# How to plot the periodic digital spectrum?

In a previous answer, Dan Boschen wrote (emphasis mine):

First let me explain the "unfolded" digital spectrum: If you allow the frequency axis of the sampled signal to extend to $$\pm \infty$$, instead of limiting to the unique digital frequency range of $$\pm > F_s/2$$ (where $$F_s$$ is the sampling rate), you will see replicas of the original spectrum that is centered about 0 (DC) to also be similarly centered around every multiple of $$F_s$$. This is why we only need to show the spectrum from $$\pm F_s/2$$ (or even $$0$$ to $$F_s/2$$ for real signals) since this replicates everywhere else. However, I find this visualization helps immensely in understanding many concepts in multi-rate signal processing as well as bridging analog and digital systems.

Indeed this was an(other) eye-opener to me. Unfortunately, all the GUI tools I have at hand at this time only plot digital spectrum in the $$0$$, $$F_s/2$$ range--and I always thought it was the full spectrum. But as mentioned by Matt in a comment below "the spectrum is periodic".

Do you know how I can plot several periods of the unfolded digital spectrum using Matlab/Octave or Python Scipy?

• There is no such thing as "the full spectrum" for discrete-time signals, because the spectrum is periodic. So how many periods do you want to plot (and why not just one)? – Matt L. Dec 6 '19 at 9:36
• Sorry for my approximate wording. " So how many periods" a few. "and why not just one" just to observe what you said: the spectrum for (a discrete-time signal only?) is periodic. Something I just learned today! – Sylvain Leroux Dec 6 '19 at 9:46
• @SylvainLeroux Another interesting observation with going between the frequency and time domain is what works in one direction also applies in the reverse direction; this can sometimes help provide an intuitive explanation if you have a better understanding of the process in one domain. For example, here we see that something that is discrete in time is periodic in frequency. It may already be clear to you to see that something that is periodic in time (a square wave for example) is discrete in frequency: A square wave has discrete frequencies at odd harmonics f, 3f, 5f,...(f: repetition rate) – Dan Boschen Dec 6 '19 at 13:07
• @SylvainLeroux: Please don't take this the wrong way: you have asked a lot of questions recently that indicate that you are missing the basic fundamentals of discrete signal & system. I'd recommend spending two days or so with a quality text book or website. This is an investment that will make life a lot easier down the road. – Hilmar Dec 6 '19 at 17:32
• I would suggest any book with Oppenheimer as an author, and expect to take months, not days, to learn it. You may do better by checking to see if Khan Academy or MIT have published lectures on DSP. To really understand this stuff you need the standard introductory course in general signal processing (which includes some DSP), possibly followed by a specific course in DSP, or if you're a good at self-study, a good DSP book. It's probably a mistake to just dive into a DSP book without having studied general signal processing. – TimWescott Dec 6 '19 at 20:07

As a simple example, consider the discrete-time signal

$$x[n]=a^nu[n],\qquad |a|<1\tag{1}$$

where $$u[n]$$ is the unit step function. The discrete-time Fourier transform (DTFT) of $$x[n]$$ is given by

$$X(e^{j\omega})=\frac{1}{1-ae^{-j\omega t}}\tag{2}$$

Note that $$\omega$$ is a normalized angular frequency:

$$\omega=2\pi f/f_s\tag{3}$$

where $$f_s$$ is the sampling frequency.

Also note from $$(2)$$ that $$X(e^{j\omega})$$ is a $$2\pi$$-periodic function in $$\omega$$, i.e., it is periodic with period $$f_s$$ in the frequency variable $$f$$. This periodicity of the spectrum is a general property of discrete-time signals.

Now you can plot the spectrum of $$x[n]$$ (i.e., $$X(e^{j\omega})$$) as a function of the frequency $$f$$, and depending on the chosen range of $$f$$ and the chosen sampling frequency, you will plot a certain number of periods. This is shown in the following Octave/Matlab script:

a = 0.9;
fs = 1;
fmin = 0;
fmax = 2.5;
f = linspace(fmin,fmax,1000);
X = 1 ./ (1 - a * exp(-1i*2*pi*f/fs) );

plot(f,abs(X))
xlabel('f')
title('|X(f)|')
grid on • Thanks a lot, Matt, for having taken the time to post such a detailed answer. I know all that must seem obvious for most of the visitors here, but it really helped me. I assume the DTFT of $x[n]$ is well known and not something you guess out of thin air. Or am I wrong? – Sylvain Leroux Dec 6 '19 at 15:03
• For reference, see 5.1.2 here for the DTFT of $x[n]$ – Sylvain Leroux Dec 6 '19 at 15:23
• The DTFT is just the version of the Fourier transform you need for discrete-time signals. Check this wikipedia-page. – Matt L. Dec 6 '19 at 16:33