frequency domain

Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. Speaking non-technically, a time domain graph shows how a signal changes over time, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency domain representation also includes information on the phase shift that must be applied to each frequency in order to be able to recombine the frequency components to recover the original time signal. The frequency domain relates to the Fourier series by decomposing a signal into an infinite or finite number of frequencies.

Common transforms

The Z-transform and Laplace transform are two transforms that transform a signal in the time domain into the complex frequency domain.

The Laplace transform is used for continuous signals in Cartesian coordinates with $s=a + jb$
The Z-transform is used for discrete signals in circular coordinates with $z = r*e^{j\omega}$

The other well known transform is the Fourier transform and the discrete Fourier transform. These two transforms are subsets of the Laplace transform and Z-Transform, respectively.

Evaluating the Laplace transform on the imaginary axis (or $s=jb$ ) is equivalent to the Fourier transform
Evaluating the Z-transform on the unit circle (or $z=e^{j\omega}$ ) is equivalent to the discrete Fourier transform

The result of transforming a time domain signal into the frequency domain is commonly called the frequency spectrum of the signal. That is, it shows the spectral content of the signal.

Magnitude and phase

In using the Laplace, Z-, or Fourier transforms, the frequency spectrum is complex and describes the frequency magnitude and phase. In many applications, phase information is not important. By discarding the phase information it is possible to simplify the information in a frequency domain representation to generate a frequency power spectrum. A spectrum analyser is a device that displays the power spectrum.

Real-life frequency domain example

A biological system that operates in the frequency domain is the auditory system, in which the basilar membrane of the inner ear is able to perform a power spectrum decomposition of incoming sound waves. The result, is that we are able to hear a collection of different frequencies played together as a collection of separate notes, rather than simply a complicated noise.