Properties of continuous Fourier Transforms Fourier Transform Notation
- Slides: 34
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 period 1. If the period (T) of a periodic signal increases, then: 1. 2. 3. the fundamental frequency (ωo = 2π/T) becomes smaller and the frequency spectrum becomes more dense 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
construct a new periodic signal f. T(t) from f(t) 1. 2. Suppose we are given a non-periodic signal f(t). In order to applying Fourier series to the signal f(t), we construct a new periodic signal f. T(t) with period T. The original signal f(t) can be obtained back
The periodic function f. T(t) can be represented by an exponential 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 functions 1. – 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 area 2. A delta t has A units
Graphic Representations of Impulse Arrow used to avoid functions drawing magnitude of impulse functions
Using delta functions The 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 F(sin
Fourier. Sinusoidal Transforms. Signals of Sinusoidal Signals • Which illustrates the last formula from the last slide (for sinus)
Periodicof. Signal Fourier Transforms a Periodic Signal
Some properties of the Fourier Transform • Linearity
Some properties of the Fourier Transform DUALITY Spectral domain Time domain
Coordinate scaling Spectral domain Time 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 2 D data
- Integral of unit step function
- Rect(t-1/2)
- Fourier transform properties solved examples
- Sin(2pift)
- Fourier transform in image processing
- Equation of fourier transform
- Ctfs fourier
- Future tense
- Present continuous past continuous future continuous
- Fourier transform
- Fourier transform of dirac delta
- Short time fourier transform applications
- Fourier series
- Parseval's identity for fourier transform
- Synthesis equation fourier series
- Phase meaning
- Fourier
- Fourier transform of ramp function
- Frequency
- Inverse fourier transform of unit step function
- Gaussian fourier transform
- Fourier
- The fourier transform and its applications
- Inverse fourier transform
- Fourier transform of an integral
- Short time fourier transform
- Polar fft
- Fourier transform of product of two functions
- Comb function fourier transform
- Sinc fourier transform
- Impulse train fourier transform
- Discrete fourier transform
- Overlap save method
- Circ function fourier transform
- Fourier transform duality examples