EE 422G Notes: Chapter 8 Instructor: Zhang Chapter 8 Discrete-Time Signals and Systems 8-1 Introduction Most “real” signals and natural (physical) processes: continuous – time A: System Design Problem How the computer sees “the rest”? an equivalent (Physical Process + Sensor +A/D + D/A)=> discrete-time system The Equivalent discrete-time system Modeled by a discrete-time model Page 8-1
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EE 422G Notes: Chapter 8 Instructor: Zhang
Chapter 8 Discrete-Time Signals and Systems
8-1 Introduction
Most “real” signals and natural (physical) processes: continuous – time
A: System Design Problem
How the computer sees “the rest”? an equivalent(Physical Process + Sensor +A/D + D/A)=> discrete-time system
The Equivalent discrete-time system Modeled by a discrete-time model
System Design (Design of the computer control Algorithm):Based on discrete-time model description. Needs for discrete-time system analysis and design tool: Z-Transform (Similar position as Laplace Transform for continuous-time system.)
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EE 422G Notes: Chapter 8 Instructor: Zhang
B. How does the computer understand the progress and behaviors of the process being monitored and controlled? By sampling the output of the Continuous-time system! => How can we ensure that the sampled signal is a sufficient representation of its continuous-time origin. i.e., how fast we have to sample?
A question we must answer before z-transform based analysis!
C. Two basic parts of the chapter Part one: Theoretical frame work for determining how fast we have to sample. Part two: z-transform
Part one: How fast
8.2A Analog-to-Digital Conversion
1. Sample OperationNeeds to Know: (1) Sampling period: T (2) X (t) is sampled at t=nT (3) What do we mean by x (n) (4) Sampling function: p (t)
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EE 422G Notes: Chapter 8 Instructor: Zhang
(5) Sampled signal xs = x (t) p (t)
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EE 422G Notes: Chapter 8 Instructor: Zhang
2. Mathematical Description of Sampling Process
Sampled signal : xs(t) = x(t)p(t)
Objective: Derivation of xs(t)’s Fourier Series Expression (Time Domain)
Derivation :
Sampling function: A Periodical function, (thus can be expressed using Fourier series), with
period T on fundamental frequency
With Fourier series coefficients:
3. Spectrum of sampled signal
Objective: Find the spectrum of the sampled signal xs(t).
Derivation :
Take Fourier Transform for
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Fourier Series Description of Sampling Function
General Equation for any periodical signal
general equation for Fourier coefficient of any periodical signal
general equation: always true for any r (width of the sampling pulse).
For n
EE 422G Notes: Chapter 8 Instructor: Zhang
4. Spectral Characteristic of ‘Real Signal’
Most ‘real’ signals: continuous with time Highest frequency fh can be found X(f) = 0 if
5. How ‘sampling process’ modifies the spectrum
Consider a
If
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Modifier
EE 422G Notes: Chapter 8 Instructor: Zhang
If
No spectrum modification
6. How fast we have to sample in order to keep the spectrum:
Condition
Should we consider ? Of course not!( implies that the real process changes faster than the sampling rate.)
Consider only => => Answer : Sampling rate: at least twice as the highest frequency of the “original process”
Sampling Theorem: ………….
7. What about if r 0 ? (show Figure 8-4)
8. Practical sampling rate:
8-2B Data Reconstruction
1. What’s Data Reconstruction? Original x(t) t 0 (anytime) Its samples xs(t) t = 0, T, 2T, … (Discrete time)Can we tell x(t) between sampled points ( nT < t < (n+1)T ) based on xs(t)?
Data Reconstruction problem!
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Constant with n
Similar as
EE 422G Notes: Chapter 8 Instructor: Zhang
2. Data Reconstruction Method
What’s a filter? A system which processes the input to generate an output. It could be an algorithm (mathematical equation/operation set) or circuit/analog computer, depending on the form of xs(t) (digital number or analogy signal .)
Let’s see how a filter works!
Output
Weighted sum of the ‘should be point x(kT) and its surrounding points
|h(0)| should > h() 0 and |h()| decreases as What is ? (In addition to being an algorithm)
Let’s see:
or
Consider xs(t)*h(t) :
Reconstruction Filter: with h(t) as impulse response!
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Discrete-time algorithm
a system with impulse response (system parameter) h() and input xs(t)!
The reconstruction algorithm
EE 422G Notes: Chapter 8 Instructor: Zhang
Output of the reconstruction Filter (y(t)): Convolution of xs(t) and h() !
3. Design of Reconstruction Filter: Ideal case
Assumption : fs>2fh (xs(t) was generated at a frequency higher than the Nyquist rate).
1/2 fs > fh fh: highest frequency of the original signal
Ideal Filter
Question : why do we need this low-pass filter to reconstruct x(t) from xs(t)?answer : xs(t) contains frequencies higher than , but x(t)does not!
Question : Will any spectrum (other than x(t)’s introduced by sampling operation remains after the filter?
Answer: No. , has ensured that no overlapping between x(t)’s frequencies and the undesired frequencies in xs(t) introduced by sampling!
Implementation of Ideal Reconstruction Filter (Given the Impulse response of the filter) Inverse Fourier transform =>
Characteristic of the Ideal Reconstruction Filter: Non causal! Output at t ( y(t) ) must be generated using xs() > t => Not good for real-time application!
How to reconstruction x(t) from nT < t < nT + T ? Answer :
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EE 422G Notes: Chapter 8 Instructor: Zhang
for example t = nT + 0.5T
l points before t = nT + 0.5T (k=n-l+1,…,n) l points after t = nT + 0.5T (k=n+1,…,n+l)
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EE 422G Notes: Chapter 8 Instructor: Zhang
Part Two
8-3A The z-Transform
1. DefinitionFor Laplace transform, we are given a function x(t), For z-Transform, we are given a sampling sequence: x(0) , x(T), x(2T), …
Definition: z-transform of a given sequence x(0) , x(T), x(2T), …
is
Why do we define such a transform?
x(t)
If we want to compute this Laplace transform by computer
On the other hand
Relationship between z- and s-planeBasic Relationship :
(1) (note )
l.h.p. (s- plane) inside the unit circle (z- plane)
(2)
s: r.h.p. z: outside the unit circle
(3)
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x(nT): samples of
EE 422G Notes: Chapter 8 Instructor: Zhang
s: j axis. z: unit circle
(4) s = 0 ( )
z = 1
2. Basic z-Transform pairs
Example 8-4: z-transform of unit pulse :
Solution :
Example 8-5 z-Transform of unit step sequence u(n):
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Note the difference between and
unbounded bounded
Similar as Laplace transform
EE 422G Notes: Chapter 8 Instructor: Zhang
Solution :
How to understand?
Step function u(t) :
Does give the same spectrum if T 0 ?
T 0 :
z-Transform gives the same spectrum as Laplace transform if the sampling rate
Example 8-6 : z-transform of unit exponential sequence
Solution:
Is this result reasonable?
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EE 422G Notes: Chapter 8 Instructor: Zhang
Why? Because
Example 8-6 B
=>
Summary: Basic z-transform pairs
Would these be sufficient? No!
3. Extended z – transform pairs
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1 1 0
EE 422G Notes: Chapter 8 Instructor: Zhang
4. Find z-transform using symbolic tool box
Example 8-7
Solution:
Analysis:
(1) n : odd =>
(2) n : even =>
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EE 422G Notes: Chapter 8 Instructor: Zhang
Very complex!
Using Symbolic ToolBoxsyms a n z % Declare symbolicxn = a^n*cos(n*pi/2); % Define x(n)xz = ztrans (xn, n, z); % Determine X(z)xz (enter)xz =
z^2/(a^2+z^2)
MatLab: always in terms of z instead of z-1.
8-3B Properties of z - transform
1. Linearity
2. Initial Value why?
3. Final value
Why? But,
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1
3
5
EE 422G Notes: Chapter 8 Instructor: Zhang
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EE 422G Notes: Chapter 8 Instructor: Zhang
8-3C Inverse z-Transform Two Basic Methods:
(1) Express X(z) into “Definition Form”
(very simple, use long division or MatLab: n = 8 X = dimpulse(num,den, n) (enter) gives the first n terms)
(2) Express X(z) into partial-fraction from
partial-fraction each term has an inverse transform expansion
what Terms? 1
What about if you have ?
11 1
1
1
1
kzze T
What about if you have ?
What about if you have ?
Let’s see: 21
11
121
1
21
)1(
1)1()1(
1
ze
zBeBAz
ze
B
ze
Az
ze
T
T
TTT
Can we now find A and B? What is the inverse z-transform of
What to do if you have ?
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EE 422G Notes: Chapter 8 Instructor: Zhang
)(1
1
)(1
1
2
1
1
1
1
1
2
1
)1)(1(
1
1
1
11
11112
jzjz
jzjzjzjzz
oddn
evennjjj
zZnTx
nnnnn
0
)1(]1)1[()(
2
1])()[(
2
1)
1
1()(
2/
21
Important: before doing partial-fraction expansion, make sure the z-transform is in proper rational function of !
Question: Is this system shift-invariant? Yes! Question: Is this example telling us ? Yes! Question: Is
or always true for different systems?No! only for time-invariant systems!
Shift-invariant system: if true for any n0 .
2. Causal and noncausal systems
Physical Description: A system is causal or nonanticipatory if the system’s response to an input does not depend on future values of the input.
Mathematical Description:
Causal system:
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What about for n > n0 ?
EE 422G Notes: Chapter 8 Instructor: Zhang
Why? Although x1(nT) may not be the same as x2(nT) for n > n0 , such difference does not affect the output determined by input up to n = n0 .
3. Linear SystemLinear System Linear Systems: can be modeled as
or
response of the shift-invariant linear system at t=kT to an impulse input applied at t=0. (Or the response at to an impulse input applied at ) Causal systems:
Linear+causal+
Example:
Given x(0) = 1, x(T) = 2, x(2T) = 2, x(3T) = 1, … h(0) = 3, h(T) = 2, h(2T) = 1, h(3T) = 0, … MatLab: x = [1 2 2 1 1]; h = [3 2 1]; y = conv(x,h); y
3 8 11 9 7 3 1Example 8-13:
Can you write a program (algorithm) to calculate y(nT) = x(nT)*h(nT) ?Example 8-13: Symbolic Tool Box
syms n z % Declare Symbolicxn =(1/2)^n % x(n)hn = (1/3)^n % h(n) xz = ztrans(xn, n, z) % z-transform of x(n)
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Convolution
EE 422G Notes: Chapter 8 Instructor: Zhang
hz = ztrans(hn, n, z) % z-transform of h(n)yz = xz*hz % multiply, not convolutionyn = iztrans (yz, z, n); % Do you know why?yn (enter)yn = 3*(1/2)^n-2*(1/3)^n % y(nT)=3(1/2)n – 2(1/3)n
* Analytic solution of convolution
i.e.
Example:
Find x(nT)*h(nT)Solution:
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n<kn-k<0
EE 422G Notes: Chapter 8 Instructor: Zhang
4. Stable system
Consider linear shift-invariant systems only.
Definition of BIBO stable:for all bounded x(nT).
Derivation of Criterion
x(kT) bounded =>
Criterion:
How to use this criterion: Ah: h(0), h(1), … h(N), 0, 0, …
(causal)
causal + Limited N => stable
Why!
for any fixed n in ,
for example,
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Limited number of terms
EE 422G Notes: Chapter 8 Instructor: Zhang
In general
Conclusion: limited terms of h => stable!
Example : stable?
What about ?
How to use this criterion: B If we know
Z(h(nT)) = H(z)
h(0) h(1) h(2)
Why? |0.2| < 1 !
What about
Not BIBO stable!
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Limited Terms
EE 422G Notes: Chapter 8 Instructor: Zhang
In general, deg(num) < deg(den)
(poles inside the unit circle!)
Example 8-14:
Solution:
Stable
8-4B Difference Equations
1. Difference Equations
Problem: determine the output of the system at the present time : t = nT y(nT)
What information to use:(1) input: current input u(nT)
previous input u(kT) (k < n) future input u(kT) (k > n)causal system : no future input!Previous input
We do not need all of them use u(n-1), … , u(n-m)(2) output: previous output (its history): y(kT) (k < n) ? Yes.
future output y(kT) (k >n) ? No, no future output
previous outputs We do not need all of them! y(n-1), …. , y(n-r) would be sufficient!
Mathematical Equation
y(nT) : depends on
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Difference Equation
EE 422G Notes: Chapter 8 Instructor: Zhang
linear system
weights: Larger weight: more important in determining y(nT)
Would the weights be the same? No! (r, m): system’s order
different systems: different order and weights (parameters)
2. z-transfer function
Different Equation
z-transform =>
z-transfer functionY(z) = H(z)X(z)
Why H(z) is the z-transform of impulse response h(nT) ?
8-4C Steady-State Frequency Response of a Linear Discrete-Time System
x(t)’s spectrum x(nT)’s spectrum
y(t)’s spectrum y(nT)’s spectrum
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EE 422G Notes: Chapter 8 Instructor: Zhang
System’s frequency response
What is Y(z)/X(z) ? H(z) = Y(z)/X(z)
System frequency response
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EE 422G Notes: Chapter 8 Instructor: Zhang
Property of frequency response T: sampling period
: sampling frequency
Frequency Response H: periodic function with period when the frequency increase by , the system’s frequency
response does not change.
Example: Input 1: T = 1 secondInput 2 : Generate the same output amplitude?
Normalized Frequency
: frequency period
Frequency Response in terms of r (argument)
Amplitude Response or Phase Response or
Question: what are their physical meaning?
Example 8-15: y(nT) = x(nT) + x(nT-2T)Solution :
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EE 422G Notes: Chapter 8 Instructor: Zhang
Comment: z-transform: good for analysis difference equation: computer program