# 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:

### Example

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 :

### Example

Find the Fourier integral representation of

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

Therefore, the integral representation is: