To preface, this isn't a homework question but rather a self-study question to help me to understand the basics of finding the DTFT and magnitude of the DTFT based on a discrete time signal sampled from a continuous time signal.

I was wondering if anyone has any methodology to approaching questions where you're asked to plot the magnitude of the DTFT of a discrete-time signal, when $x(t)$ is sampled according to $x[n] = x(nT_s)$`, with $F_s$ equal to a given value.

For example if $$1/T_{s}=8W$$ and : $$x(t) = T_{s}\frac{1}{4W}\frac{\sin(2\pi Wt)}{\pi t}\frac{\sin(2\pi 2Wt)}{\pi t}$$

What are the first questions that you would ask yourself that will help you to plot the magnitude of the DTFT. I am having trouble translating the DTFT to the plots themselves.

  • 2
    $\begingroup$ $W$ is frequency or bandwidth; are you sure that $T_s$ (the sampling interval) equals $8W$? $\endgroup$ – Matt L. Oct 10 '16 at 19:34
  • $\begingroup$ The discrete-time signal is created by sampling a continuous time signal, x(t) according to x[n] = x(nTs) with Fs = 1/Ts = 8W. I made a typo above when I said Ts=8W. $\endgroup$ – Gary Oct 12 '16 at 15:37

So the example that you've listed appears to be somewhat odd. Assuming Ts is the time step, you're saying that the continuous signal x(t) is dependent on the time step you've chosen. That just seems odd.

It also seems that you've formatted your question in the sense that you understand the Fourier Transform (FT), but you're just having a difficult time relating the FT to the associated Discrete Time Fourier Transform (DTFT) after sampling the continuous signal. I'm going to speak at a relatively high level to explain how to go from the FT to the DTFT.

The first thing you should realize is that the DTFT will be related to the FT of the continuous signal. But how so? The sampling theorem (which is the most fundamental concept in DSP) tells us that the bandwidth of a discrete signal (i.e. sampled signal) is equal to that of the sample rate$^1$. We typically write this as -Fs/2 to Fs/2 where Fs is your sample rate. Furthermore, we know that analog frequencies that are integer multiples of the sample rate will be interpreted as equivalent frequencies. This is aliasing.

For example, say Fs = 1kHz. Let's randomly select a frequency of 0.5kHz. All frequencies that are integer multiples of the sample rate at this frequency will be seen as 'the same thing' when sampled. 0.5kHz, 1.5kHz, 2.5kHz, etc. etc. will all appear to be 'the same signal' when sampled. This is also true of other frequencies such as 0.25kHz. In this case, 0.25kHz, 1.25kHz, 2.25kHz, etc. will all appear to be the same. Note that this phenomenon happens at EVERY frequency.

This is what we refer to as aliasing: different frequencies be interpreted as the same signal due to sampling. A common misperception is that aliasing only happens in one direction: that being that high frequencies are aliased onto lower frequencies. However this isn't true. Looking at our above examples, you can see that aliasing only comments on the fact that different signals are interpreted the same. It doesn't say anything about direction. Thus you could also state that lower frequency signals are aliased onto high frequency signals. Because of this aliasing of high frequency signals onto low frequency signals, and low frequency signals onto high frequency signals, the frequency spectrum of sampled signal becomes periodic. So let's put this together into a simple list of steps on how to get the DTFT from the FT.

1) Obtain a copy of the FT of the continuous signal. You will need to do this analytically.

2) Slice up the FT along the frequency spectrum at intervals of Fs. Let's say your first interval would range from -Fs/2 to Fs/2 (this is centered at zero).

3) After you've 'cut up' your FT into intervals, you can take all of those and add them together at a single interval (i.e. do the super position of them). By doing this step, we are essentially introducing aliasing into one interval.

4) Now repeat step 3 for every interval. By doing this, you're accounting for the fact that aliasing doesn't just occur from high frequency to low frequency, but it essentially occurs everywhere!

5) Look at what you've done. By doing the super position at every interval, you'll see that your frequency spectrum is now periodic (and still continuous!).

This is essentially what the DTFT is doing. Visualizing these steps is very useful, I might add illustrations if I get time. Now assuming that you've got a plot of the FT, you now have a plot of the DTFT.

[1] It would be more accurate to say that the relevant bandwidth is that of the sample rate.


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