# Sampling of a band-limited analog signal: DFT and aliasing

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 $0 \leq k \leq 399$.

1. Which DFT coefficients are free of aliasing?
2. The DFT coefficient $X[50]$ represents the spectrum of the analog signal at what frequency $f$? (Give your answer in $\textrm{Hz}$).

1. All of the DFT coefficients are free of aliasing. The sampling rate is more that twice the maximum signal frequency.
2. 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.

• That is incorrect. When you append zeros you are not up-sampling your signal. Sampling rate is always the same, you only change the spacing of frequency bins and interpolating samples between them. – jojek Feb 26 '15 at 9:22

The provided answers are correct. Appending zeros is not upsampling and therefore the sampling rate is still $R_\mathrm{s}=100\, \text{Samples/s}$. Accordingly, the frequency spacing after taking the N-DFT is $f_\mathrm{d}=(R_\mathrm{s}/N)=0.25\, \text{Hz}$.
Upsampling can be achieved by inserting $N_\mathrm{U}$ zeros after every sample followd by lowpass filtering.