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