EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Frequency Response of Discrete-Time Systems

Complex Exponentials Two-sided complex exponential z n when input into LTI systems Output will be same complex exponential weighted by H(z) Provided that z is in region of convergence for H(z) When we specialize the z-domain to frequency domain, the magnitude of H(z) will control which frequencies are attenuated or passed

Frequency Response for LTI Systems Continuous time Discrete time Real-valued impulse response: H(e -j ω ) = H*(e j ω ) Input Output

Response to Sampled Sinusoids Start with a continuous-time sinusoid Sample it every T s seconds (see slide 7-6) We show discrete-time sinusoid with Resulting in Discrete-time frequency is equal to continuous- time frequency multiplied by sampling period

Example Calculate the frequency response of the system given as a difference equation as Assuming zero initial conditions we take the z- transform of both sides Since pole is inside unit circle,

Example Group real and imaginary parts Absolute value (magnitude response) Find angle (phase response)

Magnitude and Phase Responses Output of system for input cos(   k) is simply Magnitude response Phase response

Discrete-time Frequency Response As in previous example, frequency response of a discrete-time system is periodic with 2  Why? Frequency response is function of complex exponential which has period of 2  : Absolute value of discrete-time frequency response is even and angle is odd symmetric Discrete-time sinusoid is symmetric around 

Aliasing and Sampling Rate Continuous-time sinusoid can have a frequency from 0 to infinity By sampling a continuous-time sinusoid, Discrete-time frequency  unique from 0 to  We only can represent frequencies up to half of the sampling frequency. Higher frequencies exist would be “wrapped” to some other frequency in the range.

Re Im Effect of Poles and Zeros of H(z) The z-transform of a difference equation can be written in general form as Complex number as vector in complex plane z and z i are both complex numbers Their difference is also a complex number (vector in complex plane)

Re Im x x oo Effect of Poles and Zeros of H(z) Each difference term in H(z) may be represented as a complex number in polar form Magnitude is distance of pole/zero to chosen point (frequency) on unit circle Angle is angle of vector with respect to horizontal axis

Digital Filter Design Poles near unit circle indicate filter’s passband(s) Zeros on/near unit circle indicate stopband(s) Biquad with zeros z 0 and z 1, and poles p 0 and p 1 Transfer function Magnitude response Distance from point on unit circle e j  and pole location p 0 |a – b| is distance between complex numbers a and b

Digital Filter Design Examples Transfer function Poles (X) & zeros (O) in conjugate symmetric pairs For coefficients in unfactored transfer function to be real Filters below have what magnitude responses? Re(z) Im(z) X O O X Re(z) Im(z) O O X X Re(z) Im(z) O O X X lowpass highpass bandpass bandstop allpass notch? Poles have radius r Zeros have radius 1/r Zeros are on the unit circle

DSP First Demonstrations IIR Filters (Chapter 8) Three-domain demonstrations IIR filter with one pole IIR filter with one pole and one zero Radial movement of poles (second-order section) Z-to-Freq demonstration Figure 4: Movie takes points on unit circle to create magnitude response (double click on plot to start animation) Last figure: Movie takes slice of 3-D view of the z- domain to create magnitude response (double click on plot to start animation)

DSP First Demonstrations BIBO stable causal systems All poles inside unit circle Filter amplitude response at  Enhance it by placing a pole close to e j  Suppress it by placing a zero close to or at e j  Poles/zeros at origin Do not affect amplitude response Add a phase of (–  T ) which is a pure delay Poles and zeros can cancel each other’s effect if placed close together

Effect of Poles/Zeros x --  -- -  /2  -- x o x o x x x o    