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I have been reading a textbook on the "fundamentals" of signal processing but the author of the textbook has not given any explanation for the triangular graph located at the bottom of the figure.

enter image description here

I understand the general concept of aliasing, but I do not understand what exactly is being shown in the graph featuring the "spectral band of interest". Why do we need it and why is it triangular in the first place? I see that the horizontal lines connect two points, of which one belongs to the graph of the correct frequency and one belongs to the graph of the "alias" frequency. Beyond that I am lost, even as to why there is only one line linking each correct-frequency+alias pair of points but not more.

I would be much obliged for any help in understanding this graph.

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  • $\begingroup$ The point of the bottom plot is that the spectrum is periodic. Its triangular shape is unimportant; it's just an abstract representation. $\endgroup$
    – MBaz
    Commented Mar 14, 2023 at 18:00
  • $\begingroup$ It's not a great pictures since the spectra don't match the time domain signal. I understand that can be confusing. $\endgroup$
    – Hilmar
    Commented Mar 14, 2023 at 18:12
  • $\begingroup$ Just for clarity: the triangle is NOT a spectrum; it's just supposed to show how frequencies above $f_s/2$ get aliased down to the Nyquist zone $[-f_s/2,f_s/2]$. $\endgroup$
    – Matt L.
    Commented Mar 14, 2023 at 19:37

1 Answer 1

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  • The "spectral band of interest" is just the term used here to define the base band (based on the sampling rate of $f_s = 6\,\texttt{KHz}$, so it extends from $-f_s/2$ to $f_s/2$ = $-3\,\texttt{KHz}$ to $3\,\texttt{KHz}$).
  • The triangular shape is just a convenient (in the author's mind, at least!) way to show the periodic and symmetric nature of the spectrum. The intention is to make it easier to visualize how the $7\,\texttt{KHz}$ frequency component aliases to $1\,\texttt{KHz}$ and the $4\,\texttt{KHz}$ component to $-2\,\texttt{KHz}$.

Note that it is not the actual spectrum:
The actual spectrum is comprised of the single points at $7,5,1$ and $-2\,\texttt{KHz}$

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