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A continuous-time signal x with bandwidth \$\frac{20π}{3}\$ rad/sec is sampled with sampling period \$T_s = 0.1\$ seconds to obtain a discrete-time signal \$x_s[n] = x(nT_s)\$ for all n. The discrete-time Fourier transform \$X_s\$ of the sampled signal is shown below: enter image description here

Let \$X\$ be the continuous-time Fourier transform of the input signal \$x\$. Sketch \$X(ω)\$ vs \$ω\$ for \$−10π ≤ ω ≤ 10π\$. Label all features of your plot.

Apparently, we can deduce using the equation \$X_s(\omega)=\frac{1}{T_s}\sum_{k=-\infty}^{\infty}X\left(\frac{\omega-2\pi k}{T_s}\right)\$ that the plot should look something like this:

enter image description here

Could someone please clarify how you would do this?

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2 Answers 2

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This seems to be a homework, so I'll only say that sampling causes aliasing (or frequency folding). If the original signal was properly band limited (is it the case here?), will it have aliasing if the sampling is removed?

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For understanding aliasing, just simulate this circuit.
A complex signal is "difficult", shown here as the sum of 3 sinusoidal signals.
Hope this can help.

enter image description here

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