Chapter 8 Discrete-Time Signals and Systems
Chapter 8 Discrete-Time Signals and Systems
1. Introduction
Most “real” signals and natural (physical) processes: continuous – time
A : System Design Problem
[pic]
[pic]
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.)
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 Operation
Needs 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)
(5) Sampled signal xs = x(t)p(t)
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 :[pic]
Sampling function: A Periodical function,
(thus can be expressed using Fourier series), with
period T on fundamental frequency [pic]
With Fourier series coefficients:
[pic]
[pic]
3. Spectrum of sampled signal
Objective: Find the spectrum of the sampled signal xs(t).
Derivation :
Take Fourier Transform for
[pic]
[pic]
4. Spectral Characteristic of ‘Real Signal’
Most ‘real’ signals: continuous with time
⇨ Highest frequency fh can be found
⇨ X(f) = 0 if [pic]
5. How ‘sampling process’ modifies the spectrum
[pic]
Consider a [pic]
[pic]
If [pic]
[pic]
If [pic]
[pic]
No spectrum modification
6. How fast we have to sample in order to keep the spectrum:
[pic]
Condition [pic]
Should we consider [pic] ? Of course not!
( [pic] implies that the real process
changes faster than the sampling rate.)
Consider [pic] only =>
[pic] => [pic]
Answer : [pic]
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)
[pic]
8. Practical sampling rate:
[pic]
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!
2. Data Reconstruction Method
[pic]
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 [pic]
[pic] Weighted sum of the ‘should be point x(kT)
and its surrounding points
|h(0)| should > h(() ( ((0
and |h(()| decreases as (((
What is [pic] ? (In addition to being an algorithm)
Let’s see:
[pic] or [pic]
Consider xs(t)*h(t) :
[pic]
Reconstruction Filter: with h(t) as impulse response!
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
[pic]
Question : why do we need this low-pass filter to reconstruct x(t) from xs(t)?
answer : xs(t) contains frequencies higher than [pic], but x(t)does not!
Question : Will any spectrum (other than x(t)’s introduced by sampling operation
remains after the filter?
Answer: No. [pic], 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 =>
[pic]
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 :
[pic]
for example t = nT + 0.5T
[pic]
l points before t = nT + 0.5T (k=n-l+1,…,n)
l points after t = nT + 0.5T (k=n+1,…,n+l)
Part Two
8-3A The z-Transform
1. Definition
For 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 [pic]
• Why do we define such a transform?
x(t) [pic]
If we want to compute this Laplace transform by computer
[pic]
On the other hand
[pic]
• Relationship between z- and s-plane
Basic Relationship : [pic]
[pic]
(1) [pic] (note [pic])
l.h.p. (s- plane) ( inside the unit circle (z- plane)
(2) [pic]
s: r.h.p. ( z: outside the unit circle
(3) [pic]
s: jω axis. ( z: unit circle
(4) s = 0 ([pic]) [pic]
z = 1
[pic]
2. Basic z-Transform pairs
• Example 8-4: z-transform of unit pulse [pic]:
[pic]
Solution :
[pic]
Example 8-5 z-Transform of unit step sequence u(n):
[pic]
Solution :
[pic]
[pic]
How to understand?
Step function u(t) : [pic]
Does [pic] give the same spectrum if T( 0 ?
[pic]
T( 0 : [pic]
⇨ z-Transform gives the same spectrum as Laplace transform if the sampling rate ((
Example 8-6 : z-transform of unit exponential sequence
[pic]
Solution:
[pic]
• Is this result reasonable?
[pic]
[pic]
Why? Because
[pic]
• Example 8-6 B
[pic]
=> [pic]
• Summary: Basic z-transform pairs
[pic]
Would these be sufficient? No!
3. Extended z – transform pairs
[pic]
4. Find z-transform using symbolic tool box
Example 8-7
[pic]
Solution:
[pic]
Analysis:
(1) n : odd => [pic]
(2) n : even => [pic]
[pic]
Very complex!
Using Symbolic ToolBox
syms a n z % Declare symbolic
xn = 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) [pic]
MatLab: always in terms of z instead of z-1.
8-3B Properties of z - transform
1. Linearity
[pic]
2. Initial Value [pic]
why? [pic]
3. Final value [pic]
Why? [pic]
But,
[pic]
8-3C Inverse z-Transform
Two Basic Methods:
1) Express X(z) into “Definition Form”
[pic]
(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
[pic]
( (
partial-fraction each term has an inverse transform
expansion
what Terms?
1
[pic] What about if you have [pic]?
[pic]
[pic] What about if you have [pic]?
[pic] What about if you have[pic]?
Let’s see: [pic]
Can we now find A and B? What is the inverse z-transform of [pic]
What to do if you have [pic]?
[pic]
[pic]
Important: before doing partial-fraction expansion, make sure the z-transform is in proper rational function of [pic] !
Example 8.9
[pic]
Solution : [pic]
Heaviside’s Expansion Method:
[pic]
1) [pic]
[pic]
2) [pic]( [pic]
[pic] [pic] (
[pic]
Example 8-9B MatLab Method
(1) Find partial-fraction expansion
[pic]
b = 1;
a = [1 –1.2 0.2];
[r, p, k] = residuez(b,a);
[pic]
(2) Directly Find Inverse Transform
syms n, z; % Declare symbolic
xz = 1/(1-1.2*z^(-1)+0.2*z^(-2)); % define X(z)
xn = iztrans(xz,z,n); % compute x(n)
xn (
xn = 5/4-(1/4)*(1/5)^n ( x(nT) = 1.25-0.25(0.2)n
Example 8-10
[pic]
Solution :
Question: Define [pic] (or [pic])
any relationship between
[pic] and [pic]?
[pic]
[pic]
[pic]
[pic]
[pic] [pic]
n = 0 5 + 1.25 - 6.25 = 0 1.25 - 0.25 = 1
n = 1 0 + 1.25 - 6.25*0.2 = 0 1.25 - 0.25*0.2 = 1.2
n = 2 0 + 1.25 - 6.25*0.22 = 1 1.25 - 0.25*0.22 = 1.24
n = 3 0 + 1.25 - 6.25*0.23 =1.2 1.25 - 0.25*0.23 = 1.248
Why? 6.25*0.2*0.2=0.25 =>[pic]
y(n+2) = x(n)!
Does [pic] always imply [pic] has two-step-delay
than [pic]? Yes!
z-1 : Delay operator! (Must Assume X(nT)(the sequence to be z^(-1) processed)=0 for n 0 )
[pic]
We want to establish the relationship between Z(x(nT-kT)) and Z(x(nT)) !
Let’s see what’s [pic]:
(1)[pic]? Yes!
2) [pic]
[pic]
[pic] [pic][pic]
[pic]
Question : If [pic], what’s [pic]? Answer: [pic]
8-4 Difference Equation and Discrete-Time Systems
Continuous-Time System: Differential Equation, Laplace Transform
Discrete-Time System: Difference Equation, z-Transform
Properties of Continuous-Time Systems
Properties of Discrete-Time Systems
8-4A Properties of Discrete-Time Systems
System : Processes input to generate output
How to process : system-dependent
General symbolic notation for Discrete-Time System:
y( nT ) = H [ x(nT) ]
( (
what does this operator or
notation tell us? Processor
1. Shift-Invariant System
(Time-Invariant Systems for continuous-time or general)
An example of time-varying system
The “processing algorithm” which maps input to output changes!
What do we mean by a time-invariant system?
Shift-invariant systems:
Physical:
Mathematic:
Assume x(nT): x(0), x(T), … has generated
y(nT): y(0), y(T), …
For example:
[pic] has
If we apply [pic] as input
look at if [pic]
[pic] generated
Question: Is this system shift-invariant? Yes!
Question: Is this example telling us [pic]? Yes!
Question: Is [pic]
or [pic]
always true for different systems?
No! only for time-invariant systems!
Shift-invariant system: if [pic] 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: [pic]
[pic]
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 System
Linear System [pic]
Linear Systems: can be modeled as
[pic] or [pic]
[pic] response of the shift-invariant linear system at t=kT to an impulse input applied at t=0. (Or the response at [pic] to an impulse input applied at [pic])
Causal systems: [pic]
Linear+causal+[pic]
[pic] 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 1
Example 8-13:
[pic]
Can you write a program (algorithm) to calculate y(nT) = x(nT)*h(nT) ?
Example 8-13: Symbolic Tool Box
syms n z % Declare Symbolic
xn =(1/2)^n % x(n)
hn = (1/3)^n % h(n)
xz = ztrans(xn, n, z) % z-transform of x(n)
hz = ztrans(hn, n, z) % z-transform of h(n)
yz = xz*hz % multiply, not convolution
yn = 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
[pic]
[pic]
[pic]
i.e. [pic]
Example: [pic]
[pic]
Find x(nT)*h(nT)
Solution:
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
4. Stable system
Consider linear shift-invariant systems only.
Definition of BIBO stable:
[pic] for all bounded x(nT).
Derivation of Criterion
[pic]
x(kT) bounded => [pic]
[pic]
Criterion: [pic]
How to use this criterion: A
h: h(0), h(1), … h(N), 0, 0, …
[pic](causal)
causal + Limited N => stable
Why! [pic]
for any fixed n in [pic], [pic]
for example, [pic]
[pic]
In general
[pic]
Conclusion: limited terms of h => stable!
Example : [pic] stable?
What about [pic] ?
How to use this criterion: B
If we know
Z(h(nT)) = H(z)
[pic]
h(0) h(1) h(2)
[pic]
Why? |0.2| < 1 !
What about [pic]
Not BIBO stable!
In general, deg(num) < deg(den)
[pic]
(poles inside the unit circle!)
Example 8-14: [pic] [pic]
Solution:
[pic]
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 [pic]
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 [pic]
We do not need all of them! y(n-1), …. , y(n-r) would be sufficient!
Mathematical Equation
y(nT) : depends on [pic]
linear system
[pic]
weights: [pic]
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 [pic]
z-transform =>
[pic]
[pic]
z-transfer function
Y(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 [pic]
x(nT)’s spectrum [pic]
y(t)’s spectrum [pic]
y(nT)’s spectrum [pic]
System’s frequency response
[pic]
What is Y(z)/X(z) ? H(z) = Y(z)/X(z)
System frequency response [pic]
Property of frequency response [pic]
T: sampling period
[pic] : sampling frequency
[pic]
Frequency Response H: periodic function with period [pic]
( when the frequency [pic] increase by [pic], the system’s frequency
response does not change.
Example: Input 1: [pic] T = 1 second
Input 2 : [pic]
Generate the same output amplitude?
Normalized Frequency [pic]
[pic] : frequency period [pic]
[pic]
Frequency Response in terms of r (argument)
[pic]
Amplitude Response [pic] or [pic]
Phase Response [pic] or [pic]
Question: what are their physical meaning?
Example 8-15: y(nT) = x(nT) + x(nT-2T)
Solution : [pic]
[pic]
[pic]
[pic]
[pic]
Comment: z-transform: good for analysis
difference equation: computer program
-----------------------
General Equation for any periodical signal
Fourier Series Description of Sampling Function
[pic]
The reconstruction algorithm
[pic]
a system with impulse response (system parameter) h(() and input xs(t)!
Discrete-time algorithm
general equation for Fourier coefficient of any periodical signal
general equation: always true for any r (width of the sampling pulse).
For n
[pic]
Modifier
Constant with n
Similar as [pic]
x(nT): samples of x(t)
[pic]
Note the difference between [pic]
and [pic]
unbounded
bounded
Similar as Laplace transform
[pic]
1 1 0
5
3
1
4
2
0
[pic]
What about for n > n0 ?
Convolution
[pic]
n ................
................
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