Bài giảng Signals and Systems - Chapter: The z-transform - Đặng Quang Hiếu

ET2060 - Signals and Systems
The z-transform
Dr. Quang Hieu Dang
Hanoi University of Science and Technology
School of Electronics and Telecommunications
Autumn 2012
Definition of z-transform
n z
z
z−1
x [n]
z←−→ X (z)
where z is a complex variable, z = re jω, and
X (z) =
∞∑
x=−∞
x [n]z−n
Examples: Find z-transform of
(a) x [n] = δ[n]
(b) x [n] = u[n]
Relationship to Fourier transform
◮ Fourier transform is the z-transform evaluated in the unit
circle z = e jω.
X (e jω) = X (z)|z=e jω
◮ z-transform is the Fourier transform of x [n]r−n
X (z) =
∞∑
n=−∞
x [n](re jω)−n = FT{x [n]r−n}
◮ Region of convergence (ROC) of X (z) is the set of points in
the complex plane (z-plane) for which the z-transform
summation converges (or the Fourier transform of x [n]r−n
exists!).
∞∑
n=−∞
|x [n]r−n|dt <∞
Examples
Find z-transform and draw its ROC for following sequences:
(a) x [n] = 2δ[n − 2] + δ[n]− 3δ[n + 1]
(b) x [n] = anu[n]
(c) x [n] = −anu[−n − 1]
(d) x [n] = 2nu[n]− 3nu[−n− 1]
(e) x [n] = cos(ω0n)u[n]
X (z) rational. Zeros and poles
X (z) =
N(z)
D(z)
=
b0 + b1z + · · ·+ bMzM
a0 + a1z + · · · + aNzN
◮ Zeros z0r : X (z0r ) = 0 → are roots of N(z)
◮ Poles zpk : X (zpk) =∞ → are roots of D(z)
Examples: Given x [n] = anrectN [n].
(a) Find its z-transform and ROC.
(b) Find zeros, poles and draw them in the z-plane.
Properties of ROC
(i) If X (z) converges when z = z0 then it converges for all
|z | = |z0|. Therefore, ROC generally has a ring shape:
r1 < |z | < r2.
(ii) ROC does not include poles
(iii) If x [n] is finite length, the ROC will be the entire z-plane
(may exclude 0 and/or ∞).
(iv) If x [n] is a left / right-sided signal, ROC?
(v) If x [n] is a two-sided signal, ROC?
(vi) If X (z) is a rational function with poles zpk , ROC?
The inverse z-transform
Apply inverse Fourier transform:
x [n]r−n =
1
2π
∫
2π
X (re jω)e jωndω
We have:
x [n] =
1
2πj
∮
C
X (z)zn−1dz
where C is a counterclockwise closed path encircling the origin and
entirely in the ROC.
Properties
◮ Linearity
◮ Time-shifting: x [n − n0] z←−→ z−n0X (z)
◮ Scaling in z-domain: anx [n]
z←−→ X (z/a)
◮ Time reversal: x [−n] z←−→ X (1/z)
◮ Complex conjugate: x∗[n] z←−→ X ∗(z∗)
◮ Convolution: x1[n] ∗ x2[n] z←−→ X1(z)X2(z)
◮ Differentiation in z-domain: nx [n]
z←−→ −z dX (z)dz
◮ Initial value theorem: If the signal is causal (x [n] = 0, ∀n < 0)
then
x [0] = lim
z→∞X (z)
◮ Correlation, multiplication?
Inverse z-transform: Power series expansions
Given X (z) and its ROC, expand X (z) into power series
X (z) =
∞∑
n=−∞
cnz
−n
which satisfies the ROC condition. Consequently, x [n] = cn, ∀n.
If X (z) is a rational fucntion, use long-division!
Example: Find inverse z-transform of
X (z) =
1 + 2z−1
1− 2z−1 + z−2
when
(a) x [n] causal
(b) x [n] anti-causal
Partial-fraction expansions (1)
X (z) =
N(z)
D(z)
=
b0 + b1z + · · ·+ bMzM
a0 + a1z + · · · + aNzN
When M < N, expand X (z) as
X (z) =
N∑
k=1
Ak
z − zpk
where zpk are simple poles of X (z) and
Ak = (z − zpk)X (z)
∣∣
z=zpk
When M ≥ N, X (z) = G (z) + N′(z)D(z) where M ′ < N.
Example: Given
X (z) =
1
1− 1.5z−1 + 0.5z−2
Find x[n]?
Partial-fraction expansions (2)
A pole zpk of multiplicity ℓ, X (z) is expanded as:
A1k
z − zpk +
A2k
(z − zpk)2 + · · ·+
Aℓk
(z − zpk)ℓ
◮ How to calculate Aik?
◮ Inverse z-transform of 1(z−zpk)m ?
Example: Find inverse z-transform of
X (z) =
z
(z − 12 )2(z − 1)
Complex poles? Self study!
Transfer function H(z) of a discrete-time LTI system
x [n] y [n]h[n]
y [n] = x [n] ∗ h[n]
Apply z-transform to both sides, use convolution property:
H(z) =
Y (z)
X (z)
X (z) Y (z)H(z)
Transfer function (2)
An LTI system represented by
y [n] = −
N∑
k=1
aky [n − k] +
M∑
r=0
brx [n − r ]
Apply z-transform to both sides
H(z) =
∑M
r=0 brz
−r
1 +
∑N
k=1 akz
−k
→ Pole-zero system.
Causality and stability
◮ Causal: ROC{H(z)} outside a circle.
◮ Stable: ROC{H(z)} include unit circle (z = e jω).
◮ H(z) rational, causal, stable: All poles of H(z) are within the
unit circle.
Transfer function and system’s diagram
Find the constant-coefficient difference equation for the LTI system
described in the following diagram
X (z) Y (z)b
b
z−1
z−1
−1
−2
2 3
b
z−1
0.5
−1
Unilateral z-transform
X+(z) = ZT+{x [n]} =
∞∑
n=0
x [n]z−n
Properties are similar to the bilateral case, except:
◮ Time shifting
ZT+{x [n − k]} = z−k [X+(z) +
k∑
n=1
x [−n]zn], k > 0
ZT+{x [n + k]} = z−k [X+(z)−
k−1∑
n=0
x [n]z−n], k > 0
◮ Final-value theorem
lim
n→∞ x [n] = limz→1
(z − 1)X+(z)
Solution to constant coefficient difference equation
Example: Solve the difference equation (find y [n], n ≥ 0):
y [n]− 3y [n − 1] + 2y [n − 2] = x [n]
where input is x [n] = 3n−2, with inital conditions:
y [−2] = −4
9
, y [−1] = −1
3