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$$x_a(t) = 10 + 4\cos(30\pi t +\frac\pi 3) - \sin(250\pi t + \frac 14) -3\cos(500t + \frac \pi4) $$

$\textbf{Question:}$

This signal is first sampled at a rate of $f_s = 100$ samples per second to get a discrete array $x[n]$. Find $x[n]$. That $x[n]$ is transmitted to the receiver. The receiver only knows that the received discrete signal is obtained from a continuous-time signal by sampling at a rate of 100 samples/s. In an attempt to recover the original continuous signal from the received discrete signal, the receiver converts the received discrete signal first to an impulse train using $T_s = 1/100$ seconds and then uses an ideal low-pass filter to generate the $x_r(t)$; the passband of the filter is $|\omega| < 100\pi$ rd/s, and its gain is 1/100. Find $x_r(t)$; make sure that it is in the form of a superposition of real valued sinusoids.

$\textbf{My Attempt:}$

I found $x[n]$ as: $$x[n]=10 + 4\cos(\frac{3\pi}{10}n + \pi/3) - \sin(\frac{5\pi}{2}n + 1/4) -\cos(5n + \pi/4)$$

I know that $\mathcal{X_p}(j\omega)$, which is obtained by taking the Fourier Transform of the original signal $x_a(t)$ multiplied with the impulse train, is equal to $\mathcal{X}(j\Omega)|_{\Omega = \omega T}$ which is the Discrete Time Fourier transform of $x[n]$.

So now in order to get $x_r(t)$ should I just evaluate $$\mathcal{X}(j\Omega)|_{\Omega = \omega T} \cdot \mathcal{H}_{LP}(j\omega) = X_r(j\omega)$$ and then take the inverse fourier transform of $X_r(j\omega)$ to get $x_r(t)$ or is there some other simpler way?

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Since the sampling as well as the reconstruction are ideal, i.e., you use the theoretical model of an impulse train combined with an ideal lowpass filter, there is no need to look into the intermediate steps of the reconstruction process. You know that the only effect you need to take into account is aliasing.

Any input signal with a frequency smaller than $100\pi$ (radians) will remain unchanged. This is the case for the first two terms of the continuous-time signal $x_a(t)$. The remaining two terms will be aliased. If you understand how aliasing works - that's the point of the exercise - then you'll be able to quickly figure out the resulting signal, all frequencies of which must be in the range $\omega\in[0,100\pi)$.

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  • $\begingroup$ So simply convert $x[n]$ to an impulse train modulated signal. Then convolve that with $h_{LP}(t)$. Also to check if there is aliasing, can't I just reduce the angular frequencies of all the sinusoids to be $\in (-\pi , \pi)$ and that would let me know which sinusoidal components are the same in $x_a(t)$ and $x_r(t)$? $\endgroup$
    – Ahsan Yousaf
    Commented Dec 18, 2022 at 13:27
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    $\begingroup$ @AhsonYousef: Yes, you just need to subtract multiples of $2\pi$ from the frequencies of the discrete-time signal until all of them are inside $[-\pi,\pi]$. Equivalently, you can just compute the aliased frequencies from the original continuous time signal. $\endgroup$
    – Matt L.
    Commented Dec 18, 2022 at 14:50

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