# How can I find $x_r(t)$ in this case?

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

$$\textbf{Question:}$$

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?

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 $$\omega\in[0,100\pi)$$.
• 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 $x_r(t)$? Dec 18, 2022 at 13:27
• @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. Dec 18, 2022 at 14:50