# analog-to-discrete-to-analog system sampling problem

This is not a homework problem, I am solving practice problems for my exam. Consider the analog-to-discrete-to-analog system shown in figure 1. The CT signal $$x_a(t)$$ is sampled at a frequency of $$F_s = 2000$$ Hz ($$T_s = 0.5$$ msec). The resulting impulse train is then converted to a discrete time sequence $$x_d[n]$$. The Lowpass DT filter $$H_d(e^{j\Omega}$$ is subsequently used to filter $$x_d[n]$$ giving $$y_d[n]$$. Finally, a CT version of the output $$y_a(t)$$ is created, using an ideal DT-to-CT converter (at the same sampling frequency $$F_s = 2000$$ Hz).

Note : $$H_d(e^{j\Omega}$$, which is obviously periodic, is shown for only one period.

a) For the CTFT of $$x_a(t)$$ given by $$X_a(\omega)$$ in the figure with $$B = 2000\pi$$ rad/sec, sketch the $$X_d(e^{j\Omega})$$, the DTFT of the DT sequence $$x_d[n]$$.

b) Sketch $$Y_a(\omega)$$, the CTFT of the CT signal $$y_a(t)$$. Again, assume that we are using the same frequency as that of sampling $$F_s = 2000$$ Hz.

I followed the following steps:

Step 1: Multiplied $$x_a(t)$$ by an infinite pulse train and then switched to frequency domain so I can have an expression for $$X_d(\omega)$$ in CTFT.

Step 2: After conversion, the "envelope" (which in this case is the triangular shapes) are not drawn but instead we have pulses, something like this: (Excuse my very bad drawing skills) However, only the red lines exist in this case since we are now in discrete time.

Step 3: I believe in step 2 I have found how $$X_d(e^{j\Omega})$$ should look like. But then here I get stuck about how should I filter the signal. I get that the filter is periodic but how do I use this in here? Isn't the width of the filter much smaller than the width of the signal?

• Perhaps pedantic: but why aren't you using z-domain representation? There are a lot of tools available. BTW: I do agree with Schwarz's solution since it addresses your particular question. – rrogers Jul 30 at 20:28

The main consequence of time-domain discretisation is periodicity in the frequency domain. Your solution for exercise 2 should be, as well. For this, you may need to check whether the spectrum will overlap at $$\frac{F_s}{2}$$.
$$y_d[n]=x_d[n] * h_d[n] = \mathcal{F}_\text{DTFT}^{-1}\{X(e^{j\Omega}) \cdot H(e^{j\Omega})\}$$