Homework Seven

Figure out what happens if your sampled signal, x(t), has frequency components only for $\textstyle {\frac {1}{2}}f_{s}<f<f_{s}$. Can you recover the original signal from it? If so, find the expression for x(t) in terms of x(nt).

Nick Christman

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Imagine we have an original signal that has the following frequency plot, centered about the "zero frequency" origin:

Figure 1: Original signal for which we want to analyze.

In order to analyze this signal we must look at both the positive and negative frequency aspects -- therefore, we will split the signal into two parts, positive and negative:

Figure 2: Signal separated into positive and negative aspects.

To "send the signal" we must essentially move it to a higher frequency and, remembering that x(t) has frequency components for $\textstyle {\frac {1}{2}}f_{s}<f<f_{s}$, we get the following signal, $\textstyle X(f)$:

Figure 3: Preparing signal to be sent and sampled.

Finally, we need to sample the signal at a rate of $\textstyle f_{s}={\frac {1}{T}}$. Theoretically, this leads to the following frequency plot:

Figure 4: Signal being sampled at a rate of 1/T.

In order to get the original signal, we simply need to create a lowpass filter that will essentially encompass the the original (desired) signal and filter out any high frequency components:

Figure 5: The use of a lowpass filter to obtain the original signal as shown in Figure 1.

Figure 6: Original signal obtained using a lowpass filter.

So what have we accomplished? We have taken a signal (Figure 1), prepared it to be sampled (Figure 2 & 3), sampled it at a sampling rate of $\textstyle f_{s}={\frac {1}{T}}$ (Figure 4), and used a lowpass filter to collect the original (now sampled) signal (Figure 5 & 6).