As frequency increases, phase delay should increase.

The time delay as a function of frequency can be expressed as: $t_d = \frac{L}{C_{ph}} - \frac{1}{f}$

$L$ and $C_{ph}$ are constants of $0.0017$ and $2628$ respectively.

The range of $f$ is $500\mathrm{kHz}$ to $1 \mathrm{MHz}$, the sampling frequency, $f_s$ is $78.39\mathrm{MHz}$.

Thus, when the frequency is $500 \mathrm{kHz}$, the delay should be $1.37\mathrm{\mu s}$ or $107$ samples.

For the pass band, $f$, I have used a filter response of $e^{-i2\pi fd}$, where $d$ is the delay in samples required.

I have designed stop bands with a filter response of zero between $0 \mathrm{Hz}-100\mathrm{kHz}$ and $1.4 \mathrm{MHz}-f_s$.

In MATLAB my code looks like this:

n = 50; fs = double(fs); L = double(L); Cph = 2628; f1 = linspace(500e3/fs,1e6/fs,100); f1d = L/Cph - (1./(f1*fs)); %Our array is backwards though innit f1d = f1d * -1; f1dz = f1d * fs; h1 = exp(-1i*2*pi*f1.*f1dz); fstop1 = linspace(0,100e3/fs, 10); hstop1 = zeros(size(fstop1)); fstop2 = linspace(1.4e6/fs, 1, 10); hstop2 = zeros(size(fstop2)); d=fdesign.arbmagnphase('n,b,f,h', n, 3, fstop1, hstop1, f1, h1, fstop2, hstop2); %d=fdesign.arbmagnphase('n,b,f,h', n, 1, f1, h1); D = design(d,'equiripple'); fvtool(D,'Analysis','phasedelay');

This is what I get:

enter image description here

Markers are at 500k and 1M.

What am I doing wrong?

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3 Answers 3

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I notice two problems in your code:

  • at $f=500\,\text{kHz}$ you require a phase delay of $107$ samples, but your filter length is only $50$ taps; you can't have a delay that's greater than the filter length of an FIR filter.
  • it looks like you normalize all frequencies by the sampling frequency; however, in Matlab a normalized frequency $f=1$ corresponds to the Nyquist frequency, i.e., $f_s/2$, not to $f_s$.
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  • $\begingroup$ Your second point explains lots of things! Thanks man! :) $\endgroup$ Commented Jul 15, 2016 at 13:53
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I went down the ifft route. Based off this post: What effect does a delay in the time domain have in the frequency domain?

My code looks like this:

N = length(h1_data(:,1)); %N = (2^nextpow2(N)); L = h1_cord(18,1) - h1_cord(2,1); Cph = 2628; %Generate real frequency vectors (negative and positive freal = linspace(nyq, -nyq, N); fd = (L/Cph) - (1./freal); fd = fs * fd; fd = fd * -1; %fd(1) = fd(2); f = exp((-1i*2*pi*(1:N).*fd)/N); %f(N/2+1:end) = fliplr(f(1:N/2)); %f = [f fliplr(f)]; %Overwrite with freqs we don't care about fs = 1/(h1_time(2) - h1_time(1)); nyq = fs/2; bin = nyq/(N/2); f0bin = round(250e3/bin); f1bin = round(1.25e6/bin); f(1:f0bin) = 1; f(f1bin:N/2) = 1; f(N/2+1:f1bin+N/2) = 1; f(N-f0bin:end) = 1; f(1) = 1; g = real(ifft(f)); b = [g(N/2+1:end) g(1:N/2)]; b = b .* hamming(N)'; fvtool(b,1);
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  • $\begingroup$ Chris, if this answers your question, please give it the check mark when the system lets you! Thanks for taking the time to give an update. $\endgroup$ Commented Jul 15, 2016 at 15:04
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A possible alternative to designing a FIR filter constrained to both frequency and phase response relationships using IFFT methods is to design an IIR filter using FDLS (Frequency-Domain Least Squares). See: Berchin's FDLS arbitrary filter design algorithm and http://robotics.itee.uq.edu.au/~elec3004/2014/lectures/Precise%20Filter%20Design%20(chapter).pdf

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