BANDLIMITED SIGNALING OVER AWGN CHANNELSinfo413/lecture note/C2 Bandlimitied...(from T. Noguchi, Y....

ECSE413B ♦ THO LE-NGOC ♦ BAND-LIMITED SIGNALING SCHEMES IN AN AWGN ENVIRONMENT PAGE- 1

BANDLIMITED SIGNALING OVER AWGN CHANNELS

POWER SPECTRA OF LINEARLY MODULATED SIGNALS Bandpass signal { }tj cetvts ω)(Re)( = , v(t): complex baseband signal, ωc: center (carrier) frequency

Power spectral density (psd): [ ])()(21)( cvcvs ffSffSfS −−+−=

Consider a complex baseband signal ∑∞

−∞=

−=n

n nTtgItv )()(

g(t): Tx sprectrum-shaping pulse, G(f): Fourier transform of g(t) In: symbol, real for M-PAM, complex for M-PSK, M-QAM, M-APK

In: wide-sense stationary with mean μi { } ∑∞

−∞=

−=⇒n

i nTtgtvE )()( μ

and, autocorrelation function: { }mnnII IIEm +=Φ *

21)(

PAM: ⎩⎨⎧

=+≠

=Φ0mfor 0mfor

)( 22

2

ii

iII m

σμμ

for {In}: real, mutually uncorrelated

M-ary DEMOD

+ s(t) r(t)=s(t)+n(t)

white Gaussian noise, n(t), with power spectral density

of No/2 (per Hz)

ADDITIVE WHITE GAUSSIAN NOISE (AWGN) CHANNEL

M-ary MODULATOR

No/2

frequency, f [Hz]

Bit-

to-s

ymbo

l CO

NVE

RTE

R

I/P BIT STREAM {bn}

sym

bol-t

o-bi

t CO

NVE

RTE

R

O/P BIT STREAM

{bn’}

BIT-TO-SYMBOL CONVERSION: Input binary sequence: {bn},

⎩⎨⎧

====

=p-10}Pr{bwith0

p1}Pr{bwith1b

i

in

{ai} {ai

’}

CHANNEL WITH CONSTANT

ATTENUATION C(f)=α≠0

Tx AFE

Rx AFE

THE PHYSICAL CHANNEL INCLUDES THE ANALOG Tx & Rx SECTIONS AND THE PHYSICAL MEDIUM, WHICH CAN BE A WIRELESS OR WIRELINE MEDIUM WITH C(f)=α≠0. THERMAL NOISE @ Rx INPUT IS ADDED TO THE ATTENUATED Tx SIGNAL & THE RESULTING Rx SIGNAL CAN BE AMPLIFIED FOR AN APPROPRIATE LEVEL FOR THE DEMOD


autocorrelation function of v(t):

{ } { }

444444 3444444 21

43421

T periodin t with Periodic:)(

)()(*).(

)()(*)(

21)()(*

21),( *

mTt

mTnTtgnTtgm

nTmTtgnTtgm

IIEtvtvEtt

nmII

n mII

nmnvv

−+

⎥⎦

⎤⎢⎣

⎡+−−−Φ=

−−+−

Φ

=+=+Φ

∑∑

∑ ∑

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=+

τγ

τ

τττ

Φvv(t+τ;t): periodic in t with period T, E{v(t)}: periodic with period T ⇒ v(t): cyclostationary (periodically stationary in wide sense) averaging Φvv(t+τ;t) over a single period to remove t

44444 344444 21∫∑

∑ ∫∑∫∞+

∞−

∞

−∞=

∞

−∞=

+−

−−

∞

−∞=−

−+Φ=

−+Φ=+Φ=Φ

dumTugugmT

dumTugugT

mdtttT

m

n

TnT

TnTm

T

T vvvv

)(*)(*)(1

)()(*1).(),(1)(

II

2/

2/II

2

2

τ

τττ

(time-autocorrelation function of g(t): Tjm

gg efGmT ϖτ −ℑ⎯→⎯−Φ 2)()( )

PAM: ⎩⎨⎧

=+≠

=Φ0mfor 0mfor

)( 22

2

ii

iII m

σμμ ⇒

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

+= ∑∞+

−∞=

−

43421

1/Tofperiodwithfinperiodic

222)(1)(m

mTjIIv efG

TfS ωμσ

∑+∞

−∞=

−Φℑ Φ=⎯⎯⎯ →⎯m

mTjIIv emfG

TfSvv ωτ )()(1)( 2))((

22 22

2( ) ( ) ( )

Continuous Discrete spectral linesspectrum

I Iv

m

m mS f G f G fT T TTσ μ

δ+∞

=−∞

⎛ ⎞= + −⎜ ⎟⎝ ⎠∑

14243 1444442444443

∑∑+∞

−∞=

+∞

−∞=

− −=mm

mTj

Tmf

Te )(1 δω

To remove the discrete spectral lines, we need μI=0 zero mean

sequence or:

2

0mGT

⎛ ⎞ =⎜ ⎟⎝ ⎠

for all m


TIME-LIMITED g(t) ⇒ G(f) with INFINITE BW EXAMPLE: PAM signal using rectangular pulse:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -2 0 2 4

-40

-35

-30

-25

-20

-15

-10

-5

0

-4 -2 0 2 4

2sinc( )fT 2sinc( )fT 2sinc( )fT

1020log sinc( )fT

-T/2 T/2

A g(t)

)(sincsin2

)(2/

2/2/

2/

2

fTATfT

fTATfj

AdtAefGTt

TtT

T

ftj

==

−==

−=

+=+

−

−∫

ππ

ππ


⇒ FOR BANDLIMITED TRANSMISSION, WE WANT G(f) with LIMITED

BANDWIDTH ⇒ g(t) is no longer time-limited

-B/2 +B/2

a G(f)

)(sincsin2

)(2/

2/2/

2/

2

BtaBBt

BtaBtj

adfaetgBt

BtB

B

ftj

==

==−=

+=+

−

+∫

ππ

ππ


Tx AND Rx PARTITIONING:

Tx Pavg=E{|v(t)|2}= ∫∞+

∞−dffG

TI 22

)(σ⇒Tx Es=T Pavg= ∫

+∞

∞−dffGI

22 )(σ

Define h(t)= g(t)∗gR(t), at the Rx filter output:

[ ]

[ ]{noise G.

einteferenclIntersymbo

)(

sampleMain

)0()()(

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

0

k

mm

mkkkn

n

Rn

nR

nmThIhInTnkhIkTy

tgtnnTthItgtntvty

++=+−=

⊗+−=⊗+=

∑∑

∑∞+

≠−∞=

−

∞

−∞=

∞

−∞=

44 344 21

321

nk: Gaussian noise with zero mean and variance: ∫∞+

∞−= dttgN

Ro

n22 )(

2σ

RECALL: for time-limited g(t), i.e., g(t)=0 for t∉[0,T], the Rx uses a matched filter: gR(t)=g(T-t), ⇒ Tj

RR efGfGtg ω−ℑ =⎯→⎯ )()()( * (THE MATCHED FILTER PROVIDES MAXIMUM output SNR) gR(t)=0 for t∉[0,T] ⇒h(t)= g(t)∗gR(t)=0 for t∉[-T,T] (or [0,2T]) ⇒ the intersymbol interference (ISI) term is 0, i.e., no ISI For bandlimited signaling, g(t)≠0 for t∉[0,T]. However, the ISI term can be eliminated when h(mT)=0 for all m≠0 ⇒ WE WANT BANDLIMITED G(f), GR(f)=G*(f)e-jωT AND H(f)= G(f)GR(f)

WITH h(mT)=0 for all m≠0

Symbol sequence {In}

v(t)

WGN n(t), N0/2

y(t) {y(kT)} detectionTx filter

G(f) Rx matched filter GR(f)

22

)( fGT

spectrumTx Iσ=

with μI=0


DERIVATION: MATCHED FILTER & MAXIMUM output SNR: y(t)= v(t)∗gR(t)+n(t)∗gR(t) power of noise part n(t)∗gR(t) :

N= ∫ ∫∫∫∞+

∞−

−∞+

∞−

∞+

∞−

∞+

∞−== dtdfetgfGNdffGfGNdffGN ftj

RRo

RRo

Ro π2**2 )()(

2)()(

2)(

2

N= ∫∫∞+

∞−

∞+

∞−= dttgNdttgtgN

Ro

RRo 2* )(

2)()(

2

signal part: a(t)=v(t)∗gR(t)= ∑∞

−∞=

−n

n nTthI )( where h(t)= g(t)∗gR(t)

a(kT)= ∫∑ ∫∞+

∞−

+∞

−∞=

∞+

∞−−=−− ττττττ dggIdTnkggI Rk

nRn )()()]([)(

for time-limited g(t), i.e., g(t)=0 for t∉[0,T]

Signal power: E{a2(kT)}= { }222

2 ,)()( kIRI IEdttgtg =−∫+∞

∞−σσ

Output signal-to-noise power ratio: SNR=∫

∫∞+

∞−

+∞

∞−−

dttg

dttgtg

NR

R

o

I2

2

2

)(

)()(

2/σ

is maximum when gR(t)=g(T-t) or GR(f)=G*(f)e-jωT

note: The Cauchy-Schwarz inequality: ∫∫∫+∞

∞−

+∞

∞−

+∞

∞−≤ dttydttxdttytx 22

2* )()()()(

and the equality holds when y(t)=Cx(t), C: any constant

BANDLIMITED SIGNALING IN AWGN WITH ZERO ISI

⇒ zero ISI when: ⎩⎨⎧

≠=

=000

)( 0

mmh

mTh in time domain

⇒ Nyquist theorem: H(f) satisfies constant:T0-m

hTmfH =⎟⎠⎞

⎜⎝⎛ +∑

∞

∞=

in frequency domain


DERIVATION:

∑∫

∑∫∑∫∫∞+

−∞=

+

−

+∞

−∞=

+

−

+∞

−∞=

+

−

∞+

∞−

+==

+===

k

T

T

mTj

k

T

T

mTj

k

Tk

Tk

mTfjmTfj

TkHZwithdeZ

deTkHdfefHdfefHmTh

)()(,)(

)()()()(

2/1

2/1

2

2/1

2/1

22/)12(

2/)12(

22

λλλλ

λλ

λπ

λπππ

Z(λ) is periodic function of λ with period 1/T, hence can be represented

in a Fourier series, i.e., ∑+∞

−∞=

=n

nTjnezZ λπλ 2)( with ∫−

−=T

T

nTjn deZTz

2/1

2/1

2)( λλ λπ

⇒ h(mT)=z-m/T, i.e.,

for ⎩⎨⎧

≠=

=000

)( 0

mmh

mTh we need Z(λ)= constant :0-k

ThTkH =⎟⎠⎞

⎜⎝⎛ +∑

∞

∞=

λ

Consider simple, ideal, strictly bandlimited ⎪⎩

⎪⎨

⎧

>

≤=

2for 0

2for

)(0

Bf

BfHfH

what is the narrowest bandwidth (B) such that h(mT)=0 for m≠0?

or H(f) satisfies constant -m

TTmfH =⎟⎠⎞

⎜⎝⎛ +∑

∞

∞=

⇒ time-domain: h(0)=aB, h(m/B)=0 frequency-domain:

⇒B=1/T: minimum BW for zero ISI ⇒ spectral efficiency: 1 symbol/s/Hz = m b/s/Hz for linear modulation M=2m

BUT THE IDEAL “BRICK-WALL” FILTER WITH RECTANGULAR FREQUENCY RESPONSE IS NOT PHYSICALLY REALIZABLE!

⎟⎠⎞

⎜⎝⎛ +

21 BTT

1⎟⎠⎞

⎜⎝⎛ −

21 BT

0 +B/2 -B/2

21

2B

TB

−= want

-B/2 +B/2

a H(f)

)(sincsin2

)(2/

2/2/

2/

2

BtaBBt

BtaBtj

adfaethBt

BtB

B

ftj

==

==−=

+=+

−

+∫

ππ

ππ


raised-cosine filter TRANSFER FUNCTION:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+>

+≤≤

−

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−+

−≤≤

=

Tf

Tf

TTfTT

TfT

fH

210

21

21

21cos1

2

)1(210

)(

α

ααααπ

α

or:

[ ]{ }⎪⎩

⎪⎨

⎧+≤≤−−−

−≤≤=

elseweherexx

xTxR

02/)1(2/)1(2/}2/)1{(cos

2/)1(0),( 2 ααααπ

αα

where x=(f-fc)T

IMPULSE RESPONSE:

⇒ )/(sinc21

cos),( 2 S

S

S Tt

Tt

Tt

tr πα

πα

α

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

Responses of a raised-cosine filter to an input sequence of:

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

with linear phase response

T

( ), ( 2 ), ( 3 ), ( 8 )t t T t T t T∂ ∂ − ∂ − ∂ −


SPECTRA OF Tx SIGNALS USING RRC FILTERS:

PEAK INSTANTANEOUS POWER RELATIVE TO POWER OF A QPSK SIGNAL USING TIME-LIMITED RECTANGULAR PULSE SHAPE

(from T. Noguchi, Y. Daido, J.A. Nossek, “Modulation Techniques for Microwave Digital Radio”, IEEE Communications Magazine, October 1986, pp. 21-30)


EYE DIAGRAMS:

PAM 4PAM 2

Examples of eye patterns for

2-PAM 4PAM


EYE DIAGRAMS FOR RAISED-COSINE FILTER:

RAISED-COSINE (RC) FILTER


EYE DIAGRAMS FOR RAISED-COSINE FILTER:


DESIGN OF Tx AND Rx FILTERS FOR BANDLIMITED TRANSMISSION IN AWGN CHANNELS WITH ZERO ISI

For maximum output SNR, select: GR(f)=G*(f)e-jωT For zero ISI in y(kT), select: H(f)=GR(f)G

*(f)=R(f,α): raised-cosine (RC) ⇒ optimum partition: linear-phase Tx and Rx filters with root raised-

cosine (RRC) amplitude response, i.e., G(f)=[ R(f,α)]1/2ej2πfτ and GR(f)=[ R(f,α)]1/2ej2πfd where τ and d: constant group delay Tx signal: ∑

∞

−∞=

−=n

n nTtgItv )()( , avg. Tx power: Pavg=T

dffRT

dffGT

III22

22

),()( σα

σσ== ∫∫

+∞

∞−

+∞

∞−

Avg energy per symbol: Es=TPavg= σI2.

Rx sample y(kT)=Ik+nk since h0=h(0)=1, Ik: signal sample nk: Gaussian noise sample with zero mean, variance:

2)(

22 o

Ro NdffGN

=∫∞+

∞−

{ }

)]([2

)]([2

)]([)(2

)()(2

)()()(2

)()()()(

TklNTklhNdttTklgtgNdxxlTgxkTgN

dxdyylTgxkTgyxNdxdyylTgxkTgywxwEnnE

ooRR

oRR

o

RRo

RRlk

−∂=−=−−=−−=

−−−∂=⎭⎬⎫

⎩⎨⎧

−−=

∫∫

∫ ∫∫ ∫∞+

∞−

∞+

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

Example: bandlimited M-ASK Ik∈ {A(2n-1-M), n=1,2,…,M} Es=TPavg= σI

2= A2(M2-1) /3 Rx sample y(kT)=Ik+nk Ik: signal sample with dmin=2A or [dmin]

2=12Es/( M2-1)

Performance: same as the time-limited case: ⎥⎦

⎤⎢⎣

⎡

−⎥⎦⎤

⎢⎣⎡ −=

02 1311

NE

Merfc

MP s

e

Symbol sequence {In}

v(t)

WGN n(t), N0/2

y(t) {y(kT)} detectionTx filter

G(f)Rx matched filter GR(f)

22

)( fGT

spectrumTx Iσ= with μI=0


OPTIMUM PARTITION OF Tx/Rx FILTERS FOR BANDLIMITED TRANSMISSION IN AWGN CHANNELS WITH ZERO ISI:

CASE OF INPUT RECTANGULAR PULSES

GENERAL CASE: p(t)↔P(f): Fourier transform of p(t)

)(/),()()()(function transfer channeloverall

),()(filter

)(/),()(filter

xPxRxHxHxH

xRxHRcv

xPxRxHTx

RT

R

T

α

α

α

==

=

=

Signal generated using rectangular pulses:

∑+∞

−∞=

−=n

sn nTtpIts )()(

v(t)

WGN n(t), N0/2

y(t)

{y(kT)} detectionTx filter

G(f)Rx matched filter GR(f)

symbol sequence {In}

v(t)

WGN n(t), N0/2

y(t)

{y(kT)} detectionTx filter

HT(f)Rx matched filter HR(f)

)(sinc)(

,00,1

)(

2SS

s

fTCfW

elsewhereTt

tp

π=⎩⎨⎧ <≤

=

)(sinc/),()()()(function transfer channeloverall

),()(filter

)(sinc/),()(filter

xxRxHxHxH

xRxHRcv

xxRxHTx

RT

R

T

πα

α

πα

==

=

=

x=(f-fc)Ts, fc: carrier frequency Ts: symbol interval

H(x) is normalized for H(0)=1 in this plot


EXAMPLE OF FILTERING REQUIREMENTS: INTELSAT V TDMA/QPSK Carrier frequency: fC=140 MHz Tx bit rate: fb=120.832 MHz Nyquist frequency: fN = fS/2=30.208 MHz Raised cosine filter with α=0.4


EFFECTS OF AMPLITUDE AND PHASE DISTORTIONS ON RECEIVER PERFORMANCE (from T. Noguchi, Y. Daido, J.A. Nossek, “Modulation Techniques for Microwave Digital Radio”, IEEE Communications Magazine, October 1986, pp. 21-30)

BANDLIMITED SIGNALING OVER AWGN CHANNELSinfo413/lecture note/C2 Bandlimitied...(from T. Noguchi, Y....

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Transcript of BANDLIMITED SIGNALING OVER AWGN CHANNELSinfo413/lecture note/C2 Bandlimitied...(from T. Noguchi, Y....