Properties of continuous

Fourier Transforms

Fourier Transform Notation

For periodic signal

Fourier Transform can be used for BOTH time and frequency domains

For non-periodic signal

FFT for infinite period

Example: FFT for infinite period1. If the period (T) of a periodic signal

increases,then:1. the fundamental frequency (ωo = 2π/T) becomes

smaller and2. the frequency spectrum becomes more dense 3. while the amplitude of each frequency component

decreases.

2. The shape of the spectrum, however, remains unchanged with varying T.

3. Now, we will consider a signal with period approaching infinity.

Shown on examples earlier

1. Suppose we are given a non-periodic signal f(t). 2. In order to applying Fourier series to the signal f(t), we construct a new periodic

signal fT(t) with period T.

construct a new periodic signal fT(t) from f(t)

The original signal f(t) can be obtained back

The periodic function fT(t) can be represented by anexponential Fourier series.

period

Now we integrate from –T/2 to +T/2

How the frequency spectrum in the previous formula becomes continuous

Infinite sums become integrals…

Fourier for infinite period

Notations for the transform pair

• Finite or infinite period

Singularity functions

Singularity functions1. – Singularity functions is a particular class of

functions which are useful in signal analysis.

2. – They are mathematical idealization and, strictly speaking, do not occur in physical systems.

3. – Good approximation to certain limiting condition in physical systems.

4. For example, a very narrow pulse

Singularity functions – impulse function

t 0

Properties of Impulse functions

1. Delta t has unit area2. A delta t has A units

Graphic Representations of Impulse functionsArrow used to avoid drawing magnitude of impulse functions

Using delta functionsThe integral of the unit impulse function is the unit step function

The unit impulse function is the derivative of the unit step function

Spectral Density Function F()

Spectral Density Function F()Input function

Existence of the Fourier transform for physical systems

• We may ignore the question of the existence of the Fourier transform of a time function when it is an accurately specified description of a physically realizable signal.

• In other words, physical realizability is a sufficient condition for the existence of a Fourier transform.

Parseval’s Theorem for Energy Signals

Parseval’s Theorem for Energy Signals

Example of using Parseval Theorem

Fourier Transforms of some signals

Fourier Transforms of some signals

Fourier Transforms and Inverse FT of some signals

Fourier Transforms of Sinusoidal Signals

F(sin

F

• Which illustrates the last formula from the last slide (for sinus)

Sinusoidal SignalsFourier Transforms of Sinusoidal Signals

Periodic SignalFourier Transforms of a Periodic Signal

Some properties of the Fourier Transform

•Linearity

Some properties of the Fourier Transform

DUALITY

Time domain

Spectral domain

Coordinate scaling

Time domain

Spectral domain

Time shifting. Transforms of delayed signals

• Add negative phase to each frequency component!

Frequency shifting (Modulation)

Differentiation and Integration

• These properties have applications in signal processing (sound, speech) and also in image processing, when translated to 2D data