IFFT, resampling, complex conjugate symmetry, zero padding, interpolation, aliasing, oscillation

Hello, I have measurements (complex values) in the frequency domain, from 20 MHz to 20 GHz. I would like to convert them into time domain using IFFT. My Data has three columns, the first column is frequency (20000000, 32487500, 44975000, ...,2e10), 1601 points and the second and third columns are corresponding real and imaginary data. Size of matrix is (1601 x 3)

Here is my code.

%% uniform sampling in time domain

OT = 1./Data_mix(:,1); % original time

OF = Data_mix(:,1); % original Frequency

NT = flipud(linspace(OT(end,1), OT(1,1), 1000)'); % new time in uniform spacing

NT = [NT; 0]; % 1001 x 1

T = NT(2)-NT(3); % dt, 1.952e-12

FS = 1/T; % need to be at least twice the maximum frequency ?

N = size(NT,1); % 1001

df = FS/N; % Frequency increment

f = [0 : df : FS-df]'; % 1001 x 1

c_ust = 1;

Data_R = interp1(OF, Data_mix(:,2), f,'nearest','extrap'); % real

Data_I = interp1(OF, Data_mix(:,3), f,'nearest','extrap'); % imag

Data_ust = Data_R + 1j*Data_I; % Data with uniform sampling time 1001

%% making complex conjugate symmetry

Data_cc = [real(Data_ust(1,1)); Data_ust(2:end-1,1); real(Data_ust(end,1)); conj(Data_ust(end-1:-1:2,1))]; % complex conjugate symmetric data

%% interpolation in time domain

IF = 16; % interpolation factor (2, 4, 8, 16, or 32...)

Ncc = 2*N-2; % 2000

fcc = df*(0:Ncc-1)'; % frequency points

MM = IF*Ncc; % interpolation factor

dt1 = T/MM;

t1 = dt1*(0:MM-1)'; % time points after interpolation 2000 * 16 = 32000

f1 = df*(0:MM-1)'; % frequency points after interpolation 2000 * 16 = 32000

%% zero-paddings

if mod(Ncc,2)==0 % N even

Data_zp = [Data_cc(1:Ncc/2,1); 0.5*Data_cc(1+Ncc/2,1); zeros(MM-Ncc-1,1); 
           0.5*Data_cc(1+Ncc/2,1); Data_cc(2+Ncc/2:Ncc,1)];

else % N odd

Data_zp = [Data_cc(1:(Ncc+1)/2,1); zeros(MM-Ncc,1); Data_cc((Ncc+3)/2:Ncc,1)];

end

%% IFFT

Data_TD = (MM/Ncc)*ifft(Data_zp);

figure, plot(t1, Data_TD,'o-'); grid on; xlim([0 5e-13]);

The result shows like this, which has unwanted oscillation. Please teach me what is wrong. Thank you!

[Santhana Raj]

WHat about the actual sampling frequency of the data captured? You have used dt=1.952e-12. But is this the same sampling frequency of the captured data?

I think the sampling frequency should be

(Data_mix(1,1)-Data_mix(2,1))*1601.

This is assuming the N(1601) to be the same as the no of points in the FFT done in the captured data.

[Hyungjoo Lee]

Dear Santhana, Thank you for your thoughtful answer.

Original Frequency (OF, raw measurements, 1601 points) has uniform spacing (20000000, 32487500, 4497500, 57462500, ..., 19988e10, 2e10) so df is 12487500 Hz. This means Original Time (OT = 1./OF) has non-uniform spacing as (5e-8, 3.0781e-8, 2.2235,-8, ..., 5.0031e-11, 5e-11 sec) and dt is not a constant. Thus, I made a new time (NT, 1001 points) using linspace as (5e-8, 4.995e-8, 4.99e-8, ..., 1e-10, 5e-11, 0) with dt is 5e-11 (dt=1.952e-12 was a typo, sorry). So, my FS was FS = 1/dt = 1/5e-11 = 2e10 [Hz].

Now, you suggested that FS should be (Data_mix(1,1)-Data_mix(2,1))*1601 = (20000000 - 32487500)*1601 = -1.9992e10 (or + 1.9992e10). I've tried but it does not fix my problem.

For the no of points in the IFFT, the N is (8000) because

OT = 1601 (original raw measurement points)

NT = 1001 (uniform sampling)

Ncc = 2000 (complex conjugate symmetry)

Nzp = 8000 (interpolation factor, IF = 4, Nzp = IF*Ncc)

So do you think I should use FS as dt*Nzp = 5e-11*8000 = 4e-7 [s] ? If I change FS, then df, f, Data_ust, Data_cc, and Data_zp will change correspondingly. Also, I still have those unwanted oscillations. Please advise.

Thank you.

[Hyungjoo Lee]

I found that the reason of the oscillation was due to my measurements, which are not periodic functions, Thank you.