I saw this question on one of the sites related to DFT:
The analog signal $x(t)$ is band-limited to $40\textrm{ Hz}$. Suppose the signal is sampled at the rate of $100\textrm{ samples/second}$ and that at this rate $200\textrm{ samples}$ are collected. Then $200$ zeros are appended to the $200\textrm{ samples}$ to form a $400$-point vector. Then the $400$-point DFT of this vector is computed to get $X[k]$ for $0 \leq k \leq 399$.
- Which DFT coefficients are free of aliasing?
- The DFT coefficient $X[50]$ represents the spectrum of the analog signal at what frequency $f$? (Give your answer in $\textrm{Hz}$).
The answer's provided were:
- All of the DFT coefficients are free of aliasing. The sampling rate is more that twice the maximum signal frequency.
- The DFT bin width is $100/400$ or $0.25\textrm{ Hz}$. The $50^{\rm th}$ DFT coefficient corresponds to the frequency $50$ times $0.25\textrm{ Hz}$ or $12.5\textrm{ Hz}$
I'm not really sure if this is right because when you append zeros you are upsampling and the sampling rate becomes $400\textrm{ Hz}$. So when you take a $400$-point DFT the bins are seperated by $1\textrm{ Hz}$.
Please let me know if I am right or not. If I am wrong please explain the answer to me.
Jose.