TSDT14 Signal Theory · ACF of Angle Modulation 3(3) 2016-09-23 5 TSDT14 Signal Theory - Lecture 8...

TSDT14 Signal TheoryLecture 8

Analog Modulation – Part 2

Repetition of Sampling and PAM (deterministically)

Sampling and PAM of Stochastic Processes

Mikael Olofsson

Department of EE (ISY)

Div. of Communication Systems

Broadband Angle Modulation

2016-09-23 2

TSDT14 Signal Theory - Lecture 8

Modulation of a Gaussian process

Assumptions: is an ergodic Gaussian process, mean 0.

is uniform on [0,2π) and indep of .

Signal:

Phase dev.:

Freq. dev.:

Relation: ⇒

Definitions:

⇒

ACF of Angle Modulation 1(3)

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We had:

Then:

Temporary Gaussian variable: (mean 0)

Important observation: This is real valued.

Rewrite:

We have, with f = –1/2π, using the identity :


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Recall: (mean 0)

We had:

We have:

Mean and ACF independent of time t. Wide sense stationary!

The introduced process is a normalized baseband

representation of .

PSD of Angle Modulation

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We had:

This holds if has no periodic components and has mean 0:

Recall:

Spectrum:

The only relation to is through and .

Carrier Sidebands

Bandwidth of Angle Modulation

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Define bandwidth BX as the width of the frequency band, symmetrically

around the carrier frequency that contains 90% of the power.

Broadband ⇒ very small. Ignore!

We get:

⇒

Compare Carson’s rule: Bandwidth is slightly more than 2·fd,max.

Special Cases: Broadband PM and FM

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Phase modulation:

Frequency modulation

Narrowband Angle Modulation

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We have:

Narrowband modulation, i.e. Φd(t) is small:

Spectrum:

Carrier Sidebands

We get AM in the so called quadrature component.

AWGN Impact on Angle Modulation

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Noise PSD:

Filter:

Filtered noise PSD:

Filtered noise power:

Signal power: Input SNR:

AWGN Impact on Phase Modulation

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Output signal power:Output signal:

Can be shown:

Filter:

Output noise:

Noise power:

Output SNR:

AWGN Impact on Frequency Modulation

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Output signal power:Output signal:

Can be shown:

Filter:

Output noise:

Noise power: Output SNR:

Linear Mappings

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nTx(t) y[n] = x(nT )

PAMp(t)nT

y[n]x(t ) z(t )

PAMp(t) nT

y(t )x[n] z[n]

Sampling:

Pulse-Amplitude Modulation:

(PAM)

Reconstruction:

PAMp(t)

y[n] z(t ) = y[n] p(t – nT )Σn

t → n → t :

n → t → n :

Sampling – Deterministically (from S&S)

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nTx(t) y[n] Sampling frequency:

Time domain:

If X( f ) = 0 for | f | ≥ fs/2:

| θ | < 1/2Y [θ ] = fs X(θ fs)

Frequency domain:

y[n] = x(nT )

fs = 1/T

for

Y [θ ] = Y [θ + k] for k integer

1

Sampling – Frequency Domain

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Original spectrum:

Variable substitution:

Amplitude scaling:

Periodic repetition:1WT

A( )fX

ffsW–W

–WT

( )sfX θ

θ

A

( )( )∑ −m

fmX sθ

θ

A

11/2–1/2 WT–WT–1

Afs

[ ]θY

θ11/2–1/2 WT–WT–1

[ ] ( )( )∑ −=m

fmXfY ss θθ

PAM – Deterministically (from S&S)

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Time domain:

Frequency domain:

PAM

p(t)y[n] z(t)

z(t) = y[n] p(t – nT)Σn

Z( f ) = P( f ) Y [ f T ]

1( )tp

tT–T

Example:

[ ]ny

t1–1

t1–1

( )tz

PAM – Frequency Domain

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Original spectrum:

Variable substitution:

The result:

Spectrum of the pulse:

fsW–W

Afs

[ ]θY

θ11/2–1/2 WT–WT–1

Afs

[ ]TfY

ffsfs/2–fs/2 W–W–fs

( )fP

f

ffsfs/2–fs/2 W–W–fs

P(0)Afs( )fZ

Z( f ) = P( f ) Y [ f T ]

Sampling of a Stochastic Process 1(2)

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nTX(t) Y[n] = X(nT) Assumption: X(t ) WSS

Mean:

ACF:

[ ] ( )( )∑∑ −=

−=

m

X

m

XY fmRfT

mR

TR ss

1θ

θθ

[ ] [ ]{ } ( ){ } XY mnTXnYnm === EE

[ ] [ ] [ ]{ } ( ) ( )( ){ } ( )kTrTknXnTXknYnYknnr XY =+=+=+ EE,

PSD:

Definition Sampling X(t ) WSS

This is sampling of the deterministic function rX(τ ).

Sampling of a Stochastic Process 2(2)

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nTX(t) Y[n] = X(nT) Assumption: X(t ) WSS

If RX( f ) = 0 for | f | ≥ fs/2:

| θ | < 1/2RY[θ ] = fs RX(θ fs) for

RY[θ ] = RY[θ + k] for k integer

[ ] ( )( )∑∑ −=

−=

m

X

m

XY fmRfT

mR

TR ss

1θ

θθ

PAM of a Stochastic Process 1(3)

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Time domain (naïve approach):

PAM

p(t)Y[n] Z(t)

Z(t) = Y[n] p(t – nT)Σn

Problem: Does not retain WSS.

Time domain (alternative approach):

Z(t) = Y[n] p(t – nT – Ψ)Σn

mZ(t) = mY p(t – nT)Σn

= p(t–nT) p(t+τ–mT) rY[n–m]ΣΣn m

Ψ unif. [0,T]

Ψ & Y[n] indep.

Mean:

( ) ( ){ } [ ] ( )

Ψ−−== ∑ nTtpnYtZtmn

Z EE

[ ]{ } ( ){ }∑ Ψ−−=n

nTtpnY EE

( ) ( ) ψψψ dfnTtpmn

Y Ψ

∞

∞−

∑ ∫ −−=

( ) =−−= ∑ ∫ ψψ dT

nTtpmn

T

Y

1

0ψφ

ψφ

dd

nTt

−=

−−=

( ) φφ dpT

mn

nTt

TnTt

Y ∑ ∫−

−−

=1

( ) φφ dpT

mY ∫∞

∞−

=1

( ) YmPT

01

= Independent of t.

rZ(t+τ) =

(indep.)


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ACF:

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

Ψ−−Ψ−−+=+=+ ∑∑ mTtpmYnTtpnYtZtZttrmn

Z τττ EE,

ψφ

ψφ

dd

tmT

=

−+=

[ ] [ ]{ } ( ) ( ){ }∑∑ Ψ−−Ψ−−+=n m

mTtpnTtpmYnY τEE

[ ] ( ) ( ) =−−−−+−= ∑∑ ∫ ψψψτ dT

mTtpnTtpmnrn m

T

Y

1

0

mnk −=

[ ] ( )( ) ( ) =−−−+−+= ∑ ∑ ∫ ψψψτ dmTtpTmktpkrT k m

T

Y

0

1

[ ] ( ) ( ) =−−−= ∑ ∑ ∫−+

−

φφφτ dpkTpkrT k m

tmTT

tmT

Y

1( ) ( )tptp −=~define

[ ] ( ) ( )∑ ∫∞

∞−

−−=k

Y dpkTpkrT

φφφτ ~1 [ ] ( )( )kTppkrT k

Y −∗= ∑ τ~1

Independent of t.


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ACF: ( ) [ ] ( )( )kTppkrT

rk

YZ −∗= ∑ ττ ~1

Independent of t.

Thus WSS

( ) YZ mPT

m 01

=Mean:

Recall deterministic PAM:

PSD: ( ) [ ] ( )( )

−∗= ∑ kTppkrT

fRk

YZ τ~1F

( )( ){ } ( )2~ fPpp =∗ τF

Notice: The sum is PAM of rY[k] using pulse ( p * p )(τ ).~

( ) ( ) [ ]fTRfPT

fR YZ

21=

Sampling and PAM of WSS Processes

– Summary

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nTX(t) Y[n] PAM

p(t)Y[n] Z(t)

[ ] ∑

−=

m

XYT

mR

TR

θθ

1

XY mm =

[ ] ( )kTrkr XY =

[ ] ( )nTXnY =

( ) [ ] ( )( )kTppkrT

rk

YZ −∗= ∑ ττ ~1

( ) YZ mPT

m 01

=

( ) ( ) [ ]fTRfPT

fR YZ

21=

( ) [ ] ( )Ψ−−= ∑ kTtpnYtZn

Ψ unif. [0,T]

Ψ & Y[n] indep.

www.liu.se

Mikael Olofsson

ISY/CommSys

TSDT14 Signal Theory · ACF of Angle Modulation 3(3) 2016-09-23 5 TSDT14 Signal Theory - Lecture 8...

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Transcript of TSDT14 Signal Theory · ACF of Angle Modulation 3(3) 2016-09-23 5 TSDT14 Signal Theory - Lecture 8...