Digital Signal Processing

1. 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 propertyof linear shift-invariant systems: • Steady-state response to a sinusoidalinputis sinusoidalof the samefrequencyas the input, Amplitudeand Phasedetermined by system.

3. Fourier Transformation of Discrete Systems Frequency Response • Input sequence of the form : • The outputis identicalto the inputwith 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 frequencyresponse 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 ofω(2π) • SinceH(ejω) isperiodic, only2πlengthisneeded. • Generallytheinterval0<ω<2πisused. • Real h(n), most common case • Magnitude of H(ejω) issymmetric over2π • PhaseofH(ejω) isantisymmetric over2π • Only the interval0<ω<πisneeded.

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 convergenceif : • The frequencyresponseof a stablesystem will alwaysconverge. • Inverseof the frequencyresponse– impulseresponse:

7. Fourier Transformation of Discrete Systems Fourier Transform of Discrete Signals • Example : • Calculate the impulse response, of a ideal low-pass filter, ifthe frequency response is: • The system isnot causal andunstable • This systemcan not beimplemented.

8. Fourier Transformation of Discrete Systems Introduction to Digital Filters • Filters : A system that selectively changes thewaveshape, amplitude-frequency, phase-frequency characteristicsof a signal • Digital Filters : Digital Input – Digital Output • Linear Phase – Thefrequencyresponsehastheform: • α :realnumber, A(ejω) : Realfunctionofω • Phase : • Low - Pass High - Pass • Band - Pass Band - Stop

9. Fourier Transformation of Discrete Systems Units of Frequency • Expressfrequency response intermsoffrequencyunitsinvolvingsampling interval T. • Equationsare: • H(ejω) isperiodic inωwithperiod2π/Τ • ω :radianspersecond • Replaceω with2πf, frequencyf : hertz • Example : Sampling frequency f = 10kHz, T = 100μs • H(ejω) isperiodic inf withperiod10kHz • H(ejω) isperiodic inωwithperiod20000π rad/sec

10. Fourier Transformation of Discrete Systems Real-time Signal Processing • Input Filter : AnaloguetoBandlimit Analogue inputsignalx(t) – no aliasing • ADC : Convertsx(t) intodigitalx(n) built-insample 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) producedbysampling analog x(t) • x(n) = xa(nTs) • Ts (samplingrate) = 1/Fs (samplingfrequency) • Initially, x(n) is multiplied (modulated) withasummation of delayed unit-impulse yieldsthediscrete 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 isrelatedinastraightforwardmannerto 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 fmaxthen 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 analogoutput using zeroorderhold. • The idealDAC should have an ideallow-passfilter. • The system is notcausal 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