Chap. 3: Z Transform and Properties

Chapter 3 THE Z TRANSFORM 3.1 Introduction 3.2 Deﬁnition 3.3 Convergence Properties 3.4 The Z Transform as a Laurent Series 3.5 Inverse Z Transform 3.6 Theorems and Properties 3.7 Elementary Discrete-Time Signals

July 14, 2018

Frame # 1 Slide # 1 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Fourier transform will convert a real continuous-time t signal into a function of complex variable jω. Similarly, the z transform will convert a real discrete-time signal into a function of complex variable z.

Introduction

Frame # 2 Slide # 2 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Introduction

The Fourier series and Fourier transform can be used to t obtain spectral representations for periodic and nonperiodic continuous-time signals, respectively (see Chap. 2). Analogous spectral representations can be obtained for discrete-time signals by using the z transform. The Fourier transform will convert a real continuous-time t signal into a function of complex variable jω. Similarly, the z transform will convert a real discrete-time signal into a function of complex variable z.

Frame # 2 Slide # 3 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The availability of an inverse makes the z transform very t useful for the representation of digital ﬁlters and discrete-time systems in general.

Introduction Cont’d

Frame # 3 Slide # 4 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Introduction Cont’d

The z transform, like the Fourier transform, comes along with t an inverse transform, namely, the inverse z transform. Consequently, a discrete-time signal can be readily recovered from its z transform. The availability of an inverse makes the z transform very t useful for the representation of digital ﬁlters and discrete-time systems in general.

Frame # 3 Slide # 5 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Introduction Cont’d

The most basic representation of discrete-time systems is in t terms of diﬀerence equations (see Chap. 4) but through the use of the z transform, diﬀerence equations can be reduced to algebraic equations which are much easier to handle.

Frame # 4 Slide # 6 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Objectives

Deﬁnition of Z Transform t Convergence Properties t The Z Transform as a Laurent series t Inverse Z Transform t Theorems and Properties t Elementary Functions t Examples t

Frame # 5 Slide # 7 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Z Transform

Consider a bounded discrete-time signal x(nT ) that satisﬁes t the conditions

(i) x(nT ) = 0 for n < −N1

(ii) |x(nT )| ≤ K1 for − N1 ≤ n < N2 n (iii) |x(nT )| ≤ K2r for n ≥ N2

where N1 and N2 are positive integers and r is a positive constant.

Frame # 6 Slide # 8 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Z Transform Cont’d ··· (i) x(nT ) = 0 for n < −N1

(ii) |x(nT )| ≤ K1 for − N1 ≤ n < N2 n (iii) |x(nT )| ≤ K2r for n ≥ N2

x(nT)

n K2r

nT K N 1 N − 1 2 K rn − 2

(a)

Frame # 7 Slide # 9 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Z Transform Cont’d

The z transform of a discrete-time signal x(nT ) is deﬁned as

t ∞ X X (z) = x(nT )z−n n=−∞

for all z for which X (z) converges.

Frame # 8 Slide # 10 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 In eﬀect, z transforms can be represented by zero-pole plots. t

The Z Transform Cont’d

Although the z transform of a signal x(nT ) is an inﬁnite t series, in practice it can be represented in terms of a rational function as ∞ X X (z) = x(nT )z−n n=−∞ PM M−i QM N(z) i=0 ai z i=1(z − zi ) = = = H0 D(z) N PN N−i QN z + i=1 bi z i=1(z − pi )

where zi and pi are the zeros and poles of the z transform and H0 is a multiplier constant.

Frame # 9 Slide # 11 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Z Transform Cont’d

where zi and pi are the zeros and poles of the z transform and H0 is a multiplier constant. In eﬀect, z transforms can be represented by zero-pole plots. t

Frame # 9 Slide # 12 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Example

The following z transform has the zero-pole plot shown. (z2 − 4) (z − 2)(z + 2) X (z) = = z(z2 − 1)(z2 + 4) z(z − 1)(z + 1)(z − j2)(z + j2)

j Im z

z plane j2

2 1 1 2 Re z − −

j2 −

Frame # 10 Slide # 13 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.1 Absolute Convergence

(i) x(nT ) = 0 for n < −N1 (ii) |x(nT )| ≤ K1 for − N1 ≤ n < N2 n (iii) |x(nT )| ≤ K2r for n ≥ N2

where N1 and N2 are positive constants and r is the smallest positive constant that will satisfy condition (iii), then the z transform of x(nT ), i.e., ∞ X X (z) = x(nT )z−n n=−∞ exists and converges absolutely if and only if

r < |z| < R∞ with R∞ → ∞

Frame # 11 Slide # 14 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Absolute Convergence Cont’d

Region of x(nT) convergence z plane

n K2r R ∞ K 1 r

nT K 1 N N2 − 1 n K2r −

( a) ( b)

Frame # 12 Slide # 15 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Absolute Convergence Cont’d

The proofs of the Absolute Convergence Theorem and the theorems that follow can be found in the textbook.

Frame # 13 Slide # 16 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The z transform is given by t ∞ X X (z) = x(nT )z−n n=−∞ If we compare the above two series for X (z), we conclude that the z transform is a Laurent series of X (z) about the origin, i.e., a = 0, with an = x(nT )

The Z Transform as a Laurent Series

The Laurent series of a function X (z) about point z = a t assumes the form ∞ X −n X (z) = an(z − a) n=−∞ (See Appendix A.)

Frame # 14 Slide # 17 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Z Transform as a Laurent Series

The Laurent series of a function X (z) about point z = a t assumes the form ∞ X −n X (z) = an(z − a) n=−∞ (See Appendix A.) The z transform is given by t ∞ X X (z) = x(nT )z−n n=−∞ If we compare the above two series for X (z), we conclude that the z transform is a Laurent series of X (z) about the origin, i.e., a = 0, with an = x(nT )

Frame # 14 Slide # 18 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 The Z Transform as a Laurent Series Cont’d

Since the z transform is a speciﬁc Laurent series, it follows t that it inherits all the properties of the Laurent series, which are stated in the Laurent theorem as detailed in the slides that follow.

Frame # 15 Slide # 19 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Laurent Theorem

(a) If F (z) is an analytic and single-valued function on two concentric circles C1 and C2 with center a and in the annulus between them, then it can be represented by the Laurent series

∞ X −n F (z) = an(z − a) n=−∞

where I 1 n−1 an = F (z)(z − a) dz 2πj Γ The contour of integration Γ is a closed contour in the counterclockwise sense lying in the annulus between circles C1 and C2 and encircling the inner circle.

Frame # 16 Slide # 20 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Laurent Theorem Cont’d

z plane

C Ŵ 2

C1 a

(a)

Frame # 17 Slide # 21 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Laurent Theorem Cont’d

(b) The Laurent series converges and represents F (z) in the open annulus obtained by continuously increasing the radius of C2 and decreasing the radius of C1 until each of C1 and C2 reaches a point where F (z) is singular.

z plane

C1 Ŵ a

(b)

Frame # 18 Slide # 22 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Laurent Theorem Cont’d

(c) A function F (z) can have several, possibly many, annuli of convergence about a given point z = a and for each one a Laurent series can be obtained. z plane

III II I a

(c)

Frame # 19 Slide # 23 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Laurent Theorem Cont’d

(d) The Laurent series for a given annulus of convergence is unique.

z plane

III II I a

(c)

Frame # 20 Slide # 24 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Example

The function represented by the zero-pole plot at the left has three unique Laurent series as shown at the right.

z plane jIm z z plane

j2 III

I a 2 1 1 2 Re z − −

j2 −

(a)

Frame # 21 Slide # 25 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Inverse Z Transform

The absolute-convergence theorem states that the z t transform, X (z), of a discrete-time signal x(nT ) satisfying the conditions

(i) x(nT ) = 0 for n < −N1

(ii) |x(nT )| ≤ K1 for − N1 ≤ n < N2 n (iii) |x(nT )| ≤ K2r for n ≥ N2

exists and converges absolutely if and only if

r < |z| < R with R → ∞

Frame # 22 Slide # 26 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 One of these series converges in the outer annulus (i.e., the t largest one) which is deﬁned as

R0 < |z| < R with R → ∞

where R0 is the radius of a circle passing through the most distant pole of X (z) from the origin.

Inverse Z Transform Cont’d

The Laurent theorem states that a function X (z) has as many t distinct Laurent series about the origin as there are annuli of convergence.

Frame # 23 Slide # 27 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Inverse Z Transform Cont’d

The Laurent theorem states that a function X (z) has as many t distinct Laurent series about the origin as there are annuli of convergence.

One of these series converges in the outer annulus (i.e., the t largest one) which is deﬁned as

R0 < |z| < R with R → ∞

where R0 is the radius of a circle passing through the most distant pole of X (z) from the origin.

Frame # 23 Slide # 28 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 From the Laurent theorem, there is a unique Laurent series of t X (z) that converges in the outer annulus of convergence

R0 < |z| < R with R → ∞

Therefore, the z transform of x(nT ) is the unique Laurent t series that converges in the outer annulus and, furthermore, r = R0.

Inverse Z Transform Cont’d

Summarizing: From the absolute convergence theorem, the z transform t converges in the annulus

r < |z| < R with R → ∞

Frame # 24 Slide # 29 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Therefore, the z transform of x(nT ) is the unique Laurent t series that converges in the outer annulus and, furthermore, r = R0.

Inverse Z Transform Cont’d

Summarizing: From the absolute convergence theorem, the z transform t converges in the annulus

r < |z| < R with R → ∞

From the Laurent theorem, there is a unique Laurent series of t X (z) that converges in the outer annulus of convergence

R0 < |z| < R with R → ∞

Frame # 24 Slide # 30 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Inverse Z Transform Cont’d

Summarizing: From the absolute convergence theorem, the z transform t converges in the annulus

r < |z| < R with R → ∞

From the Laurent theorem, there is a unique Laurent series of t X (z) that converges in the outer annulus of convergence

R0 < |z| < R with R → ∞

Therefore, the z transform of x(nT ) is the unique Laurent t series that converges in the outer annulus and, furthermore, r = R0.

Frame # 24 Slide # 31 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 From the Laurent theorem, we have t 1 I x(nT ) = X (z)zn−1 dz 2πj Γ where contour Γ encloses all the poles of X (z)zn−1. In DSP, this contour integral is said to be the inverse z t transform of X (z).

Inverse Z Transform Cont’d

We conclude that signal x(nT ) can be obtained from its z t transform X (z) by ﬁnding the coeﬃcients of the Laurent series of X (z) that converges in the outer annulus.

Frame # 25 Slide # 32 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 In DSP, this contour integral is said to be the inverse z t transform of X (z).

Inverse Z Transform Cont’d

We conclude that signal x(nT ) can be obtained from its z t transform X (z) by ﬁnding the coeﬃcients of the Laurent series of X (z) that converges in the outer annulus. From the Laurent theorem, we have t 1 I x(nT ) = X (z)zn−1 dz 2πj Γ where contour Γ encloses all the poles of X (z)zn−1.

Frame # 25 Slide # 33 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Inverse Z Transform Cont’d

Frame # 25 Slide # 34 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Notation

Like the Fourier transform and its inverse, the z transform and t its inverse are often represented in terms of operator notation as X (z) = Zx(nT ) and x(nT ) = Z−1X (z) respectively.

Frame # 26 Slide # 35 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 In these theorems t Zx(nT ) = X (z) Zx1(nT ) = X1(z) Zx2(nT ) = X2(z)

and a, b, w, and K represent constants which may be complex.

Z Transform Theorems

The general properties of the z transform can be described in t terms of a small number of theorems, as detailed in the slides that follow.

Frame # 27 Slide # 36 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Z Transform Theorems

The general properties of the z transform can be described in t terms of a small number of theorems, as detailed in the slides that follow. In these theorems t Zx(nT ) = X (z) Zx1(nT ) = X1(z) Zx2(nT ) = X2(z)

and a, b, w, and K represent constants which may be complex.

Frame # 27 Slide # 37 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Similarly, the inverse z transform of a linear combination of z t transforms is given by

−1 Z [aX1(z) + bX2(z)] = ax1(nT ) + bx2(nT )

Theorem 3.3 Linearity

The z transform of a linear combination of discrete-time t signals is given by

Z[ax1(nT ) + bx2(nT )] = aX1(z) + bX2(z)

Frame # 28 Slide # 38 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.3 Linearity

The z transform of a linear combination of discrete-time t signals is given by

Z[ax1(nT ) + bx2(nT )] = aX1(z) + bX2(z)

Similarly, the inverse z transform of a linear combination of z t transforms is given by

−1 Z [aX1(z) + bX2(z)] = ax1(nT ) + bx2(nT )

Frame # 28 Slide # 39 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.4 Time Shifting

For any positive or negative integer m, t Zx(nT + mT ) = zmX (z)

In eﬀect, multiplying the z transform of a signal by a negative or positive power of z will cause the signal to be delayed or advanced by mT s.

Frame # 29 Slide # 40 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.5 Complex Scale Change

For an arbitrary real or complex constant w t Z[w −nx(nT )] = X (wz)

Evidently, multiplying a discrete-time signal by w −n is equivalent to replacing z by wz in its z transform. Similarly, multiplying a discrete-time signal by v n is equivalent to replacing z by z/v in its z transform.

Frame # 30 Slide # 41 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.6 Complex Diﬀerentiation

The z transform of an arbitrary signal nT1x(nT ) is given by t dX (z) Z[nT x(nT )] = −T z 1 1 dz Complex diﬀerentiation provides a simple way of obtaining the z transform of a discrete-time signal that can be expressed as a product nT1x(nT ).

Frame # 31 Slide # 42 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.7 Real Convolution

The z transform of the real convolution summation of two t signals x1(kT ) and x2(nT ) is given by

∞ ∞ X X Z x1(kT )x2(nT − kT ) = Z x1(nT − kT )x2(kT ) k=−∞ k=−∞

= X1(z)X2(z)

The real convolution summation is used frequently for the representation of digital ﬁlters and discrete-time systems (see Chap. 4).

Frame # 32 Slide # 43 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.8 Initial-Value Theorem

The initial value of a signal x(nT ) represented by a z t transform of the form N(z) PM a zM−i X (z) = = i=0 i D(z) PN N−i i=0 bi z occurs at KT = (N − M)T and its value at nT = KT is given by

x(KT ) = lim [zK X (z)] z→∞

Frame # 33 Slide # 44 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.8 Initial-Value Theorem Cont’d ··· N(z) PM a zM−i X (z) = = i=0 i D(z) PN N−i i=0 bi z

Corollary: If the degree of the numerator polynomial, N(z), in t a z transform is equal to or less than the degree of the denominator polynomial D(z), then we have

x(nT ) = 0 for n < 0

i.e., the signal is right-sided.

Frame # 34 Slide # 45 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.9 Final-Value Theorem

The value of x(nT ) as n → ∞ is given by t x(∞) = lim [(z − 1)X (z)] z→1 The ﬁnal-value theorem can be used to determine the steady-state response of a discrete-time system.

Frame # 35 Slide # 46 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.10 Complex Convolution

If the z transforms of two discrete-time signals x1(nT ) and t x2(nT ) are available, then the z transform of their product, X3(z), can be obtained as I 1 z −1 X3(z) = Z[x1(nT )x2(nT )] = X1(v)X2 v dv 2πj Γ1 v I 1 z −1 = X1 X2(v)v dv 2πj Γ2 v

where Γ1 (orΓ2) is a contour in the common region of convergence of X1(v) and X2(z/v) (or X1(z/v) and X2(v)).

Frame # 36 Slide # 47 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 It is also the basis of the window method for the design of t nonrecursive digital ﬁlters (see Chap. 9).

Theorem 3.10 Complex Convolution Cont’d

The complex convolution theorem can be used to obtain the z t transform of a product of discrete-time signals whose z transforms are available.

Frame # 37 Slide # 48 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.10 Complex Convolution Cont’d

The complex convolution theorem can be used to obtain the z t transform of a product of discrete-time signals whose z transforms are available.

It is also the basis of the window method for the design of t nonrecursive digital ﬁlters (see Chap. 9).

Frame # 37 Slide # 49 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Parseval’s formula is often used to solve a problem known as t scaling which is associated with the design of recursive digital ﬁlters in hardware form (see Chap. 14).

Theorem 3.11 Parseval’s Discrete-Time Formula

If X (z) is the z transform of a discrete-time signal x(nT ),

t then ∞ X 1 Z ωs |x(nT )|2 = |X (ejωT )|2 dω ω n=−∞ s 0

where ωs = 2π/T .

Frame # 38 Slide # 50 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.11 Parseval’s Discrete-Time Formula

If X (z) is the z transform of a discrete-time signal x(nT ),

t then ∞ X 1 Z ωs |x(nT )|2 = |X (ejωT )|2 dω ω n=−∞ s 0

where ωs = 2π/T . Parseval’s formula is often used to solve a problem known as t scaling which is associated with the design of recursive digital ﬁlters in hardware form (see Chap. 14).

Frame # 38 Slide # 51 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Theorem 3.11 Parseval’s Discrete-Time Formula Cont’d

If T is normalized to 1 s, Parseval’s formula simpliﬁes to:

t ∞ X 1 Z 2π |x(nT )|2 = |X (ejωT )|2 dω 2π n=−∞ 0

Frame # 39 Slide # 52 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Elementary Discrete-Time Signals

Function Deﬁnition ( 1 for n = 0 Unit impulse δ(nT ) = 0 for n 6= 0 ( 1 for n ≥ 0 Unit step u(nT ) = 0 for n < 0 ( nT for n ≥ 0 Unit ramp r(nT ) = 0 for n < 0

Exponential u(nT )e αnT , (α > 0)

Exponential u(nT )e αnT , (α < 0)

Sinusoid u(nT ) sin ωnT

Frame # 40 Slide # 53 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Elementary Discrete-Time Signals Cont’d

1.0 1.0

0 nT 0 nT (a) (b)

1.0

nT 0 nT (c) (d)

1.0

0 nT 0 nT

(e) ( f ) (a) Unit impulse, (b) unit step, (c) unit ramp, (d) increasing exponential (e) decreasing exponential, (c) sinusoid. Frame # 41 Slide # 54 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Examples

Find the z transforms of the following signals: (a) unit-impulse δ(nT ) (b) unit-step u(nT ) (c) delayed unit-step u(nT − kT )K (d) signal u(nT )Kw n (e) exponential signal u(nT )e−αnT (f ) unit-ramp r(nT ) (g) sinusoidal signal u(nT ) sin ωnT

Frame # 42 Slide # 55 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 (b) As in part (a)

Zu(nT ) = u(0) + u(T )z−1 + u(2T )z−2 + ··· = 1 + z−1 + z−2 + ··· = (1 − z−1)−1 z = z − 1 (c) From the time-shifting theorem (Theorem 3.4) and part (b), we have Kz−(k−1) Z[u(nT − kT )K] = Kz−k Zu(nT ) = z − 1

Examples Cont’d

Solutions (a) From the deﬁnitions of the z transform and δ(nT ), we have Zδ(nT ) = δ(0) + δ(T )z−1 + δ(2T )z−2 + ··· = 1

Frame # 43 Slide # 56 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 (c) From the time-shifting theorem (Theorem 3.4) and part (b), we have Kz−(k−1) Z[u(nT − kT )K] = Kz−k Zu(nT ) = z − 1

Examples Cont’d

Solutions (a) From the deﬁnitions of the z transform and δ(nT ), we have Zδ(nT ) = δ(0) + δ(T )z−1 + δ(2T )z−2 + ··· = 1

(b) As in part (a)

Zu(nT ) = u(0) + u(T )z−1 + u(2T )z−2 + ··· = 1 + z−1 + z−2 + ··· = (1 − z−1)−1 z = z − 1

Frame # 43 Slide # 57 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Examples Cont’d

Solutions (a) From the deﬁnitions of the z transform and δ(nT ), we have Zδ(nT ) = δ(0) + δ(T )z−1 + δ(2T )z−2 + ··· = 1

(b) As in part (a)

Frame # 43 Slide # 58 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 (e) By letting K = 1 and w = e−αT in part (d), we obtain z Z[u(nT )e−αnT ] = z − e−αT

Examples Cont’d

(d) From the complex-scale-change theorem (Theorem 3.5) and part (b), we get " # 1 −n Z[u(nT )Kw n] = KZ u(nT ) w

Kz = KZu(nT )| = z→z/w z − w

Frame # 44 Slide # 59 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Examples Cont’d

(d) From the complex-scale-change theorem (Theorem 3.5) and part (b), we get " # 1 −n Z[u(nT )Kw n] = KZ u(nT ) w

Kz = KZu(nT )| = z→z/w z − w

(e) By letting K = 1 and w = e−αT in part (d), we obtain z Z[u(nT )e−αnT ] = z − e−αT

Frame # 44 Slide # 60 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Examples Cont’d

(f ) From the complex-diﬀerentiation theorem (Theorem 3.6) and part (b), we have

d Zr(nT ) = Z[nTu(nT )] = −Tz [Zu(nT )] dz d z Tz = −Tz = dz (z − 1) (z − 1)2

Frame # 45 Slide # 61 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Examples Cont’d

(g) From part (e), we deduce

u(nT ) Z[u(nT ) sin ωnT ] = Z ejωnT − e−jωnT 2j 1 1 = Z[u(nT )ejωnT ] − Zu(nT )e−jωnT 2j 2j 1 z z = − 2j z − ejωT z − e−jωT z sin ωT = z2 − 2z cos ωT + 1

Frame # 46 Slide # 62 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Standard Z Transforms

x(nT ) X (z) δ(nT ) 1 z u(nT ) z − 1 Kz−(k−1) u(nT − kT )K z − 1 Kz u(nT )Kw n z − w K(z/w)−(k−1) u(nT − kT )Kw n−1 z − w z u(nT )e−αnT z − e−αT Tz r(nT ) (z − 1)2

Frame # 47 Slide # 63 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 Standard Z Transforms Cont’d

x(nT ) X (z) Te−αT z r(nT )e−αnT (z − e−αT )2 z sin ωT u(nT ) sin ωnT z2 − 2z cos ωT + 1 z(z − cos ωT ) u(nT ) cos ωnT z2 − 2z cos ωT + 1 ze−αT sin ωT u(nT )e−αnT sin ωnT z2 − 2ze−αT cos ωT + e−2αT z(z − e−αT cos ωT ) u(nT )e−αnT cos ωnT z2 − 2ze−αT cos ωT + e−2αT

Frame # 48 Slide # 64 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7 This slide concludes the presentation. Thank you for your attention.

Frame # 49 Slide # 65 A. Antoniou Digital Signal Processing – Secs. 3.1-3.7