chapter 4 baseband pulse transmission - fju.edu.tw

Digital Communication輔仁大學電子工程系所

Chapter 4Baseband Pulse Transmission

4 - 1Chapter 4: Baseband pulse modulation Dr. Sheng-Chou Lin


Outline

•Matched filter which is the optimum system for detecting a known signalim AWGN

•Calculation of bit error rate due to channel noise

• Intersymbol Interference (ISI) which arises when channel is dispersive

•Nyquest’s criterion for distortionless baseband data transmission

•Correlative-level codingfor combating ISI

•Digital subscriber lines

•Equalization of a dispersive baseband channel

•Eye Pattern for displaying combined effects of ISI and channel noise



Introduction

•Transmission of digital data—Baseband channel: digital computer—Bandpass channel: microwave radio, satellite channelmodulation process

•Baseband transmission: low-pass channel BW > data stream BW

•Major sources of bit errors—Noise: optimum detection in presence of noise Matched filter—Intersymbol interference (ISI):> due to non-ideal low-pass channel

> can be reduced using pulse shaping and equalization

AWGN Channel ISI Channel

Shaping Filter



Matched Filter

•Optimum design of receiver filter for detecting a pulse transmitted over achannel corrupted by additive noise

x t g t w t+= 0 t T

filter input pulse signal white noise

T: an arbitrary observation interval

y t go t n t+=

g t h t w t h tfilter output

go T2

E n2 t ----------------------=

maximizing the peak pulse SNRmeasured at time t = T

useschwarz’s inequality

hopt t kg T t– =

Hopt f kG* f j2fT– exp=

: a time-reversed and delay version of g(t)

Linear receiver



Properties of Matched Filter

T2---

Tt

A2---

A2---– A2T 8

A2T 4

tT 2 3T 4 2T

go t

T

hopt tg t

•Peak pulse SNR depends on ratio of signal to power spectral density of noise

Go f Hopt fG f k G f2 j2fT– exp= =

go T Go f j2fT exp fd–

k G f2 fd–

kE= = =

Fourier of resulting matched filter output go(t)

matched filter output at time t = T

E g2 tfd–

G f2 fd–

= = signal energy = integral of square magnitude spectrum of a pulse signal

E n2 t k2No

2---------- G f2 fd

–

k2No E2

--------------= = average output noise power

maxkE 2

k2NoE 2--------------------------- 2E

No------= =peak pulse signal-to-noise ratio =

E No signal energy-to-noisespectral density ratio

=



Matched Filter for Rectangular Pulse

•Correlator configurationhopt t kg T t– =

y t h t x t g T – x t – d0

T

= =

y T g T – x T – d0

T

g x d0

T

= =

T T – =

td0T

x t g t w t+=

g t

y t y T

y T g tx ttd0

T

=

• For a rectangular pulse, matched filter may beimplemented using a circuit “Integrate-and-dump”

Integrate-and-dump circuit

rectangular pulse

Matched filter output

Integrator output



Output of Matched Filter

•Consider a binary PCM system based on polar non-return-to-zero (NRZ)—symbols 1 and 0 are represented by positive and negative rectangular pulses—Channel noise is modeled as additive white Gaussian noise w(t) of zero

mean and power spectral density No/2

Receiver for baseband transmission of binary-encoded PCM wave using polar NRZ signaling.

x t+A w t, symbol 1+A– w t, symbol 0+

=y x ttd

0

Tb A– 1

Tb----- w ttd

0

Tb+= =

Received signal Matched filter output sample at time t = Tb

kATb=1 Gaussian distributed



Output Noise Variance

•The output y represents sample value of a random variable Y—Random variable Y is Gaussian distributed with a mean A—Variance of random variable Y is Y

2 n2=

(b) Probability density function of Y when 1

Noise analysis of PCM system.

(a) Probability density function of random variable Yat matched filter output when 0 is transmitted. is transmitted.

n t 1Tb----- w ttd

0

Tb=

Output noiseE n2 t 1

T b2------ E w ttd

0

Tb w uud

0

Tb

1T b

2------ E w tw u td ud0

Tb0

Tb= =

1T b

2------No

2----- ud

0

Tb= 1

T b2------ Tb

No

2-----

No

2Tb--------

No

2----- 1

Tb----- = = =

No t u– noise variance

2W 1 Tb=equivalent noise BW



Error Rate Analysis

fY y 0 12Y

2----------------- y A+ 2–

2Y2

----------------------exp=

1No Tb

---------------------- y A+ 2–No Tb

----------------------exp=

conditional probability of errorgiven symbol = 0

p10 P Y symbol 0 fY y 0 yd0

= =0 0=

p101

No Tb---------------------- y A+ 2–

No Tb----------------------exp yd

0

=

1No Tb

---------------------- A2–No Tb--------------exp yd

A

=1

------- z2– exp zdANo Tb

-----------------

=

z y No Tb =

12--- erfc

ANo Tb

-----------------= 12--- erfc

A2Tb

No---------- 1

2--- erfc

Eb

No-----= =

erfc u1

------- z2– exp zdu=

Q x 12--- erfc

x2

-------=

Eb = A2Tb = bit energy



Probability of Error

• Average probability of symbol error is abinary symmetric channel dependssolely on Eb/No

- Exponential improvement inaverage probability of symbol

- decreases very rapidly as Eb/No isincreased

- small increase in Eb error free

p10 p01=

Pe p0 p10 p1 p01+ p10= =

12--- erfc

Eb

No-----=

p0 p1 1 2= =

average probabilityof symbol error

Eb /No: ratio of transmitted signal

per bit to noise spectral density

erfc u u2– exp2 u

-------------------------PeEb No– exp

2 Eb No --------------------------------

upper bound

Probability of error in a PCM receiver



M-ary PAM Transmission

1.5

-1.5

0.5

-0.5T0 2T 3T 4T

00 01 10 11

T=2Tb

•M-ary PAM is faster than thecorresponding binary PAM

- 1 baud = bits/sec.

- 1/T : signaling rate = bauds rate =symbols/sec.

•Given the same average Pe, M-aryPAM needs more transmit power

Log2 M

Tb Log2 M

3AA-A-3A–

Pe1M---- M 2– P n A 2P n A + =

M 1–M

------------P n A = M 1–M

------------erfcANo T

----------------=

x2 t E ak2 2

M---- 2m 1– 2A2

m 1=

M 2

A2 M 2 1–3

-------------- Pav= = = =

PeM 1–M

------------erfcANo T

---------------- M 1–M

------------erfcPav3TM2 1– No

------------------------= =

1 1M----–

erfc3Es

M 2 1– No-------------------------= Es Eb Log2 M=



Intersymbol Interference

Baseband binary data transmission system

• Intersymbol Interference (ISI)

- Due to dispersive channel

• A key question: A pulse shape of interest

- Discrete Pulse-amplitude-amplitude (PAM)

Transmitted signal (Binary PAM)s t ak g t kTs–

k=

y t ak p t kTs– k n t+=

Received signalp t g t h t c t =

p 01= p(t) is normalized

P f G fH fC f=

scaling factor. amp. changes

y(t) is sampled at time ti = iTb

y ti ak p i k– Tb k –=

n ti +=

ai ak p i k– Tb k –=k i

n ti + +=

through system Pe12-----E erfc

A +No Tb

------------------=

12-----E erfc

A2

N0Tb----------

N0Tb----------+

p 0A= ISI = SNR A2

N0Tb----------=

S/N is highPerformanceis limited byISI rather thannoise

2L com.



Nyquist’s Criterion

•Nyquist’s Criterion for distortionless baseband Binary transmission

- No ISI for bandlimited channel

Ideal magnitude response Ideal basic pulse shape

P f nRb– n –=

Tb= P f t 1–

p i k– Tb 1 , i k=0 , i k

=

Pf Rb P f nRb– n –=

p 0 1= =

Fourier Transform

sampling time

Rb 1 Tb=

t 1For zero ISI

•Consider rectangular function

P f 12W------

f2W------ =

p t c 2Wt sin=

2W 1 Tb Rb= =

Nyquest rate (Rect.) = Rb= 2W,W: Nyquest BW

Tb

1–Tb

P f nRb– n –=

t

Rb– Rb 2 RbRb 2f t

Tb Tb 1=



Rectangular Spectrum

• Two difficulities for rectangular function

- undesirable because of the abrupttransmission at the band edges

- p(t) decays as for large slowdecay; due to timing error

W

1 t tISI

timing error

y t a0 c 2Wt sin 2Wt sin

-------------------------------1– kak

2Wt k– ---------------------------

k 0+=

desired symbol Diverge1k---

k 1=

P(f) is discontinuous p(t) ~ 1/t

d/df P(f) ............... p(t) ~ 1/t2

d2/df2 P(f) ............... p(t) ~ 1/t3

dn/df n P(f) ........... p(t) ~ 1/tn+1

A series of sinc pulse corresponding to sequence 1011010

1 2W

WW– fW

W– f

discontinuous



Raised Cosine Spectrum

Frequency response

Time response

different rollofffactor

•Raised Cosine Spectrum

- overcome the two above practicaldifficulties with ideal Nyquest channel

- Extend the bandwidth from the minimumvalue W = Rb /2 to W ~ 2W

- dn/df n P(f) ........... p(t) ~ 1/tn+1flat portion; rolloff portion

P f

12W------ , 0 f f1

14W------ 1 f W–

2W 2f1–------------------------sin–

, f1 f 2W f– 1

0 , f 2W f– 1

=

1 f1 W–=Rolloff factorexcess bandwidth over W = (W-f1)/W = 1-f1 /W

Transmission bandwidth: BT = W + (Wf1) = (1+)W

0 1

BT



Characteristics of Raised Cosine

•Rolloff factor vs.Bandwidth

- Transmission bandwidth: BT = W +(W f1) = (1+)W

- =0 ideal Nyquest channel

- =1 BT = 2W ; full-cosine Rollofff

• Insensitive to sampling errors

•Deduce tails of pulse considerably

- =1 tails of p(t) are samllest

- Higher , Higher BT ; smaller ISI dueto timing error

• Time response of P(f)

• For=1

time response

Two interesting properties: useful inextracting a timing signal from the receivedsignal for synchronization

- p(t) = 0.5 at , pulse width of1/2 Amp. = Tb

- zero crossing at ,in addition to ,

price paid: BT (=1 )= 2W =2BT (=0 )

P f14W------ 1 f

2W------cos–

, f1 f 2W

0 , f 2W

=

p t 4Wt sin1 16W2t2–--------------------------=

t Tb 2=

t 3Tb 2= 5Tb 2t Tb= t 2Tb=

p t c 2Wt sin 2Wt cos1 162W 2t2–---------------------------------=

decay as 1 t2ideal Nyquest channel



A Example (BW of T1 System)

•T1 carrier; 24 channels; 8 bit PCM; Tb = 0.647msec;

—Minimum transmission bandwidth; Tb=1/2W

BT (= 0) = W =1/2Tb= 1/2(0.647ms)=> 722 kHz

—For full-cosine case =1

BT (= 1) = 2W =1/Tb= 1/(0.647ms=> 1.544 MHz

•For SSB (single-sided band): Bandwidth = 4 k Hz (Voice bandwidth =3.1k Hz); 24 Channels FDM—BT (SSB) = 4 (24)=196kHz << BT(T1)



Optimum Linear Receiver

•Two channel conditions for baseband data transmission system—Channel noise acting alone matched filter receiver—Intersymbol interference (ISI) acting alone pulse-shaping transmit filter to

realize Nyquest channel

In a real-life situation: channel noise + Intersymbol interference

•Objective: design a linear optimum receiver for linear dispersive andnoisy channel—Zero-forcing equalizer: ISI is forced to zero at all instants;> Optimal if channel is zero noise; simple because effect of channel noise is ignored

> Performance is degraded due to noise enhancement

—A refined approach mean-square error criterion (MSE)> A balanced solution to reduce both channel noise and ISI

> computational complexity

> performs as well as and often better than zero-forcing equalizer



MMSE Receiver (I)

w t

channel Receiverfilter

c t

x t y tImpulse response of receiver filter = c(t)Channel output = x(t)

x t ak q t kTb– k w t+=

q t g t h t=Noise

TX filter Channel

y t c x t – d– c t x t= =

Receiver filter output

Resulting output y(t) at t = iTb

y iTb i= ni+

i ak c q iTb kTb –– d–

k=

ni c w iTb – d–=

Error signal

ei y iTb ai– i ni+ ai–= =

transmitted symbol

J 12---- E ei

2 =mean-square-error

J 12---- E i

2 E ni2 E ai

2 + + E ini E niai E iai ––+=0 0

: stationary,i E i independent of instant time t = iTb

Rq 1 2 Rq 1 2– q kTb 1– q kTb 2– k= =

E i2 Rq 1 2 c 1 c 2 1 2dd––

=

E ni2 c 1 c 2 RW 2 1– 1 2dd

––

=

No

2----- c 1 c 2 2 1– 1 2dd

––

=

E ai2 1=

E iai E ak ak c q iTb kTb –– d–

k=

c q – d–=



MMSE Receiver (II)

J 12---- E i

2 E ni2 E ai

2 + + E ini E niai E iai ––+=

12--- 1

2--- Rq t –

No

2-----t – +

c tc t dd––

c tq t– td

––+=

c tJ 0=

Rq t – No

2----- t – +

c d– q t– =

Fourier transform

Sq fNo

2-----+ C f Qf=

Sq f 1Tb----- Q f k

Tb-----+

2

k=

C f Qf

Sq fNo

2-----+

--------------------------=Optimum Linear receiver

power spectral density of sequence

g t G f

gt G f–

Gf g t–

q kTb

A matched filter q(-t)

A transversal (tapped-delay-line) equalizer

Heq f 1

Sq fNo

2-----+

--------------------------=



Optimum Linear Receiver

Optimum linear receiver consisting of the cascade connection of matched filter and transversal equalizer.

• Practical considerations

- Channel is usually time varying in a real-life telecommunications environment

• Full transmission capacity of telephonechannel adaptive receiver

- Adaptive implementation of matchedfilter and equalizer

• Two factors contribute to pulse distortion ina public switched telephone network

- Difference in transmission of individuallinks that may be switched together

- Difference in number of links in aconnection

• Telephone channel is random

C(f)ck k N–=N



Optimum Receiver with Finite-tap Equalizer

x iT ak p iT kT– k iT + xi= =

y iT yi cjxi j–j N–=

N

= =

Equalizer output

Equalizer input

cuJ 0=

J E yi ak– 2 =

cj RX i j–

j N–=

N

RX i– RX i= =

p t q t q t– =

t w t q t– =

autocorrelation of x

matched filter

Jmin c2 1 cj p i–

j N–=

N

– c2 1 go– = =

C fP fPf No

a2

------ Pf+ Pf=

as j

pi q t q t– t It==

as j cj pi j– p i j– –No

c2

------ pj i–+j N–=

N

p i–=Jmin c

2 1 go– c2T 1 P f+ 1– fd

1 2T–

1 2T

= =

c2 T2------- 1 P w+ 1– wd

T–

T

=

C f Heq f 1P f No a

2+----------------------------------= =

P f 1T--- Q f k

T---+

2

k –=

=

P w 1T--- Q f 2k

T----------+

2

k –=

=

w t

channel matchedfilter

q t–

x t

x iT Equalizer

y iT



Adaptive Equalization

Block diagram of adaptive equalizer

LMS Algorithm

Two operating modes of an adaptive equalizer

• Two operatingmodes of anadaptive equalizer

- Training mode(Position 1)

- Tracking mode(Position 2)



Decision-Feedback Equalization

Block diagram of decision-Feedback equalizer

yn hk xn k–k=

h0 xn hk xn k–k 0 hk xn k–

k 0+ +=

feedback feedforward

• Avoid noise enhancement• nonlinear and therefore more difficult to analyze• Error propagation: not persist indefinitely; occur in burst- L= number of feedback taps. L consecutive correct decisions

decision errors will be flushed out finite duration

- P(next decision error) < 1/2

- Error rate is increased by a factor ~ 2L ; L consecutive correctdecisions



Distorted Pulse

•System parameters—QPSK modulation—Raised-cosine pulse shaping; Rolloff factor = 0.35—Two-ray mobile radio channel model—Infinite-length equalizer

-20 -15 -10 -5 0 5 10 15 20

Sampling time ( x T )

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Am

plitu

de

-20 -15 -10 -5 0 5 10 15 20


-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5in-phase channel Quadrature channel

SNR=10dB



Equalized Pulse

-20 -15 -10 -5 0 5 10 15 20


-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Am

plitu

de

-20 -15 -10 -5 0 5 10 15 20


-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-20 -15 -10 -5 0 5 10 15 20


-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Am

plitu

de

-20 -15 -10 -5 0 5 10 15 20


-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Linear

DFE

in-phase Quadrature

in-phase Quadrature

SNR=10dB

SNR=10dB



Eye Patterns (Diagram)

• Eye Patterns: An experimental tool for the channel (noise+ISI)

- Evaluate combined effect of these impairments (channel noise and intersymbolinterference) on overall system performance

- It resembles human eye for binary wave

- Interior region of eye pattern Eye opening

• An eye pattern provides a great deal of useful information about the performance

- Width of eye opening time intervalover which received signal can besampled without error from ISI; preferredtime eye is open the widest

- Sensitivity of system to timing errors rate of closure of eye as sampling time isvaried

- Height of eye opening noise margin ofthe system



Eye Patterns (Channel Noise)

Noiseless

SNR = 20dB

SNR = 10dB

• Eye diagram of system under idealized conditions

- A quaternary (M=4) baseband PAM

- Source symbols are randomly generated

- no channel noise; no bandwidth limitation

- Raised-cosine pulse-shaping (W=0.5, = 0.5 Hz)

- M-1 = 3 openings; T Tblog2M 2Tb= =

• SNR=20dB; effectof noise is hardlydiscernibleeyeis open

• SNR=10dB;openings of eyediagram arebarely visible



Eye Patterns (Bandwidth limitation)

•Eye diagram of system under a bandwidth-limitedcondition and a noiseless channel

- Channel is modeled by a low-pass butterworthfilter, squared magnitude response is

, N : order of the filter

•N = 25 and fo = 0.975Hz; BT = W(1+) = 0.75 Hz

- Channel Bandwidth > BT

- A decrease in the size of eye openings

- There is a blurred region at time = 1 sec

•N = 25 and fo = 0.5Hz

- The channel bandwidth is reduced; ChannelBandwidth < BT

- Further reduce the extent to which eyes are open

H f2 11 f fo 2N+------------------------------=

chapter 4 baseband pulse transmission - fju.edu.tw

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