Fourier Series and Transforms

Fourier Series

Fourier series provide a way of representing any periodic function as a sum of trigonometric functions. For a periodic function with period , the Fourier series is given by:

Where the coefficients and are called the Fourier coefficients, integrals calculated over the period of the function:

Note that if the function is even , then the term is always 0, and the series is comprised of cosine terms only:

Likewise for odd functions , the term is always zero, and the series is comprised of sine terms only:

The Fourier series uniquely represents a function if:

  • The integral of function over its period is finite
  • The function has a finite number of discontinuities over any finite interval
  • Most (if not all) functions/signals of any engineering interest will satisfy these conditions

Exponential Representation

The Fourier series can be rewritten using Euler's formula :

Note that T = 2L, the period of the function.

Frequency Spectrum Representation

The spectrum representation gives the magnitude and phase of the harmonic components defined by the frequencies contained in a signal

This gives two spectra:

  • The frequency spectrum, describing the magnitude for each frequency present in the signal
  • The phase spectrum, describing the phase for each frequency present in the signal

The diagram below shows the frequency spectrum for the functions and , respectively:


Find the fourier series of the following function:

is an odd function with period (), hence we only need the integral:

Since :

Can introduce a new index , such that :

The Fourier series for is therefore given by:

Fourier Transforms

Fourier series give a representation of periodic signals, but non periodic signals can not be analysed in the same way. The Fourier transform works by replacing a sum of discrete sinusoids with a continuous integral of sinusoids over frequency, transforming from the time domain to the frequency domain. A non-periodic function can be expressed as:

Provided that:

  • and are piecewise continuous in every finite interval
  • exists

This can also be expressed in complex notation:

  • is the Fourier transform of , denoted
  • is the inverse Fourier transform of , denoted

For periodic signals:

  • Fourier series break a signal down into components with discrete frequencies
    • Amplitude and phase of components can be calculated from coefficients
    • Plots of amplitude and phase against frequency give frequency spectrum of a signal
    • The spectrum is discrete for periodic signals

For non-periodic signals:

  • Fourier Transforms represent a signal as a continuous integral over a range of frequencies
    • The frequency spectrum of the signal is continuous rather than discrete
    • gives the spectrum amplitude
    • gives the spectrum phase

Fourier Transform Properties

Fourier transforms have linearity, same as z and Laplace.

Time Shift

For any constant :

If the original function is shifted in time by a constant amount, this does not affect the magnitude of its frequency spectrum . Since the complex exponential always has a magnitude of 1, the time delay alters the phase of but not its magnitude.

Frequency Shift

For any constant :


Find the Fourier integral representation of

This is the Fourier transform of . Using Euler's relation :

Therefore, the integral representation is: