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Suppose we are given an input signal s[m,n] with DTFT $S(\omega_1, \omega_2)$.

We sample it at $\omega_1 = \frac{2 \pi k}{256}$ and $\omega_2 = \frac{2 \pi l}{256}$ to get a 256 point DFT S[k,l]. Now suppose we take the IDFT of S[k,l] to get $\tilde{s}[m,n]$

I am trying to understand the relationship between $\tilde{s}[m,n]$ and s[m,n]. My understanding is that I should be able to express $\tilde{s}[m,n]$ as a summation of every 256th sample of the original signal.

So far I have been advised to try expressing the Sampled DTFT as a dirac comb, i.e.:

$S[k,l] = \sum_{k = -\infty}^{\infty} \sum_{l = -\infty}^{\infty} S( \frac{2 \pi k}{256}, \frac{2 \pi l}{256}) \delta(\omega_1 - \frac{2 \pi k}{256}, \omega_2-\frac{2 \pi l}{256}) $

and use the definition of the IDFT:

$\tilde{s}[m,n] = \frac{1}{256^2}\sum_{k=0}^{255} \sum_{l=0}^{255} S[k,l] e^{j(\frac{2 \pi km}{256}+\frac{2 \pi ln}{256})}$

Using these two pieces of information I can get the expression:

$\tilde{s}[m,n] = \frac{1}{256^2}\sum_{k=0}^{255} \sum_{l=0}^{255} [\sum_{k = -\infty}^{\infty} \sum_{l = -\infty}^{\infty} S( \frac{2 \pi k}{256}, \frac{2 \pi l}{256}) \delta(\omega_1 - \frac{2 \pi k}{256}, \omega_2-\frac{2 \pi l}{256})] e^{j(\frac{2 \pi km}{256}+\frac{2 \pi ln}{256})}$

but I am not sure how to proceed from here. Any advice would be helpful I have been racking my brain over this for a couple hours now.

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For simplicity, I prefer a 1D notation without losing generality.

The IDFT associated with the properly taken uniform (in frequency) samples of a given valid DTFT can be shown to be the following.

Case-1: Assume a finite length signal $x[n]$ of length $N$, and consider its DTFT $X(\omega)$. Now if you take $M \geq N$ uniform (frequency) samples of $X(\omega)$ according to $$ X[k] = X( \frac{2\pi}{M}k) ~~~, k=0,1,...,M-1 $$ then the M-point inverse DFT of $X[k]$ will exactly be $x[n]$ padded with $M-N$ zeros; i.e., $$ y[n] = IDFT_M\{ DFT_M\{ x[n] \} \} = x_M[n] $$ So the sequence $x[n]$ and $y[n]$ have the same values for the first $N$ samples (i.e., all samples of $x[n]$ are retained in $y[n]$), but $y[n]$ have further (effectively padded) zero samples.

If $ M < N$ , then time-aliasing in $x[n]$ will happen and the first $N-M$ samples of $y[n]$ will be in error. But the remaning $2M-N$ samples of $y[n]$ will be identical to the original $x[n]$ at the corresponding index positions.

Case-2: $x[n]$ is infinite length (and stable) sequence and $X(\omega)$ is its DTFT. Then according to $M < N$ subcondition of case-1, all samples of $y[n]$ will be aliased. No samples of $y[n]$ will be equal to $x[n]$, except other than (pure) coincidence.

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  • $\begingroup$ Correct, I am expecting aliasing in y[n], but I am trying to figure out how to prove derive the expression which relates the aliased $\tilde{s}[m,n]$ values to those of the original signal. $\endgroup$ – Filip Oct 15 '18 at 0:25
  • $\begingroup$ aliased values of y[n] are addition of last $N-M$ samples of $x[n]$ to first $N-M$ samples of $x[n]$ ; i.e. $y[0] = x[0] + x[M]$, $y[1] = x[1]+x[M+1]$,...,$y[N-M-1] = x[N-M+1]+x[N-1]$. You can see those indices by a simple plot of the periodic extension of the circularly alised inverse DFT. $\endgroup$ – Fat32 Oct 15 '18 at 0:30
  • $\begingroup$ or stated in other words: $$ \tilde{y}[n] = \sum_k x[n-kM] $$ $\endgroup$ – Fat32 Oct 15 '18 at 0:32
  • $\begingroup$ But what is the math behind this fact, to me it is not obvious why there is aliasing in the discrete time domain. $\endgroup$ – Filip Oct 15 '18 at 0:47

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