A signal can be viewed from two different standpoints; the time domain or the frequency domain.

The one which we are most used to is the time domain. This is like the trace on an oscilloscope where the vertical deflection is the signals amplitude, and the horizontal deflection is the time variable.

The second representation is the frequency domain. This is like the trace on a spectrum analyser, where the horizontal deflection is the frequency variable and the vertical deflection is the signals amplitude at that frequency.

Any given signal can be fully described in either of these domains, and we can convert between the two by using the Fourier Transform.

The frequency domain is a a beautiful mathematical landscape, nestled among the mappings of the complex plane. All manner of strange creatures may be found here, including abstract signals, samples, transfer functions, and spectra.

Traditionally, the prospective traveller arrives via a Fourier transform, which sadly confines them to the imaginary axis. Visitors wanting a more complete experience will use the Laplace transform. In modern times, the Z-transform allows one to make the journey discretely.

The traveller will quickly realise that the familiar rules and intuition of the time domain do not apply here. The residents do not concern themselves with time or other such derivative matters, and are instead concerned with the frequency and magnitude of events, and making sure everyone is on the same phase.

Navigation is difficult, as distances are deceptively logarithmic and the coordinates are complex, and often entirely imaginary. Transients do not survive long. However, the experienced traveller becomes used to these quirks, and even fond of the advantages of this mode of thinking. He no longer feels restricted to the real line, or even the stable left-hand side of the plane, and makes bold journeys far up the imaginary axis to the megahertz and gigahertz ranges, and even into the unstable regions of the right-hand plane. Some may even take up permanent residence, and become electrical engineers.

Places of interest are often marked with poles. Root loci wind down to the zeros, kept in check by the negative feedback of their transfer functions. Even if the gain is less than you anticipated, be sure to check the stability of the region with the detailed tables produced by Routh and Hurwitz. (Although your enthusiasm may become critically damped).

These days it has become fashionable to take a fast fourier transform and sample only a few locations around the unit circle, or bypassing the frequency domain completely and brute-forcing it with matrices in the time domain. But these methods miss something of the elegance of the classical methods: the winding route of the Nyquist diagram as it pirouettes around -1; the mesmerising curves and circles of the Smith chart; and the subtle intertwining of the magnitude and phase in a Bode plot.

As you take your inverse transform to return home, always remember to pick up your initial and final conditions, which, although superfluous in the frequency domain, are vital to successfully arriving in the time domain.

Node your homework. Preferably in an interesting way.

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