Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Frequency Response
Fundamental property of linear shift-invariant systems:
Steady-state response to a sinusoidal input is
sinusoidal of the same frequency as the input,
Amplitude and Phase determined by system.
Slide 1
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Frequency Response
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Input sequence of the form:
The output is identical to the input with a complex
multiplier H(e jω)
H(e jω) is called the frequency response of the
system: gives the transmission of the system for
every value of ω.
Slide 2
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Frequency Response
Example: Calculate the
frequency response of
the following FIR filter if
h(k)=1/4 k=0,1,2,3
y(k) = h(0)u(k)+h(1)u(k1)+h(2)u(k-2)+h(3)u(k3)
Slide 3
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Frequency Response Properties
Frequency response is a periodic function of ω (2π)
Since H(ejω) is periodic only 2π length in needed.
Generally the interval 0<ω<2π is used.
Real h(n)
most common case
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Magnitude of H(ejω) is symmetric over 2π
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Phase of H(ejω) is antisymmetric over 2π
Only the interval 0<ω<π is needed.
Slide 4
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Fourier Transform of Discrete Signals
Fourier Transform of discrete time signal:
The series does not always converge.
Example: x(n) unit step, real exponential sequence
There is convergence if:
The frequency response of a stable system will always
converge
Inverse of the frequency response- impulse response:
Slide 5
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Fourier Transform of Discrete Signals
Example: Calculate the impulse response, of an ideal
low-pass filter, if the frequency response is:
The system is not causal and unstable
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This system can not be implemented.
Slide 6
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Introduction to Digital Filters
Filters: A system that selectively changes the waveshape,
amplitude-frequency, phase-frequency characteristics of a
signal
Digital Filters: Digital Input – Digital Output
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Linear Phase-Τhe frequency response has the form:
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α: real number, A(ejω): real function of ω
Phase:
α) Low-Pass
c) Band-Pass
b) High-Pass
d) Band-Stop
Slide 7
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Units of Frequency
Express frequency response in terms frequency units
involving sampling interval T. Equations are:
H(ejωΤ) is periodic in ω with period 2π/Τ
ω: radians per second
Replace ω with 2πf, frequency f: hertz
Example: Sampling frequency f=10 KHz, T= 100 μs
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H(ej2πfΤ) is periodic in f with period 10 KHz
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H(ejωΤ) is periodic in ω with period 20000π rad/sec
Slide 8
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Real-time signal processing
Input Filter: Analogue, to bandlimit analogue input
signal x(t) – no aliasing
ADC: Converts x(t) into digital x(n)
build-in sample and hold circuit
Digital Processor: microprocessor – Motorola MC68000
or DSP – Texas Instrument TMS320C25
 The Bandlimited signal is sampled
Analog  Discrete time continuous amplitude signal
 Amplitude is quantized into 2B levels (B-bits)
 Discrete Amplitude is encoded into B-bits words.
Slide 9
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Sampling
Digital signal x(nT) produced by sampling analog x(t)
x(n) = xa(nTs)
Ts(sampling rate) = 1/Fs (sampling frequency)
Initially, x(n) is multiplied (modulated) with a
summation of delayed unit-impulse yields the
discrete time continuous amplitude signal xs(t):
Slide 10
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Sampling
Fourier transform relations for x(t):
Discrete–time signal transform relations are:
The relationship between the two transforms is:
Sum of infinite number of components of the frequency
response of the analog waveform
Slide 11
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Sampling
If analog frequency is bandlimited:
Then:
Digital frequency response is related in a
straightforward manner to analog frequency response
Slide 12
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Sampling
Slide 13
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
Sampling
The shifting of information from one band of
frequency to another is called aliasing.
It is controlled by the sampling rate 1/T
How high the sampling frequency should be?
Sampling Theorem
Shannon Theorem: If x(t) has fmax as its highest
frequency, and x(t) is periodically sampled so that
T<1/2 fmax then x(t) can be reconstructed
fmax Nyquist frequency
In order to reduce the effects of aliasing anti-aliasing
filters are used to bandlimit x(t)
They depend on fmax
Slide 14
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
DAC
The basic DAC accepts parallel digital data
Produces analog output using zero order hold
The ideal DAC should have an ideal low-pass filter
The system is not causal and unstable
Slide 15
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
FOURIER TRANSFORMATION
OF DISCRETE SYSTEMS
DAC
Since it is impossible to implement an ideal lowpass
filter zero order hold is used instead.
Its impulse response is:
The frequency response is:
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