chapter 2 systems and signalscmliu/courses/dsp/chap2.pdf1. introduction-- definition signals any...

Chapter 2 Systems and Signals

Introduction

Discrete-Time Signals: Sequences

Discrete-Time Systems

Properties of Linear Time-Invariant Systems

Linear Constant-Coefficient Difference Equations

Frequency-Domain Representation of Discrete-Time Signals and Systems

Representation of Sequence by Fourier Transforms

Symmetry Properties of the Fourier Transform

Fourier Transform Theorems

1

1. Introduction-- Definition

Signals

Any physical quantity that varies with time, space or any other independent variable.

Communication beween humans and machines.

Systems

mathematically a transformation or an operator that maps an input signal into an output signal.

can be either hardware or software.

such operations are usually referred as signal processing.

Digital Signal Processing

The representation of signals by sequences of numbers or symbols and the processing of these sequences.

S

2

1. Introduction-- Definition3

1. Introduction-- Classification

Continuous-Time versus Discrete-Time Signals

Continuous-time signals are defined for every value of time.

Discrete -time signals are defined at discrete values of time.

Continuous-Valued versus Discrete-Valued Signals

A signal which takes on all possible values on a finite range or infinite range is said to be a continuous-valued signal.

A signal takes on values from a finite set of possible values is said to be a discrete-valued signal.

Multichannel versus Multidimensional Signals

Signals may be generated by multiple sources or multiple sensors. Such signals are multi-channel signals.

A signal which is a function of M independent variables is calledmulti-dimensional signals.

4

1. Introduction-- Basic Elements

A/D Converter

Converts an analog

signal into a

sequence of digits

D/A Converter

Converts a sequence

of digits into an

analog signal

A/D Converter

D/A Converter

DigitalSignal

Processing

AnalogInputSignal

AnalogOutputSignal

DigitalInputSignal

DigitalOutputSignal

{3, 5, 4, 6 ...}

0

t

5

1. Introduction-- Advantages of Digital over Analog Processing

Better control of accuracy

Easily stored on magnetic media

Allow for more sophisticated signal

processing

Cheaper in some cases

6

Examples

A picture is a two-dimensional signal

I(x,y) is a function of two variables.

A black-and-white television picture is a three-

dimensional signal

I(x,y,t) is a function of three variables.

A color TV picture is a three-channel, three-dimensional

signals

Ir(x,y,t), Ig(x,y,t), and Ib(x,y,t)

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

7

2. Discrete-Time Signals: Sequences

Continuous signals

f(x), u(x), and s(t), and so on.

Sampled signals

s[n], f[n], u(n).

Shifting by k units of time

Unit sample sequence

(Kronecker Delta)Time

xa(t)

x [n]

t

n

Linear combination form

Unit step sequence

Sinusoidal Function][][ knxny

( ),

,n

n

n

0 0

1 0

x n x k n kk

[ ] [ ] [ ]

sin( ) sin( ) n fn 2

u nn

n[ ]

,

,

0 0

1 0

8

2. Discrete-Time Signals: Sequences (c.1)

Definition

Xa(t) = A cos( W t+ ), - ∞�


Discrete-Time Sinusoidal Signals

X(n) = A cos( n+ ), n =1, 2, ...

A is the amplitude of the sinusoid

is the frequency in radians per sample

is the phase in radians

f=/2 is the frequency in cycles per sample.

X(n) = A cos( n+ )

10


A discrete-time sinsoidal is periodic only if its frequency f is a rational number

– X(n+N) = X(n), N=p/f, where p is an integer

Discrete-time sinusoidals where frequencies are separated by an integer multiple of 2 are identical

– X1(n) = A cos( 0 n)

– X2(n) = A cos( (0 2) n) The highest rate of oscillation in a discrete-time

sinusoidal is attained when = or (=-), or equivalently f=1/2.

– X(n) = A cos(( 0+)n) = -A cos((0+)n

Discrete-Time Sinusoidal SignalsX(n) = A cos( n+ )

11


Exercise: Find the periods.

sin0.1k, cos 10.1k, sin0.1k, cos3k/7

An observation

For the sampling process f[k]= f(kT), the mapping between discrete-frequency and analog-frequency is one-to many.

Example

– sin1.1t, sin3.1t, sin3.1t, sin(-0.9)t, sin5.1pt, sin (-2.9)t have the sample digital envelop for sampling time T=1.

– Reason : 1T -2T = nx(2)

12


The Digital Frequency

If 1T and 2T differ by a multiple of 2, they are still considered equal.

The digital frequency can be restricted to lie in an interval of 2to eliminate this nonuniqueness.(-, ], [0, 2) or [, 3]

If (-, ] is selected, then

Example: Find the frequencies of the following sequences for T=1– sin 4.2k, cos5.2k, sin 10k, sin(-2.1k), cos20k

/T

3/Ts 5/Ts0

/Ts3/Ts /Ts

/T

Analog

Digital

13

3. Discrete-Time Systems

Systems

Mathematically a transformation or an operator that maps

an input signal into an output signal

Can be either hardware or software.

Such operations are usually referred as signal processing.

E.x.

Discrete-Time System H

n n

y n x k x k x n y n x nk

n

k

n

( ) ( ) ( ) ( ) ( ) ( )

1 1

14

3. Discrete-Time Systems-- Classification

Time-Invariant versus Time-Variant Systems

A system H is time-invariant or shift invariant if and only if

x(n) ---> y(n)

implies that

x(n-k) --> y(n-k)

for every input signal x(n) x(n) and every time shift k.

Causal versus Noncausal Systems

The output of a causal system satisfies an equation

y(n) = F[x(n), x(n-1), x(n-2), ...].

where F[.] is some arbitrary function.

Memory versus Memoryless Systems

A sysetm is referred to as memoryless system if the output y(n) at every value of n depends only on the input x(n) at the same value of n.

15

3. Discrete-Time Systems-- Classification

(c.1)

Linear versus Nonlinear Systems

A system H is linear if and only if

H[a1x1(n)+ a2 x2 (n)] = a1H[x1 (n)] + a2H[x2 (n)]

for any arbitrary input sequences x1(n) and x2(n), and any arbitrary constants a1 and a2.

Multiplicative or Scaling Property

H[ax(n)] = a H[x(n)]

Additivity Property

H[x1(n) + x2 (n)] = H[x1 (n)] + H[x2 (n)]

Stable versus Unstable Systems

An arbitrary relaxed system is said to be bounded-input-bounded-output (BIBO) stable if and only if every bounded input produces a bounded output.

Linear systems

y(n)

u1(n)

+

u2(n)

a

b

Linear systems

u1(n)

u2(n)

Linear systems

+y(n)

a

b

16

4. Linear Time-Invariant Systems

The Importance of LTI Systems

Powerful analysis techniques exist for such systems.

Many real-world systems can be closely approximated for linear, time-invariant systems.

Analysis techniques for LTI systems suggest approaches for that of nonlinear systems.

Linear Systems

Time-Invariance y(n)=H[x(n)] ==> H[x(n-k)] = y(n-k)

Linear superposition H[a1x1(n)+a2x2(n)] = a1H[x1(n)]+a2H[x2(n)]

= a1y1(n) + a2y2(n)

Specification for LSI Systems H x n H x k n k x k H n k x k h n k

k k k

[ ( )] [ ( ) ( )] ( ) [ ( )] ( ) ( )

Impulse Response

of the System

17

4. Linear Time-Invariant Systems18

4. Linear Time-Invariant Systems (c.1)

Linear Systems (c.1)

Finite Impulse Response (FIR)or Infinite Impulse Response (IIR)

Depends on the the finite and infinite number of terms of h(n)

Convolution Formula

Ex. A saving account with monthly interest rate 0.5%. The interest is added to the principal at the

first day of each month. If we deposit u[0] = $100.00, u[1] = -$50.00, u[2] = 200.00 what is

the total amount of money. What is the total amount of money on the first day of the fifth

month ?

y n x n h n x k h n kk

( ) ( ) ( ) ( ) ( )

19

4. Linear Time-Invariant

Systems (c.2)

Discrete Convolution-- Table Lists

Consider the convolution of

{1, 2, 3, 4, -1, ...} and {-1, 1, -3, 2, ...}

Discrete Convolution--

Graphical Computation

Flipping h [i] to yield h [-i]

Shifting h [-i] to yield h [k-i]

Multiplication of h [k-i] and u [i]

Summation of h [k-i]u [i] from i =0, 1, 2, ...

y k h k i u i h i u k ii

k

i

k

[ ] [ ] [ ] [ ] [ ]

0 0

1 2 3 4 -1

-1

1

-3

2

-1 -2 -3 -4 1

1 2 3 4 -1

-3 -6 -9 -12 3

2 4 6 8 -2

y[0] y[1] y[2] y[3] ...

20

5. Properties of the LTI Systems

Communicative

Parallel Sum

Cascade Form

y k h k i u i h k u k

h i u k i u k h k

i

i

[ ] [ ] [ ] [ ]* [ ]

[ ] [ ] [ ]* [ ]

x n h n h n

x n h n x n h n

[ ]*{ [ ] [ ]}

[ ]* [ ] [ ]* [ ]

1 2

1 2

][*]}[*][{

]}[*][{*][

21

21

nhnhnx

nhnhnx

h1[n]+h2[n]x[n] y[n]

h1[n]x[n] y[n]

h[n]x[n] y[n]

x[n]h[n] y[n]

h1[n]*h2[n]x[n] y[n]

h1[n]x[n] y[n]

h2[n]

+

h2[n]

21

5. Properties of the LTI Systems (c.1)

Stability of LTI Systems (BIBO, Bounded-Input-Bounded Output System)

Linear time-invariant systems are stable if and only if the impulse response is absolutely summable, i.e., if

Since that

If x[n] is bounded so that

then

S h kk

[ ]

k

k

knxkh

knxkhny

][][

][][][

x n Bx[ ]

y n B h kxk

[ ] [ ]

If S= then the bounded input

will generate

x nh n

h nh n

h n

[ ]

*[ ]

[ ], [ ]

, [ ]

0

0 0

y x k h kh k

h kS

k k

[ ] [ ] [ ][ ]

[ ]0

2

22

q -> p~q -> ~p

5. Properties of the LTI Systems (c.2)

Ex. Find the impulse response of the following systems

][]1[][

][][

][1

1][

][][

2

121

nxnxny

nxny

knxMM

ny

nnxny

n

k

M

Mk

d

23

6. Linear Constant-Coefficient Difference

Equations

Input-Output Description

From Difference Equation to Impulse Response

Not every convolution can be transformed into a simple difference equation

If the difference equation description of a system is known, then the impulse response of the system can be readily obtained.

Example

y n a y n k b x n kkk

N

k

k

M

( ) ( ) ( )

1 0

y k y k u k

or y k y k u k

[ ] . [ ] [ ]

[ ] . [ ] [ ]

1 1005 1

1005 1

24


Equations25

Ex 2.16


Equations (c.1)

Solution Specification for the LCCDE

The solution can be obtained recursively for either positive

time or negative time.

Linear, time-invariant, and causal ==> the solution is unique.

Initial Condition

Initial-rest conditions if the initial condition is zero.

Initial rest condition ==> LTI and causal

Linear Constant Coefficient Equations

by {ak, bk, N, M}

x(n) y(n)

26

7. Frequency-Domain Representation of

Discrete-Time Signals and Systems

Eigenfunctions of LTI Systems

Response to Exponential Functions

Eigenvalues of the System

Frequency Response-- Complex

Magnitude and Phase

y n h k e e h k e H e ej n k

k

j n j k

k

j j n( ) ( ) ( ) ( )( )

H e H e jH ej Rj

Ij( ) ( ) ( )

H e H e ej j j H ej

( ) ( ) ( )

Ex.

y n x n n

y nM M

x n k

d

k M

M

[ ] [ ]

[ ] [ ]

1

11 2 1

2

X(n) = A cos( n+ )

27

7. Frequency-Domain Representation of

Discrete-Time Signals and Systems28

Ex. 2.17

Method 1: Frequency Response

Method 2: Derive from Impulse Response.

8. Representation of Sequences by Fourier

Transforms

Fourier Transform Pair

Transform

Inverse Transform (Synthesis Formula)

Magnitude & Phase Spectrum

X e x nj jnn

( ) ( ) exp

x n X e e dj jn( ) ( )

1

2

X e X e ej j j X ej

( ) ( ) ( )

Question:

1. Periodic Functions for

the Transform.

2. Relationship with the

Frequency response.

3. Inverse of each other ?

4. Existence of the

transform for functions.

30


Transforms31

Proof of Fourier Pair

Subsisting analysis equation into synthesis equation

L'Hôpital's rule

In its simplest form, l'Hôpital's rule states that for

functions ƒ and g:

32

Existence of Fourier Transform33

Convergence of the infinite sum

The series can be shown to

converge uniformly to a

continuous function of


Existence of Fourier transform of infinite sequence


Relax of the condition of uniform convergence of infinite

sum


Example


Transforms (c.1)

Example

Lowpass Filters

Impulse Response

H e j c

c

( ),

,

1

0

Causal ?

Decay Factor ?

Absolutely Summable ?

Gibbs phenomenon.

h n e d

n

nn

j n

c

[ ]

sin,

1

2

37


Transforms38


Transforms (c.2)

Fourier Transform of the periodic trainX e rj

r

( ) ( )

2 2x n for all n( ) 1

X e r

x n r e d

e d e for any n

j

r

r

j n

j n j n

( ) ( )

[ ] ( )

( ) .

2 2

1

22 2

1

22

0

0

00

Absolutely Summable ?

Square Summable ?

e e rj n j nrn

0 2 20

( )

Lighthill, 1958

39


Transforms40

Fourier transform of sinusoidal signal

Extend the theory

e e rj n j nrn

0 2 20

( )

9. Symmetry Properties of the Fourier

Transform

Definition

Conjugate-symmetric sequence

xe[n]=x*e [-n]

Conjugate-antisymmetric sequence

xo[n]=-x*o [-n]

Properties

A sequence can be represented as

x[n] = xe[n] + xo[n]

x n x n x ne [ ] ( [ ] *[ ]) 1

2

x n x n x no [ ] ( [ ] *[ ]) 1

2

X e X e X e

X e X e X e

ej j j

oj j j

( ) [ ( ) *( )]

( ) [ ( ) *( )]

1

2

1

2

X e X e X ej ej

oj( ) ( ) ( )

X e X e

X e X e

ej

ej

oj

oj

( ) ( )

( ) ( )

*

*

41


Transform (c.1)

Example

Real sequence

x[n]=anu[n]

42

X e x nj jnn

( ) ( ) exp

x n X e e dj jn( ) ( )

1

2


Transform (c.2)43


Transform (c.3)44

10. Fourier Transform Theorems45

Theorem Examples46

11. Discrete-Time Random Signals

Precise description of signals are so complex

Modeling the signals as a stochastic process.

Let the means of the input and output processes are

If x[n] is stationary, then mx[n] is independent of n and

]}[{][]},[{][ nynmnxnm yx

]}[{]},[{ nymnxm yx

k

x

k

y khmknxkhnym ][][][]}[{


Autocorrelation function of the output process

If x[n] is stationary

Let l=r-k

}][][{][][

][][][][]}[][{],[

k r

k r

yy

rmnxknxrhkh

rmnxknxrhkhmnynymnn

k r

xxyyyy rkmrhkhmmnn ][][][][],[

l

hhxx

k l

xxyyyy lclmlmklhkhmmnn ][][][][][][],[

k

hh klhkhlc ][][][


Fourier Transform

Power density spectrum

Cross-correlation

)()()( jxxj

hh

j

yy eeCe

k

xx

k

xy kmkhkmnxkhnxmnynxm ][][][][][]}[][{][

2* )()()()( jjjjhh eHeHeHeC

)()(][ jxxjj

xy eeHe


Ex. White Noise

Power Spectrum

The average power

][][ 2 mm xxx

11. Concluding Remarks

Introduction

Discrete-Time Signals: Sequences

Discrete-Time Systems

Properties of Linear Time-Invariant Systems

Linear Constant-Coefficient Difference Equations

Frequency-Domain Representation of Discrete-Time Signals and Systems

Representation of Sequence by Fourier Transforms

Symmetry Properties of the Fourier Transform

Fourier Transform Theorems

51

chapter 2 systems and signalscmliu/courses/dsp/chap2.pdf1. introduction-- definition signals any...

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