difference between FFT(X) and FFT(X,N)

 Hello,

 I have the following:
  clc
  clear all
  close all

Fs=1000;
T = 1/Fs;                     % Sample time
L = 1000;                     % Length of signal
t = (0:L-1)*T;
delta_T=2.345 / 1000; %delay time in SECONDS
% Time vector
x = sin(2*pi*50*t);
w=2*pi*50; %angular freq. component
X=fft(x);
Y=X.*exp(1j *2*pi*([0:L/2 -L/2+1:-1])*L*T*delta_T);
y_1=real(ifft(Y));
y_2 = sin(2*pi*50*(t+delta_T));
%plot signals
 stem(y_2-y_1); %<--- negligible

If I do the above process efficiently, by performing 'N' point FFT where 'N' is a power of 2, then what changes should be made in the above code to produce same result?

Answer 1

[Wayne King]

Are you sure you need to use a power of two for efficiency? At any rate you can get rid of the padding in time domain. And although you won't get the perfect reconstruction you are looking for, the waveforms will be very similar.

    Fs=1000;
    T = 1/Fs;                     % Sample time
    L = 1000;                     % Length of signal
    t = (0:L-1)*T;
    delta_T=2.345 / 1000; %delay time in SECONDS
    % Time vector
    w=2*pi*50; %angular freq. component
    x = sin(2*pi*50*t);
    Lprime = 2^nextpow2(length(x));
    X=fft(x,Lprime);
    Y=X.*exp(1j *2*pi*([0:Lprime/2 -Lprime/2+1:-1])*Lprime*T*delta_T);
    y_1=real(ifft(Y));

Now if you plot y_1

plot(y_1)

You'll see the effect of the zero padding from point 1000 to 1024. So just throw away the padding

y_1 = y_1(1:L);

However, you will not obtain the kind of perfect reconstruction you were looking for, but compare y_1 and y_2.

y_2 = sin(2*pi*50*(t+delta_T));
plot(y_1)
hold on;
plot(y_2,'k');

[zozo]

Can someone please explain why this difference is occurring? Is it something to do with wrapping around of circular convolution?