Sampling Signals

Question

A Continuous Time signal is given by $() = \text{cos}(200) + 2\text{cos}(320)$. A sampled signal, $_()$, is produced by sampling $()$ at sampling frequency $f = 300 \text{ Hz}$. If we reconstructed $_()$ signal by filtering it using an ideal LPF with cutoff frequency $f_c = 250 \text{ Hz}$ and gain $T$. What would be $_()$?

Answer 1

The sinuoid at frequency 160Hz, would be aliased to 140Hz. The sinuosid at 100Hz will have another coomponent at 200Hz, so you would have four sinusoids 100, 140Hz, 200, 160 and scaled in amplitude by T.

This happens because sampling in time domain creates periodic repetitions of the spectrum in frqeuency domain. let $X(f)$ denote the spectrum of the signal $x(t)$ then after sampling the spectrum of the sampled signal is given by the expression $$\sum_{n=-\infty}^{\infty}X(f -nFs)$$

Now if you look at the spectrum of a sinuosid it has two impulses one $F_c$ other at $-F_c$, for example the sinuosid at frequency 160Hz has two impulses one at -160 and other at +160Hz, now when you plug in this spectrum in the expression above and keep getting shifted copies you will find that you get impulses at +140,+440,+740 ......and so on similarly at the negative frquencies as well.

Also the sinusoid at 100Hz will similarly get shifted copies at 100(itself, meaning $n=0$ in the above expression) then at 200Hz (-100 +300) similary at -100Hz(100-300Hz).

Now the filter cutoff is at 250Hz so all these components will be seen.

The ideal sampling and filters for the signal should have been sampling rate greater than 320Hz and filter cutoff of 160Hz