18791 Lecture #8 INTRODUCTION TO THE ZTRANSFORM Department of Electrical and Computer Engineering...

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18791 Lecture #8INTRODUCTION TO THE ZTRANSFORM
Department of Electrical and Computer EngineeringCarnegie Mellon University
Pittsburgh, Pennsylvania 15213
Phone: +1 (412) 2682535FAX: +1 (412) 2683890
[email protected]://www.ece.cmu.edu/~rms
September 22, 2005
Richard M. Stern
CarnegieMellon
Slide 2 ECE Department
Introduction
Last week we discussed the discretetime Fourier transform (DTFT) at length
This week we will begin our discussion of the Ztransform (ZT)
– ZT can be thought of as a generalization of the DTFT
– ZT is more complex than DTFT (both literally and figuratively), but provides a great deal of insight into system design and behavior
So today we will:
– Define ZTs and their regions of convergence (ROC)
– Provide insight into the relationships between frequency using ZT and DTFT relationships
– Discuss relations between unit sample response and shape of ROC
CarnegieMellon
Slide 3 ECE Department
The DiscreteTime Fourier Transform (DTFT) and the Ztransform (ZT)
The first equation aserts that we can represent any time function x[n] by a linear combination of complex exponentials
The second equation tells us how to compute the complex weighting factors
In going from the DTFT to the ZT we replace by
x[n]=12π
X(ejω)−ππ∫ ejωdω X(ejω)= x[n]e−jω
n=−∞
∞∑
ejωn=cos(ωn)+jsin(ωn)
X(ejω)
ejωn zn
CarnegieMellon
Slide 4 ECE Department
Generalizing the frequency variable
In going from the DTFT to the ZT we replace by
can be thought of as a generalization of
For an arbitrary z, using polar notation we obtain so
If both and are real, then can be thought of as a complex exponential (i.e. sines and cosines) with a real temporal envelope that can be either exponentially decaying or expanding
ejωn zn
zn ejωn
z=ρejω
znzn=ρnejωn
CarnegieMellon
Slide 5 ECE Department
Definition of the Ztransform
Recall that the DTFT is
Since we are replacing (generalizing) the complex exponential building blocks by , a reasonable extension of
would be
Again, think of this as building up the time function by a weighted sums of functions instead of
X(ejω)= x[n]e−jω
n=−∞
∞∑
ejωn zn X(ejω)
X(z)= x[n]z−n
n=−∞
∞∑
zn ejωn
CarnegieMellon
Slide 6 ECE Department
Computing the Ztransform: an example
Example 1: Consider the time function x[n]=αnu[n]
X(z)= x[n]z−n= αnz−n= (αz−1)n
n=0
∞∑
n=0
∞∑
n=−∞
∞∑
=1
1−αz−1=
zz−α
CarnegieMellon
Slide 7 ECE Department
Another example …
Example 2: Now consider the time function
Let
Then,
x[n]=−αnu[−n−1]
X(z)= x[n]z−n= −αnz−n= (αz−1)n
n=−∞
−1∑
n=−∞
−1∑
n=−∞
∞∑
l=−n;n=−∞⇒ l =∞;n=−1⇒ l=1
(αz−1)n= −(zα−1)l =1−l=1
∞∑
n=−∞
−1∑ (zα−1)l =1−
1
1−zα−1=
1
1−αz−1l=0
∞∑
CarnegieMellon
Slide 8 ECE Department
The importance of the region of convergence
Did you notice that the Ztransforms were identical for Examples 1 and 2 even though the time functions were different? Yes, indeed, very different time functions can have the same Ztransform! What’s missing in this characterization? The region of convergence (ROC).
In Example 1, the sum converges only for
In Example 2, the sum converges only for
So in general, we must specify not only the Ztransform corresponding to a time function, but its ROC as well.
X(z)= αnz−n
n=0
∞∑ z>α
X(z)= αnz−n
n=−∞
−1∑ α >z
CarnegieMellon
Slide 9 ECE Department
What shapes are ROCs for Ztransforms?
In Example 1, the ROC was We can represent this graphically as:
z>α 
CarnegieMellon
Slide 10 ECE Department
What shapes are ROCs for Ztransforms?
In Example 2, the ROC was We can represent this graphically as:
(ROC is
shaded
area)
z<α 
CarnegieMellon
Slide 11 ECE Department
General form of ROCs
In general, there are four types of ROCs for Ztransforms, and they depend on the type of the corresponding time functions
Four types of time functions:
– Rightsided
– Leftsided
– “Both”sided (infinite duration)
– Finite duration
CarnegieMellon
Slide 12 ECE Department
Rightsided time functions
Rightsided time functions are of the form
(as in Example 1). ROCs are of the form
Comment: All causal LSI systems have unit sample responses that are rightsided, although not all rightsided sample responses correspond to causal systems.
x[n]=0,n<n0
α <z
CarnegieMellon
Slide 13 ECE Department
Leftsided time functions
Leftsided time functions are of the form
(as in Example 2). ROCs are of the form except that it is
possible that they
don’t include
x[n]=0,n>n0
z<α 
z=0
CarnegieMellon
Slide 14 ECE Department
“Both”sided (infiniteduration) time functions
Rightsided time functions are of the form for all n
(as in ). ROCs are of the form , an
annulus bounded by and , exclusive.
x[n]≠0
x[n]=α n,α <1 α <z<β
CarnegieMellon
Slide 15 ECE Department
An example of a “bothsided” time function
Consider the function with
Using the results of Examples 1 and 2, we note that
The ROC is , which is the region of “overlap” of the ROCs of the ztransforms of the two terms of the time function taken individually.
αnu[n]−βnu[−n−1] α <β
X(z)=−1
1−βz−1−
1
1−αz−1=
z(α−β)(z−α)(z−α)
α <z<β
CarnegieMellon
Slide 16 ECE Department
Finiteduration time functions
Finiteduration time functions are of the form
ROCs include the entire zplane except possibly
x[n]≠0, n1<n<n2
z=0
CarnegieMellon
Slide 17 ECE Department
Stability and the ROC
It can be shown that an LSI system is stable if the ROC includes the unit circle (UC), which is the locus of points for which
Comment: this is exactly the same condition that is required for the DTFT to exist
z=1
X(ejω)
CarnegieMellon
Slide 18 ECE Department
Causality, stability and the ROC
Recall that for a system to be causal the sample response must be rightsided, and the ROC must be the outside of some circle.
Hence, for a system to be both causal and stable, the ROC must be the outside of a circle that is inside the UC.
In other words, if an LSI system is both causal and stable, the ROC will be of the form with z>α α <1
CarnegieMellon
Slide 19 ECE Department
The inverse Ztransform
Did you notice that we didn’t talk about inverse ztransforms yet?
It can be shown (see the text) that the inverse ztransform can be formally expressed as
Comments:
– Unlike the DTFT, this integral is over a complex variable, z and we need complex residue calculus to evaluate it formally
– The contour of integration, c, is a circle around the origin that lies inside the ROC
– We will never need to actually evaluate this integral in this course … we’ll discuss workaround techniques in the next class
x[n]=12πj
X(z)zn−1dzc∫
CarnegieMellon
Slide 20 ECE Department
Comparing the Ztransform with the LaPlace transform
Ztransforms:
The Ztransform uses
as the basic building block
The DTFT exists if the ROC of the Ztransform includes the unit circle
The DTFT equals the Ztransform evaluated along the unit circle,
Causal and stable LSI systems have ROCs that are the outside of some circle that is to the inside of the unit circle
LaPlace transforms:
The LaPlace transform uses
as the basic building block
The CTFT exists of the ROC of the LaPlace transform includes the jaxis,
The CTFT equals the LaPlace transform evaluated along the jaxis,
Causal and stable LTI systems have ROCs that are righthalf planes bounded by a vertical line to the left of the jaxis
zn=ρejωn=ρnejωn est=e(σ+jΩ)t =eσtejΩt
z=ejω
s=jΩ
CarnegieMellon
Slide 21 ECE Department
Mapping the splane to the zplane
Map (i.e. warp conformally) the splane into the zplane:
Comments:
– jaxis in splane maps to unit circle in zplane
– Right half of splane maps to outside of zplane
– Left half of zplane maps to inside of splane
CarnegieMellon
Slide 22 ECE Department
Summary  Intro to the ztransform
The ztransform is based on a generalization of the frequency representation used for the DTFT
Different time functions may have the same ztransforms; the ROC is needed as well
The ROC is bounded by one or more circles in the zplane centered at its origin
The shape of the ROC depends on whether the time function is rightsided, leftsided, infinite in duration, or finite duration
An LSI system is stable if the ROC includes the unit circle
The inverse ztransform can only be evaluated using complex contour integration
The zplane can be considered (in some ways) as a conformal mapping of the splane