1. Digital Signal Processing
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Intelligent Systems Laboratory
Digital Signal Processing
Prof. George Papadourakis, Ph.D.

2. 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.

3. Fourier Transformation of Discrete Systems
Frequency Response
Input sequence of the form :
The output is identical to the input with a complex multiplier H(ejω)
H(ejω) is called the frequency response of the system : Gives the transmission of the system for every value of ω.

4. Introduction to Neural Networks
Fourier Transformation of Discrete Systems
Frequency Response
Example :
Calculate the frequency response of the following FIR filter if h(k) = ¼, k = 0,1,2,3

5. 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 is needed.
Generally the interval 0<ω<2π is used.
Real h(n), most common case
Magnitude of H(ejω) is symmetric over 2π
Phase of H(ejω) is antisymmetric over 2π
Only the interval 0<ω<π is needed.

6. 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 :

7. Fourier Transformation of Discrete Systems
Fourier Transform of Discrete Signals
Example :
Calculate the impulse response, of a ideal low-pass filter, if the frequency response is :
The system is not causal and unstable
This system can not be implemented.

8. 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
Linear Phase – The frequency response has the form :
α : real number, A(ejω) : Real function of ω
Phase :
Low - Pass High - Pass
Band - Pass Band - Stop

9. Fourier Transformation of Discrete Systems
Units of Frequency
Express frequency response in terms of frequency units involving sampling interval T.
Equations are :
H(ejω) is periodic in ω with period 2π/Τ
ω : radians per second
Replace ω with 2πf, frequency f : hertz
Example : Sampling frequency f = 10kHz, T = 100μs
H(ejω) is periodic in f with period 10kHz
H(ejω) is periodic in ω with period 20000π rad/sec

10. 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) built-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

11. 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):

12. 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

13. 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

14. Fourier Transformation of Discrete Systems
Sampling

15. 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 should the sampling frequency be?
Sampling 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 .

16. Fourier Transformation of Discrete Systems
DAC
The basic DAC accepts parallel digital data. It produces analog output using zero order hold.
The ideal DAC should have an ideal low-pass filter.
The system is not causal and unstable.

17. Fourier Transformation of Discrete Systems
DAC
Since it is impossible to implement an ideal low-pass filter, zero order hold is used instead.
Its impulse response is:
The frequency response is :

18. Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Intelligent Systems Laboratory