how I can use the FFT data to compute the convolution?

Question

First,I have a vector A of length N,through process

A1 = fft(A,N);
A2 = Filter(A1);
A3 = ifft(A2,N);

Now,I need compute the convolution of vector A2 with vector B,vector B is another vector with N elements，I noticed that seemed that I could use

ifft(fft(A3,2N).*fft(B,2N)) 

to calculate convolution. fft(A3,2N) means padding A3 with N zeros.

In fact, A2 = fft(A3,N)

So, the problem is what can I do to avoid calculate A3 and fft(A3) by using A2?

Or maybe the problem is how can I use A2 to calculate fft(A3,2N)?I wan to know if I can reduce some calculations in the whole process.thanks.
Here is a matlab test

clear;
N = 4;
A = 1:4;
B = 3:6;
A1 = fft(A);
A2 = (A1).*2020+1234;%filter(no sense)
A3 = ifft(A2);
A4 = fft(A3,2*N);
B2 = fft(B,2*N);
x = xcorr(A3,B)%ans: 0.1952    0.4051    0.6958    1.0470    0.7676    0.5050    0.2424
x2 = ifft(A4.*conj(B2),2*N)%ans: 1.0470    0.7676    0.5050    0.2424         0    0.1952    0.4051    0.6958
x3 = ifft([A2(1:N/2) A2(N/2+1)/2 zeros(1, 2*N-length(A2)-1) A2(N/2+1)/2 A2(N/2+2:end)].*conj(B2),2*N)
%ans: 3.5624    4.2973    5.4391    6.3191    6.2232    5.2077    4.0659    3.4665

Answer 1

You can pad zeros in frequency domain. But you need to take care about Nyquist bin when N is even.

A3 = [A2(1:N/2) A2(N/2+1)/2 zeros(1, 2*N-length(A2)-1) A2(N/2+1)/2 A2(N/2+2:end)];

Here Nyquist bin is A2(N/2+1)/2 and zeros are padded in the middle of the spectrum

For N-odd

A3 = [A2(1:(N-1)/2+1) zeros(1, 2*N-length(A2)) A2((N-1)/2+2:end)];

This is only for the case if you need to interpolate in time-domain. Almost irrelevant to the problem you have.

Posted the solution to you problem below.

Answer 2

clear;

N = 4;
M = 2*N-1;

a = 1:4;
r = ones(1, N);           % Rectangular window

A1 = fft(a);
W = fft(r, N);

A2 = (A1).*W;
A3 = ifft(A2, N);

B = 3:6;
x = xcorr(A3,B)

A4 = fft(A3,M);           % A3 is interpolated by a factor of M
B2 = fft(B, M); 

Freq_Multiplication = (A4.*conj(B2));

x2 = fftshift(abs(ifft(Freq_Multiplication, M)))

% Zeropadding A2 is actually interpolation in time-domain. Not really a linear convolution. x3 will show constant results

Zero_pad_A2 = [A2(1:N/2) A2(N/2+1)/2 zeros(1, M-length(A2)-1) A2(N/2+1)/2 A2(N/2+2:end)];

A5 = fft([A3 zeros(1, M-length(A3))], M);

%x3 = ifft(Zero_pad_A2.* conj(B2), M) % This not performing linear convolution

x3 = fftshift(abs(ifft(A5.* conj(B2), M)))