# How to distinguish between the different frequency domains?

Sometimes the terms 'Fourier domain', 'complex frequency domain', 'Frequency domain' and 's domain' are used interchangeably. Take these answers here for example.

Can you really use them interchangeably all the time, without being technically wrong? So, could you describe what would be wrong if I would replace 's domain' by 'fourier domain' for example? Or replacing 'complex frequency domain' by 'frequency domain'?

Hopefully whoever you are talking to would actually define the terms they were using before they started slinging them around. I've heard some of these used in multiple contexts. Wikipedia has an article that tries to encompass some of the different things people might be talking about when they use the term frequency domain.

The domain we are referring to here is the well-defined mathematical term for the set of legal inputs to a function. There are multiple kinds of transforms (Laplace, Z, various kinds of Fourier transforms). The domains we are referring to are the (mathematical) domains of the function that results from the transform.

I tend to think of the s-domain as being associated with the Laplace transform: $F(s) = \mathcal{L}[f] = \int_0^{\infty} e^{-st}f(t) \mathrm{d} t$. The function $F(s)$ is potentially defined for the entire continuous complex plane, so the domain is the complex plane. (Although for most functions $f$ there would be a region of convergence.) I haven't personally heard the term s-domain in any other context. For the Z transform you might hear "Z-domain" (which is also the entire continuous complex plane).

I think if I heard a scientist use the term Fourier domain my initial assumption would be that they were (probably) talking about the Fourier decomposition of a periodic function into a Fourier series. (If I heard an engineer use the term Fourier domain I would think they were probably talking about a discrete Fourier transform, see below.) Most people who use Fourier series in their daily work use the exponential Fourier series: $c_n = F(n) = \mathcal{F}[f] = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\mathrm{d}x$. So the domain here (the subscript $n$ of $c_n$) is the integers. (If I were talking to a student I would assume they were talking about the sine/cosine version, which is also defined across the integers (but inconveniently with two different series across the non-negative integers (once for sines and once for the cosines))).

For an engineer, and especially a computer engineer like me, I would assume that we were talking about the discrete Fourier transform. This applies to periodic functions that are band limited (i.e., have very little energy in frequencies above some value.) In that case the time (or space)-domain signal can be sampled at a finite rate and the transform is: $X_k = \mathcal{F}[\{x_n\}] = \sum_{n=0}^{N-1}x_n e^{-2\pi i k n / N}$. Here again the domain (the subscript $k$ of $X_k$) is the integers, and in this case just a finite sequence of integers $\{0, N-1\}$.

Finally, if I was talking to a scientist or mathematician and they used the term Fourier domain, my initial assumption that they were talking about a Fourier decomposition might be wrong. They might instead be talking about a continuous Fourier transform: $F(\omega) = \mathcal{F}[f] = \int_{-\infty}^{\infty}f(t)e^{-2\pi i \omega t} \mathrm{d} t$. The domain of the function $F$ is the entire real line. Note that there is some similarity with the Laplace transform but with the Fourier transform we are typically only looking at $s=2\pi i \omega$, the complex unit circle, (although sometimes it might be useful to look at the rest of the plane), and for the Laplace transform we are only looking at positive time.

Complex frequency domain could refer to the continuous Fourier transform, I suppose, but I would tend to think of it as referring to the same thing as the s-domain.

All these transforms are related, but in any conversation you need to be clear whether the time domain is continuous or discrete and whether the time domain function/signal repeats or not. You should also make clear whether the time domain function's range is real or complex. Then in the frequency domain you might be talking about complex frequencies (like in the Laplace transform or Z transform) or real frequencies, and, if real frequencies, continuous or discrete. Sometimes you can infer from context, but sometimes not. The range in the frequency domain is more often the complexes rather than the reals.

You should start every conversation with a boring list of definitions.

• Thanks. But how would you answer a general public question like: "Are there multiple frequency domains? Why do you need multiple frequency domains in the first place? What makes them different from each other?" How would you answer that? Apr 23 '13 at 19:49
• Are these comments obsolete now you edited?
– Raphael
Apr 24 '13 at 6:27
• Mine were (I deleted them). I hope @user1095332 agrees, but I don't know if they've yet had a chance to confirm that my edits answer their question. Apr 24 '13 at 11:16