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Chapter 6Chapter 6Bandpass Random Processes Bandpass Random Processes

Huseyin BilgekulEEE 461 Communication Systems II

Department of Electrical and Electronic Engineering Eastern Mediterranean University

Bandpass Random Processes PSD of bandpass random processes BP Filtered White Noise Sinusoids in Gaussian Noise

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Homework AssignmentsHomework Assignments

• Return date: December 13, 2005.

• Assignments:

Problem 6-38

Problem 6-41

Problem 6-45

Problem 6-48

Problem 6-50

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Equivalent Representations of Bandpass Equivalent Representations of Bandpass SignalsSignals

• Remind: Equivalent representations of a bandpass signal

Px(f)

f0 fcfc

( ) ( )

Envelope and Phase form

Inphase and Quadrature (IQ) form

Complex

Re cos

Envelope of (

cos si

( ))

n

( )

c

c c

j j t

jc

t

t

gg t x

v t

t jy t g t e R t e

g t e R t t t

v t x t t y t

t

t

v

Re ( )cos ( )

Im ( )sin ( )

x t g t R t t

y t g t R t t

2 2

1

( ) ( ) ( )

( )( ) ( ) tan ( )

( )

R t g t x t y t

y tt g t

x t

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Bandpass Random ProcessBandpass Random Process• If x(t) and y(t) are jointly WSS processes, the real bandpass process

Will be WSS stationary if and only if:

Re cos sincj tc cv t g t e x t t y t t

• If v(t) is a Gaussian random process then g(t), x(t) and y(t) are Gaussian processes since they are linear functions v(t). However R(t) θ(t) are NOT Gaussian because they are NONLIEAR functions of v(t).

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Bandpass Random ProcessBandpass Random Process• What happens to a signal at a receiver? How does the PSD of the

signal after a BPF correspond to the signal before the BPF?

• Remind: Equivalent representations of a bandpass signal

Px(f)

f0 fcfc

( ) ( )

Envelope and Phase form

Inphase and Quadrature (IQ) form

Complex

Re cos

Envelope of (

cos si

( ))

n

( )

c

c c

j j t

jc

t

t

gg t x

v t

t jy t g t e R t e

g t e R t t t

v t x t t y t

t

t

v

Re ( )cos ( )

Im ( )sin ( )

x t g t R t t

y t g t R t t

2 2

1

( ) ( ) ( )

( )( ) ( ) tan ( )

( )

R t g t x t y t

y tt g t

x t

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BPF System BPF System • Bandpass random process can be written as:

• With the impulse response: Re cos sincj t

c cv t g t e x t t y t t

Ideal LPFH0(f)

v(t)

BP ProcessIdeal LPF

H0(f)

x

x x

x

+

Basebandx(t)

Inphase

Basebandy(t)

Quadrature

2cos(ct+)

2sin(ct+) sin(ct+)

cos(ct+)

0 02 cosh t h t t

v(t)

BP Process

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Impulse ResponseImpulse Response

• Impulse Response

• Transfer Function

• So, x(t) and y(t) are low-pass random processes, what else can be deduced?

• Assume theta is uniformly distributed phase noise

0 02 cosh t h t t

0 0c cH f H f f H f f

1

H0(f )1

2H (f )

0 fcfc

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PSD of BP Random ProcessesPSD of BP Random Processes• PSD of x(t) and y(t)

Pv(f )

ffcfc

Px(f) or Py(f)

f0 fcfc

0

v c v c o

x y

o

P f f P f f f BP f P f

f B

Pv(ffc)

f0 fcfc

LPFPv(ffc)

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PSD of BP Random ProcessesPSD of BP Random Processes

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PSD of BP Random ProcessesPSD of BP Random Processes

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Properties of WSS BP ProcessesProperties of WSS BP Processes• If the narrowband noise is Gaussian, then the in-phase x(t) and quadrature y(t)

components are jointly Gaussian

• If the narrow band noise is wide-sense stationary (WSS), then the in-phase and quadrature components are jointly WSS.

• In-phase and quadrature components have the same PSD.

• In-phase and quadrature components of narrowband noise are zero-mean

– Noise comes original signal being passed through a narrowband linear filter

• Variance of the processes is the same (area under PSD same)

________ ________ ________

22 2 2 1( )

21

(0) (0) (0) (0)2v x y g

v t x t y t g t

R R R R

( ) ( ) 0x t y t

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Properties of WSS BP Processes Properties of WSS BP Processes ContinuedContinued

• Bandpass PSD from baseband PSD.

• PSD of I and Q from bandpass PSD

1

4v g c g cP f P f f P f f

0

v c v c o

x y

o

P f f P f f f BP f P f

f B

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BP White Noise ProcessBP White Noise Process

• The PSD of a BP white noise process is No/2. What is the PSD and variance of the in-phase and quadrature components?

• From the SNR calculations, it is clear that the variance of the white noise is:

cos sin

0

0

x c y c

n c n c

x y

o

x y

n t n t t n t t

S f f S f f f BP f P f

f B

N f BP f P f

f B

___ ____ ____2 2 22 2

2

c

c

f Bo

x yf B

Nn df NB n n

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Sinusoids in Gaussian NoiseSinusoids in Gaussian Noise

• Signal is a sinusoid mixed with narrow-band additive white Gaussian noise (AWGN)

• Can be written in terms of ENVELOPE and PHASE terms as:

tntAty c cos

2 2

21

cos sin

cos

Envelope

tan Phase

x c y c

c

x y

y

x

y t A n t t n t t

R t t t

R t A n t n t

n tt

A n t

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• In-phase and Quadrature terms of noise Gaussian with variance 2.

• Similar transformation to that used for calculating the dart board example, the joint density can be found in polar coordinates.

• Marginal density of the Envelope is Rician type.

• Approaches a Gaussian if A >>

2 2 22 cos / 2

2,

2

R AR A

R

Rf R e

Sinusoids in Gaussian NoiseSinusoids in Gaussian Noise

2 2 2 2

2 2 2

/ 2 cos /

2

/ 2

02 2

1,

2

R A ARR R

R A

Rf R f R d e e d

R ARe I