Transcript Introduction to Digital Signal Processing

Lecture 2:
Z-Transform
XILIANG LUO
2014/9
Fourier Transform
Convergence
 A sufficient condition: absolutely summable
 it can be shown the DTFT of absolutely summable sequence converge
uniformly to a continuous function
Square Summable
 A sequence is square summable if:
∞
𝑥[𝑛]
𝑛=−∞
2
<∞
 For square summable sequence, we have mean-square convergence:
Z-Transform
a function of the complex variable: z
If we replace the complex variable z by 𝑒 𝑗𝜔 , we have the Fourier Transform!
Z-Transform &
Fourier Transform
Complex z-plane
Region of Convergence
 The set of z for which the z-transform converges is called ROC of the ztransform.
 Absolutely summable criterion:
ROC
 ROC consists of a ring in the z-plane
Closed-Form in ROC
 When X(z) is a rational function inside ROC, i.e.
 P(z), Q(z) are polynomials in z
 Zeros: values of z such that X(z) = 0
 Poles: values of z such that X(z) = infinity
Z-Transform Example:
Right-Sided
Z-Transform Example:
Left-Sided
Diff. Sum, Same Z-Transform?
 One is right-sided exponential sequence
 One is left-sided exponential sequence
 But they share the same algebraic expressions for their Z-Transforms
 This emphasizes the importance of the region of convergence!!
ROC Properties
ROC Properties
ROC Properties
Inverse z-Transform
 From the z-Transform, we can recover the original sequence using the
following complex contour integral:
1
𝑥𝑛 =
2𝜋𝑗
𝑋 𝑧 𝑧 𝑛−1 𝑑𝑧
𝐶
C is a closed contour within the ROC of the z-transform
Inverse z-Transform Methods
 Inspection
 familiar with the common transform pairs
 Partial Fraction Expansion
 Power Series Expansion
z-Transform Properties
 1. Linearity
 2. Time Shifting
 3. Multiplication by an Exponential Sequence
z-Transform Properties
 4. Differentiation of X(z)
 5. Conjugation of a Complex Sequence
 7. Time Reversal
z-Transform Properties
 7. Convolution of Sequences
z-Transform and LTI Systems
 LTI system is characterized by its impulse response h[n]
x[n]
h[n]
y[n]
𝑦 𝑛 = 𝑥 𝑛 ⋆ ℎ[𝑛]
𝑌 𝑧 = 𝑋 𝑧 × 𝐻(𝑧)
H(z) is called the system function of this LTI system!
Cauchy-Riemann Equations
 If function f(z) is differentiable at z0=x0+y0, then its component
functions must satisfy the following conditions:
𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖𝑣(𝑥, 𝑦)
𝜕𝑢 𝜕𝑣
=
𝜕𝑥 𝜕𝑦
𝜕𝑢
𝜕𝑣
=−
𝜕𝑦
𝜕𝑥
Analytic Functions
 A function f(z) is analytic at a point z0 if it has a derivative at each
point in some neighborhood of z0.
 So, If f(z) is analytic at a point z0, it must be analytic at each point in
some neighborhood of z0.
Taylor Series
 Theorem: Suppose that a function f is analytic throughout a disk: |zz0|<R0, centered at z0 and with radius R0, then f(z) has the power
series representation:
+∞
𝑓 𝑧 =
𝑎𝑛 𝑧 − 𝑧0
𝑛=0
𝑎𝑛 =
𝑓
𝑛
(𝑧0 )
𝑛!
𝑛
𝑧 − 𝑧0 < 𝑅0
Laurent Series
 If a function is not analytic at a point z0, one cannot apply Taylor’s
theorem at that point!
 Laurent’s Theorem: Suppose a function f is analytic throughout an
annular domain centered at z0:
𝑅1 < 𝑧 − 𝑧0 < 𝑅2
Let C denote any positively oriented simple closed contour around z0
and lying in the domain, then, at each point in the domain, f(z) has the
series representation:
+∞
𝑓 𝑧 =
𝑐𝑛 𝑧 − 𝑧0
𝑛=−∞
𝑛
Laurent Series
+∞
𝑓 𝑧 =
𝑐𝑛 𝑧 − 𝑧0
𝑛
𝑛=−∞
1
𝑐𝑛 =
2𝜋𝑖
𝐶
𝑓 𝑧 𝑑𝑧
𝑧 − 𝑧0 𝑛+1
Homework Problems
3.52:
3.56:
3.57:
3.59:
Next
 Sampling of Continuous-Time Signals
 Please read the textbook Chapter 4 in advance!