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.

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.

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.

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

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.

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 :

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.

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

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

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

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):

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

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

Fourier Transformation of Discrete Systems Sampling

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 .

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.

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 :

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