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 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download