Z-Transformation - I

Digital signal processing: Lecture 5

z-transformation - I

• Fourier transform & inverse Fourier transform: – Time domain & Frequency domain representations • Understand the “true face” (latent factor) of some physical phenomenon. – Key points: • Definition of Fourier transformation. • Definition of inverse Fourier transformation. • Condition for a signal to have FT.

• The sampling theorem – If the highest frequency component contained in an

analog signal is f0, the sampling frequency (the Nyquist rate) should be 2xf0 at least.

• Frequency response of an LTI system: – The Fourier transformation of the impulse response – Physical meaning: frequency components to keep (~1) and those to remove (~0). – Theoretic foundation: Convolution theorem.

• The z-transformation. • z－変換 • Convergence region of z- • z－変換の収束領域 transform. －変換とフーリエ変換 • Relation between z- • z transform and Fourier • z－変換と差分方程式 transform. • 伝達関数 or システム • Relation between z- 関数 transform and difference equation. • Transfer function (system function).

Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/4 z-transformation（z-変換） • Laplace transformation is important for analyzing analog signal/systems. • z-transformation is important for analyzing discrete signal/systems. • For a given signal x(n), its z-transformation is defined by

∞ Z[x(n)] = X (z) = ∑ x(n)z−n (6.3) n=0

where z is a complex variable.

• The z-transform defined above is called one- sided z-transform (片側ｚ-変換）. • One-sided z-transform is meaningful for causal signals (因果的信号). • For non-causal signals, the summation should start from minus infinity, and the z-transform so defined is called two-sided z-transform. • We study only one-sided z-transform here.

Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/6 Convergence region 収束領域 • Fourier transform exists (converges) if and only if the signal is absolutely summable. • z-transform exists for some value of z. • The region in which the z-transform exists is called the convergence region. • z-transformation is a more powerful tool for analyzing signals (see Fig. 6.1 in p. 89).

Z[δ (n)] = δ (0) + δ (1)z−1 + ... + δ (n)z−n + ... = 1, for any z

−1 −n Z[u0 (n)] = u0 (0) + u0 (1)z + ... + u0 (n)z + ...

− − 1 z = 1+ z 1 + ... + z n + ... = = , for | z |> 1 1− z−1 z −1

Z[α n ] = 1+ α1z−1 + ... + α n z−n + ... 1 z = = , for | z |> α 1−αz−1 z −α

• Fourier transform is the z- z-plane j transform on the unit circle. • The values of the z-transform on the unit circle equals to those of the Fourier transform -1 0 1 of the same signal.

-j

jω = ω X (e ) X (z) |z=e j or |z|=1 (6.16)

• It is important to specify the convergence region in the results. • If we use the right table, we can find the z-transform of more complicated signals.

r(n) is the ramp signal

Linearity: z - transform is a linear transform ∀a,b

if Z[x1(n)] = X1(z)

and Z[x2 (n)] = X 2 (z)

then Z[ax1(n) + bx2 (n)] = aX1(z) + bX 2 (z)

Translation : if x(n) = 0 for n < 0 and Z[x(n)] = X (z) then m−1 Z[x(n + m)] = z m[X (z) − ∑ x(n)z −n ] n=0 Z[x(n − m)] = z −m X (z) The second equation is especially useful for system analysis, as will seen later

Given : y(n) = −a1 y(n −1) − a2 y(n − 2) + b0 x(n) + b1x(n −1) + b2 x(n − 2)

From linearity and translation property of z - transform, we have

−1 −2 −1 −2 Y (z) = −a1z Y (z) − a2 z Y (z) + b0 X (z) + b1z X (z) + b2 z X (z)

−1 −2 b0 + b1z + b2 z Y (z) = −1 −2 X (z) (6.21) 1+ a1z + a2 z

• The purpose of this example is to show how to find the z-transform of the system output when the system is given as a difference equation. • The linearity and the translation properties of z- transform are used here. • It is interesting to see another way to relate the input and the output of a system. • Actually, z-transform is one way for solving the constant coefficient difference equations.

• Signal multiplied by an exponential signal – If X(z) is the z-transformation of x(n), then Z[anx(n)]=X(a-1z) • Example 6.5 z−1 sinω Z[sinωn ⋅u (n)] = 0 1− 2z−1 cosω + z−2

z − ( ) 1 sinω n α Z[α sinωn ⋅u0 (n)] = z − z − 1− 2( ) 1 cosω + ( ) 2 α α

• Differentiation of X(z) – If X(z) is the z-transformation of x(n), then dX (z) Z[nx(n)] = −z dz • Example 6.6

d 1 z−1 Z[r(n)] = Z[nu (n)] = −z [ ] = 0 dz 1− z−1 (1− z−1)2

• Convolution theorem Matlab6.3 (revised) for z-transformation syms a n x z; h=a^n; H=ztrans(h,n,z) y(n) = x1(n)* x2 (n) x=1; ⇒ = X=ztrans(x,n,z) Y (z) X1(z)X 2 (z) HX=H*X y=symsum(h*x,0,n) Y=ztrans(y,n,z) • See Example 6.7 and simplify(Y) == Example 6.8. simplify(HX)

• Initial value

x(0) = limz→∞ X (z)

• Final value

−1 x(∞) = limn→∞ x(n) = limz→1(1− z )X (z)

• See example 6.9.

Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/18 The System function of an LTI system (or transfer function 伝達関数） system input : x(n) ⇒ X (z) system output : y(n) ⇒ Y (z) impulse response : h(n) ⇒ H (z) time domain : y(n) = x(n)*h(n) z - plane : Y(z) = X (z)H (z)

• H(z)=Y(z)/X(z) is called the transfer function or system function of the system (p. 102). • H(z) is the z-transform of h(n).

(2) (2) Suppose that the difference equation is given by (T=1とする）

y(n) = x(n) - a1y(n-1)

Find the transfer function of the corresponding digital filter. （hint: see Example 6.4)

(3) Confirm Example 6.8 using the program (with revisions) given in p. 96 of the textbook.

• Quizzes: T1 – For a given signal x(n), what is the 1.2 one-sided z-transformation of 1 x(n)? 0.8

0.6

T5 0.4 T2

– What is the relation between z- 0.2

transform and Fourier transform? 0

– What is the transfer function (伝達関数） of a digital filter?

T4 T3

Name: Student ID: .