1 1111 Lecture 16 Random Signals and Noise (III) Fall 2008 NCTU EE Tzu-Hsien Sang.

11111

Lecture 16 Random Signals and Noise (III)

Fall 2008

NCTU EE

Tzu-Hsien Sang

2

Outline

• Terminology of Random Processes

• Correlation and Power Spectral Density

• Linear Systems and Random Processes

• Narrowband Noise

33

Linear Systems and Random Processes

– Without memory: a random variable a random variable

– With memory: correlated outputs

• Now, we study the statistics between inputs and outputs, e.g., my(t), Ry(), …

• Assume X(t) stationary (or WSS at least) H() is LTI.

3

H X(t) = X(t,) Y(t) = Y(t,)

444

).()()()()()(

).()()()()()(

).()()(

).()()(

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

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

:Functionn Correlatio

).0()]([)(

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

: ofMean

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Y(t)

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SSy

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too. WSSis

,condition) initial (no LTI is and WSSis If (1) :Remarks

).()()()()()(

).()()()()()(

).()()()(

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

)}]()(){([)]()([)(

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66

• Gaussian Random Process: X(t) has joint Gaussian pdf (of all orders).

– Special Case 1: Stationary Gaussian random process

Mean = mx; auto-correlation = RX().– Special Case 2: White Gaussian random process

RX(t1,t2) = (t1-t2) = () and SX(f) = 1 (constant).

6

0

Show that if ( ) is stationary Gaussian, so is ( ).

Case I: ( ) is white ( independent random variables).

( ) ( ) ( ) ( ) ( ) lim ( ) ( )

weighted sum of Gausk

X t Y t

X t

Y t h t X t X h t d X k h t k

1 2 2 1 1

sian random variables.

( ) has a 1st-order Gaussian distribution. Similarly, the higher

order joint pdf of ( ) is joint Gaussian. For example,

( , ) ( | ) ( ) (would be jointly GaY Y Y

Y t

Y t

f y y f y y f y

ussian).

777

1

2 1

Case II: ( ) is Gaussian but not white (more realistic case).

Claim: ( ) is produced by passing a white Gaussian random process

through ( ).

Then, define ( ) ( ) ( ), we have ( )

x t

x t

h t

h t h t h t y t

2 ( ) ( ).

Back to Case I. Therefore, ( ) is stationary Gaussian.

Note: We often use lower-case letters ( and instead of and )

for denoting random processes. You need to judge by the context

h t Z t

y t

x y X Y

whether it is a deterministic signal or a random process.

88

• Properties of Gaussian Processes(1) X(t) Gaussian, H() stable, linear Y(t) Gaussian.(2) X(t) Gaussian and WSS X(t) is SSS.(3) Samples of a Gaussian process, X(t1), X(t2), …, are

uncorrelated They are independent.(4) Samples of a Gaussian process, X(t1), X(t2), …, have

a joint Gaussian pdf specified completely by the set of means and auto-covariance function .

• Remarks: Why do we use Gaussian model?– Easy to analyze.– Central Limit Theorem: Many “independent” events

combined together become a Gaussian random variable ( random process ).

99

• Example: RC filter with white Gaussian input.

0

0

0

3

3

2 0

2

3

0

Input: is white Gaussian and zero-mean with ( ) .2

1 1Filter: ( )

1 2 1

1where =3 dB cutoff frequency = .

21

The output: ( ) ( ) ( ) .2 1 ( )

( ) .4

i

i

n

n n

RCn

NS f

H ffj fRC j f

fRC

NS f S f H f

ff

NR e

RC

1010

0 0

0

0

0

2 2 00

2 0 00 20

0

20

Output power: ( ) (0) .4

Another approach: ( ) ( ) .2 1 4

Mean: ( ) 0 (0) 0.

Another approach: ( ( )) lim ( ) 0.

The first-order pdf: ( ,

n n

n

n

n

Nn t R

RCN Ndx

n t S f dfRC x RC

n t H

n t R

f y t

2

0

2

0

1) .

2

Note: ( ) does not "completely" describes the behaviour of

a random process.

RCy

NeNRC

R

1111

• Noise equivalent Bandwidth

It is just a way to describe a band-limited noise with the bandwidth of an ideal band-pass filter.

2

2

( )1.

2 { ( ) }N

h t dtB

h t dt

1212

Narrowband Noise

• Q: Besides certain statistics, is there a more “waveform-oriented” approach to describe a noise (or random signal)?

2W

-f0 f0 f

0

Interpretation: A baseband radom process is shifted to a higher frequency.

( ) ( ) cos 2 .n t R t f t

0W f

1313

0

0

However, a bandpass random signal can have a random phase:

( ) ( ) cos(2 ( )).

In general, if we allow a time-invariant phase bias , then

(Envelope-phase representation) ( ) ( ) cos( ( ) )

o

n t R t f t t

n t R t w t t

0 0

2 2 1

r (Quadrature-component representation)

( ) ( ) cos( ) ( )sin( )

( )with ( ) ( ) ( ) and ( ) tan ( ).

( )

c c

sc s

c

n t n t w t n t w t

n tR t n t n t t

n t

1414

0 0

How to produce (t) and ( ):

( ) ( ) cos( ) ( )sin( ) (in mean-square error)

Remarks: Here, we assume is a random variable, independent

of ( ), uniformly distributed over (0, 2 )

c s

c c

n n t

n t n t w t n t w t

n t

1 2

or (- , ). If is

not a random variable, ( ) and ( ) are not WSS. We cannot

use LTI theory to predict the outputs of LPF's.

z t z t

1515

Properties of Quadrature-component representation

(1) ( ) ( ) ( ) 0.

Proof: (i) Let ( ) ( ) ( ).

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

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

c s

X

n t n t n t

X t n t n t

R E X t X t n t n t n t n t

n t n t n t n t n t n t n

2

2

2 2

2

1

( ) ( )

( ) ( ( )) . ( WSS)

( 0) ( ) [ ( ) ( ( )) ]

(0) ( ( )) 0- ( ( )) 0.

The only possibility is ( ( )) 0.

(ii) { ( )

n

X X n

n

t n t

R n t

S f R d R n t d

S n t d n t d

n t

E z t

0

1

} 2 ( ) cos( ) 0.

{ ( )} { ( )} (0) 0. Similarly, { ( )} 0.c s

n t t

E n t E z t H E n t

1616

0 0

0 0

0 0

1

(2) ( ) ( ) { ( ) ( )}

( ) ( ), .

0, otherwise

( ) { ( ) ( )}.

Proof: (i) ( ) 2 ( )

c s

c s

n n n n

n n

n n n n

S f S f Lowpass S f f S f f

S f f S f f W f W

S f j Lowpass S f f S f f

z t n t

1

0

1 1

0 0

0 0 0

0 0 0

cos( ).

( ) { ( ) ( )}

{4 ( ) ( ) cos( ) cos( ( ) )}

2 { ( ) ( )}cos 2 { ( ) ( ) cos(2 2 )}

2 ( ) cos 2 { ( ) ( )} {cos(2

Z

n

t

R E z t z t

E n t n t t t

E n t n t E n t n t t

R E n t n t E t

1

0

0 0

0 0

1

0 0

2 )}

2 ( ) cos .

Thus, ( ) ( ) [ ( ) ( )]

( ) ( ).

(t) is the low-pass portion of ( ).

( ) { ( ) ( )}.

Similarly, c

n

z n

n n

c

n n n

R

S f S f f f f f

S f f S f f

n z t

S f Lowpass S f f S f f

0 0( ) { ( ) ( )}.sn n nS f Lowpass S f f S f f

1717

1 2

1 2

1 2

0 0

0

0 0

1

Proof: (ii) ( ) { ( ) ( )}

{4 ( ) ( ) cos( )sin( ( ) )}

2 ( )sin .

Thus, ( ) [ ( ) ( )].

( ) { ( ) ( )}

{ ( ) (

c s

z z

n

z z n n

n n c s

R E z t z t

E n t n t t t

R w

S f j S f f S f f

R E n t n t

E h u Z t

1 2

1 2

1 2

2

1 2

*

) ( ) ( ) }

( ) ( ) { ( ) ( )}

( ) ( ) ( )

( ) ( ) ( ).

( ) ( ) ( ) ( )

(

c s

Z Z

Z Z

n n Z Z

u du h v Z t u dv

h u h v E Z t u Z t v dudv

h u h v R u v dudv

h h R

S f H f H f S f

H f

1 2

2

2

0 0

0 0

) ( )

( ) [ ( ) ( )]

{ ( ) ( )}.

Z Z

n n

n n

S f

j H f S f f S f f

j Lowpass S f f S f f

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