# Reconstruction of an oversampled signal with sub-nyquist analog filters

I have a baseband, bandlimited signal of 100 Hz. I sampled it at 2000 Hz. Now for reconstruction if I pass them (the samples) through an analog filter of say 200 Hz, would the reconstructed signal be same as the original analog signal. In books they pass it through a LPF with cutoff frequency of 1000 Hz. I am questioning like what happens if I pass through various LPFs of cutoff frequencies above 200 Hz i.e. 300 Hz, 500 Hz, 1000 Hz. Would there be differences or not?

• @JohnMarvin: As per the suggestion, adding my comment in the question area. In books they consider fs/2 as the LPF cutoff frequency and beautifully explain how the reconstruction happens and how you get the exact values of x(t) at the sampling instants with only the sinc function at the sampling instant contributing and all other sincs going through zeros at this instant. How can we mathematically explain similarly when we do not take fs/2 as the LPF cutoff frequency? – Seetha Rama Raju Sanapala Sep 12 '15 at 22:08

If you sample your signal at 2000 Hz, in frequency domain, you will have your signal shifted to 2000 Hz, and its multiples. As your signal has bandwidth of 100 Hz, the minor frequency that you don't want will be 1900 Hz. So in theory, thinking about an ideal low pass filter, you can filter in any frequency between 100 Hz and 1900 Hz.

So, if use a low pass filter with cutoff frequency in this range, you will be able to reconstruct your signal. Nothing different happens.

It is easier to answer your question when you think about what is happening in frequency domain when you samples your signal: the spectrum becomes periodic in the frequency of sampling.

See the figure bellow. In (a) you have the spectrum of your original signal. In (c), you have the sampled signal spectrum.

When you sample at $f_{S}$, a replica of the signal will appear there and in $-f_{S}$, and in its multiples (not show in the figure). So, you need to eliminate this components using a low pass filter to reconstruct your original signal. So, you can use a low pass filter with cutoff frequency anywhere between $f_{max}$ and $f_{S} - f_{max}$.

• In books they consider fs/2 as the LPF cutoff frequency and beautifully explain how the reconstruction happens and how you get the exact values of x(t) at the sampling instants with only the sinc function at the sampling instant contributing and all other sincs going through zeros at this instant. How can we mathematically explain similarly when we do not take fs/2 as the LPF cutoff frequency? – Seetha Rama Raju Sanapala Sep 6 '15 at 22:18
• Now you got me. When you analyze in frequency domain, we can see that, somehow, it is going to be reconstruct, although it is obvious that the sinc functions are not zero at the other samples. I would say that the sum of all the sinc functions in an specific sample, except the one who is interpolating that sample, will be zero, but I am not sure about this, and I don't know how to prove it mathematically. So, let's wait and see if someone know the answer. I suggest you to add this comment in your main question, so people are more likely to see it. In the mean time, I will do some research. – JohnMarvin Sep 7 '15 at 16:32