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I am reading the book

Signals and Systems Laboratory with Matlab Book by Alex Palamides and Anastasia Veloni

I was going through chapter 6 (Fourier transform) and I came across a confusing thing which is symbol of angular frequency Ω, which is upper case omega and in other books it is usually used for unit of resistance(ohm) while lower case omega ω is usually used for angular frequency. So here author/writer forgot this fact? enter image description here

lennon310
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DSP_CS
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1 Answers1

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Authors of signal processing books usually don't write about resistance, so the symbol $\Omega$ can be used for other purposes. One common purpose is to denote angular frequency in continuous time. In this way one can make a distinction between angular frequency in continuous time and normalized angular frequency in discrete time. I would guess that for the latter the authors use $\omega$. In that case we have

$$\begin{align}\Omega&=2\pi f\\\omega&=2\pi f/f_\mathrm{s}=2\pi fT\end{align}$$

where $f_\mathrm{s}$ is the sampling frequency, and $T=1/f_\mathrm{s}$ is the sampling interval. Furthermore,

$$\omega=\Omega T$$

A famous book which uses this notational convention is Oppenheim and Schafer's Discrete-Time Signal Processing. So the authors of the book you're reading are in good company.

robert bristow-johnson
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Matt L.
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  • In nutshell ,you mean in signal processing , Ω denotes angular frequency and ω denotes normalized angular frequency? – DSP_CS May 10 '23 at 09:54
  • @engr: Often that's the case, but not always, because authors are of course free to choose whatever notation they like. But whenever continuous time as well as discrete time are discussed in the same book, then this convention is pretty common. – Matt L. May 10 '23 at 10:01
  • @engr Yes that's pretty common. I personally tend to use the opposite convention, using $\Omega$ for normalized angular frequency, and $\omega$ for non-normalized. – Andy Walls May 10 '23 at 10:02
  • Something I might add here -- as a writer of a book or article you get to choose what symbols you use. There's no International Commission on Mathematical Symbols. You can even choose to use one set of symbols in Chapter 3, and a different one in Chapter 6 (and I've seen this done, to my confusion). This means that as a consumer of scientific literature, it's your job to understand (sometimes after significant frustrated puzzling) what the heck the author meant. I do try for consistency in my symbols when I'm writing, out of sympathy, but I don't always succeed. – TimWescott May 12 '23 at 19:44