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

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Definition of the z -Transform

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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) :

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

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

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

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

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

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

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

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

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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

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Properties of the z-Transform

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

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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,

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Shift

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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

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Discrete-Time Approximation Backward approximation Forward approximation Trapezoid approximation

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Multiplication by an Exponential Sequence

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Differentiation of X(z)

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Complex Conjugation

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Reversal

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Initial Value Theorem

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Convolution of Sequences 1/2

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2/2 Convolution of Sequences Proof:

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z-Transform of Linear Systems

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Linear Time-Invariant System

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N th -Order Difference Equation z-Transform

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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.

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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 ).

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

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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

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

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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.

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

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

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

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