Z-Transform and Fourier Transform Insights
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:
For square summable sequence, we have mean-square convergence:
Z-Transform
a function of the complex variable: z
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 z-
transform.
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:
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]
h[n]
x[n]
y[n]
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:
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: |z-
z0|<R0, centered at z0 and with radius R0, then f(z) has the power
series representation:
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:
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:
Laurent Series
Homework Problems
3.59:
3.57:
3.52:
3.56:
Next
Sampling of Continuous-Time Signals
Please read the textbook Chapter 4 in advance!
Delve into the world of Z-Transform and Fourier Transform, exploring convergence, square summable sequences, complex Z-plane, region of convergence, and more. Understand the distinction between right-sided and left-sided exponential sequences and their shared properties in Z-Transforms.
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Presentation Transcript
Lecture 2: Z-Transform XILIANG LUO 2014/9
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: ?[?]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 ???, we have the Fourier Transform!
Z-Transform & Fourier Transform
Region of Convergence The set of z for which the z-transform converges is called ROC of the z- transform. 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!!
Inverse z-Transform From the z-Transform, we can recover the original sequence using the following complex contour integral: 1 ? ? ?? 1?? 2?? ? ? = ? 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] y[n] x[n] h[n] ? ? = ? ? [?] ? ? = ? ? ?(?) 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: ? ? = ? ?,? + ??(?,?) ?? ??=?? ?? ?? ??= ?? ??
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: |z- z0|<R0, centered at z0 and with radius R0, then f(z) has the power series representation: + ? ? ?0 < ?0 ? ? = ??? ?0 ?=0 ??=??(?0) ?!
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: ?1< ? ?0 < ?2 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: + ? ? ? = ??? ?0 ?=
Laurent Series + ? ? ? = ??? ?0 ?= 1 ? ? ?? ? ?0 ??= 2?? ?+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!
