How to apply a Farrow structure delay filter(FIR) to a signal in Matlab?

For a signal, which is very common, just like this one

t=linspace(0,100e-6,1024);
k=10e13;
y=exp(-1i*k*pi*t.^2);

The function in the signal processing toolbox of MATLAB is used to generate a delay filter with Farrow structure, and the filter coefficients are obtained.

L=18;
d = fdesign.fracdelay(abs(Delay),'N',L);
farrow = design(d, 'lagrange', 'FilterStructure', 'farrowfd');
farrow_coff=farrow.Coefficients;

Is it because the previous delay filter is not correct, the function filter can not correctly filter to obtain the desired results?

Or please give me a simulation example for reference.

Answer 1

[Star Strider]

I cannot run your code because:

Unrecognized function or variable 'Delay'.

and ‘L’ is missing as well.

However, if you need to do phase-neutral digital filtering, use the filtfilt function instead of filter.

[chen jiyuan]

Thank you very much for your answer. What I want to use this code is in wideband digital array beamforming. In this processing, each channel have to delay (fraction delay) to sum the final beam.

The flow chat is shown here.

The code is here

clear all;close all;
%% Paras Set
f0=15e9; 
B=2e9;
tau=100e-6; %pulse duration 
k=B/tau;
fmax=f0+B;
fmin=f0;
M=64;%number of Array 
fs=1e6;

N_r=tau*fs;%sampling points

R=15e2+10;%target distance
R_ref=15e2;%referee distance

c=3e8;
lambda0=c/fmax;
d=0.5*lambda0;% array space

theta_s=10;
theta_num=1024;
theta=linspace(-90,90,theta_num);

%% Echo signal and reference signal
%First, the array elements are calculated, and then the whole array is synthesized
t=linspace(0,tau,N_r);
f=linspace(fmin,fmax,N_r);
delay_r=2*R/c;
delay_ref=2*R_ref/c;
s_ref=exp(1i*2*pi*fmin*(t-delay_ref)+1i*2*pi*k*(t-delay_ref).^2); %refence signal

s_sum=0;
for m=1:M
    
    for i=1:theta_num
        delay_m=(m-1)*d*sind(theta_s)/c;
        delay=delay_r+delay_m;%Echo signal delay consists of two parts
        delay_com=(m-1)*d*sind(theta(i))/c; 
%         h_delay=dirac(t-(m-1)*d*sind(theta_s)/c);
         s_r=exp(1i*2*pi*fmin*(t-delay)+1i*2*pi*k*(t-delay).^2);  %Radar echo signal
         s_array=s_r.*conj(s_ref);%Mixing operation
         for j=1:N_r
            %%%%Phase compensation factor            
            s_com(j)=exp(1i*pi*(2*k*delay_com*t(j)+2*fmin*delay_com-k*delay_com.^2-2*k*delay_com*delay_ref));         
         end
        s(i,:)=s_array.*s_com;%compensate the phase
        
        % how to achieve time delay for each array?? dose not work

%         farrow_coff=Farrow_Coff(delay_com,4);
%         s(i,:)=fftfilt(s(i,:),farrow_coff);
%         s(i,:)=Farrow_Delay(s(i,:),farrow_coff);

    end
    s_sum=s_sum+s;%Synthesize the whole array
end

%%

figure
% plot(theta,s_sum(:,N_r));
mesh(theta,f,abs(s_sum.'));
% view(2);

figure
plot(theta,abs(s_sum(:,1)));hold on
plot(theta,abs(s_sum(:,N_r)));
plot(theta,abs(s_sum(:,N_r/2)));

I cannot realize how to delay each channel, I hope to get your help, thank you.

After delay compensation, all channels have the same delay, that is, the array has the same beam pointing at all frequencies.

[Star Strider]

I initially thought that this was a straightforward signal processing problem. For that, I refer you to Fractional Delay Filters Using Farrow Structures however I do not have the DSP System Toolbox so I cannot help with the implementation.

I understand how these could be useful for beamforming with antennas or acoustic systems, however my experience with antenna deisgn is minimal at best, and limited to antennas for the amateur radio shortwave bands (3 - 30 MHz) part of the spectrum. I have no experience with acoustic applications.

An open-access article discussing this is Robust design of Farrow-structure-based steerable broadband beamformers with sparse tap weights via convex optimization . Others are behind paywalls or other bostructions, so I will not refer to them here.

[chen jiyuan]

thank you

[Star Strider]

As always, my pleasure!