Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering

Definition of the z -Transform

Overview on Transforms The Laplace transform of a function f(t) : The z -transform of a function x(k) : The Fourier-series of a function x(k) :

Example 1: a right sided sequence k x(k)x(k)... for, is For a signal

Example 2: a lowpass filter Suppose a lowpass filter law is where 1/3

Example 2: a lowpass filter 2/3 Rearranging the equation above, Signals Transfer function

Example 2: a lowpass filter 3/3 Signals Transfer function The block-diagram representation:

Example 3: a highpass filter A highpass filter follows: where 1/2 Transfer function

z -Transform Pairs Discrete-time domain signal z-domain signal 1/2

z -Transform Pairs 2/2 Discrete-time domain signal z-domain signal

Example 4: a decaying signal Suppose a signal is for. Find. for z -transform Inverse z -transform

Example 5: a signal in z -domain Suppose a signal is given in the z-domain: for z -transform Inverse z -transform From the z-transform table, The signal is equivalent to

Properties of the z-Transform

Linearity of z-Transform where a and b are any scalars.

Example 6: a signal in z-domain Suppose a signal is given in the z-domain: for z -transform Inverse z -transform Since the z-transform is a linear map, Arranging the right hand side,

Shift

Example 7: arbitrary signals z -transform Inverse z -transform Any signals can be represented in the z-domain: k y(k)y(k) 5 z -transform Inverse z -transform k y(k)y(k) 3 2 1

Discrete-Time Approximation Backward approximation Forward approximation Trapezoid approximation

Multiplication by an Exponential Sequence

Differentiation of X(z)

Complex Conjugation

Reversal

Initial Value Theorem

Convolution of Sequences 1/2

2/2 Convolution of Sequences Proof:

z-Transform of Linear Systems

Linear Time-Invariant System

N th -Order Difference Equation z-Transform

Stable and Causal Systems Re Im 1 The system G(z) is stable if all the roots (i.e., d i ) of the denominator are in the unit circle of the complex plane.

Stable and Causal Systems Re Im 1 The system G(z) is causal if the number of poles is greater than that of zeros (i.e., M N ).

Example 8: a non-causal filter Suppose a transfer function is given By applying the inverse z-Transform Therefore, the system is causal if

Example 9: open-loop controller Suppose the dynamic equation of a system is Approximating the dynamic equation by The transfer function from U(z) to Y(z) is 1/2

Example 9: open-loop controller A promising control algorithm is 2/2 However, the control algorithm is non-causal.

Frequency Response of H(z) The z-transform of a function x(k) : The Fourier-transform of a function x(k) : (Recall: Similarity of the z-Transform and Fourier Transform) The frequency response is obtained by setting where T is the sampling period.

Example 10: frequency response of a low pass filter Suppose a lowpass filter 1/2 By substituting for z, The magnitude is

2/2 Since, Example 10: frequency response of a low pass filter

IIR Filters and FIR Filters An IIR (Infinite Impulse Response) filter is A FIR (Finite Impulse Response) filter is