Inspired by Dr. Sarwate's answer to the question How replicas are formed in Frequency domain when a signal is sampled in Time Domain? , I decided to see for myself the underlying concept.
I interpreted Dr. Sarwate's answer that sampling was a sequential activity. All one needed to do was to multiply an impulse with the signal meant to be sampled. And this needed to be repeated with delayed/shifted versions of the impulse.
Multiplication in time domain is convolution in frequency domain and vice-versa. I thought that with the impulse shifting by every sample, its spectral response would also undergo a similar shift in frequency domain. But it seems not to be the case.
I wrote a toy program in octave and the plots of the frequency response havent brought any joy. What have I misunderstood here?
## Num Samples
N = 10;
## Row vector representing samples (all zeroed out)
x = zeros(1,N);
y = zeros(1,N);
z = zeros(1,N);
## Undelayed impulse (For lack of a better word)
x(1) = 10;
## Delayed by 1 sample
y(2) = 10;
## Delayed by 2 samples
z(3) = 10;
## Obtain spectrum of undelayed impulse
s_spectrum_x = fft(x);
s_spectrum_abs_x = abs(s_spectrum_x);
## Obtain spectrum of impulse delayed by 1 sample
s_spectrum_y = fft(y);
s_spectrum_abs_y = abs(s_spectrum_y);
## Obtain spectrum of impulse delayed by 2 samples
s_spectrum_z = fft(z);
s_spectrum_abs_z = abs(s_spectrum_z);
>>disp(s_spectrum_abs_x);
10 10 10 10 10 10 10 10 10 10
>>disp(s_spectrum_abs_y);
10 10 10 10 10 10 10 10 10 10
>>disp(s_spectrum_abs_z);
10 10 10 10 10 10 10 10 10 10
Shouldn't the initial bins of y and z be 0?