1Dept. of EE, NDHU

Chapter Three

Baseband Demodulation/Detection

2Dept. of EE, NDHU

Error Probability Performance

• Error probability function

where is the time cross-correlation coefficient between two signals

• Antitpodal signal

– equals to -1, then

• Orthogonal signal

– equals to 0, then

))1(

()2

(00 N

EQ

N

EQP bd

B

)2

(0N

EQP b

B

)(0N

EQP b

B

3Dept. of EE, NDHU

Error Probability of Binary Signaling

• Unipolar signaling

• Detection of unipolar baseband signaling0binary for 0 0)(

1binary for 0 )(

2

1

TttS

TtAtS

TAaa

Ta

TAdttnAAEtsTzETaT

2210

2

2

011

)2/1(2

thresholdoptimal the

,0)(

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

)(0N

EQP b

B

4Dept. of EE, NDHU

Error Probability of Binary Signaling

• Bipolar signaling

• Detection of bipolar baseband signaling0binary for 0 )(

1binary for 0 )(

2

1

TtAtS

TtAtS

02

thresholdoptimal the 210

21

aa

aa

)2

(0N

EQP b

B

5Dept. of EE, NDHU

Bit Error Performance of Unipolar and Bipolar Signaling

6Dept. of EE, NDHU

Intersymbol Interference in the Detection Process

)()()()( fHfHfHfH rct

7Dept. of EE, NDHU

Nyquist Channels for Zero ISI

8Dept. of EE, NDHU

Pulse Shaping to Reduce ISI

• Goals and Trade-offs

– Compact signaling spectrum is to provide the higher allowable data rate

– Time pulse would become spread in time, which induces ISI

• The Raised-Cosine filter

where W is the absolute bandwidth and W0=1/2T represents the minimum Nyqu

ist bandwidth and the -6 dB bandwidth

Wffor

WfWWforWW

WWfWWffor

fH

0

2 )2

4(cos

2 1

)( 00

02

0

20

000

])(4[1

])(2cos[)2(sin2)(

tWW

tWWtWcWth

9Dept. of EE, NDHU

Raised-Cosine Filter Characteristics

10Dept. of EE, NDHU

Two Types of Error-Perfformance Degradation

11Dept. of EE, NDHU

Example 3.3 Bandwidth Requirements

(a) Find the minimum required bandwidth for the baseband transmission of

a four-level PAM pulse sequence having a data rate of R=2400 bits/s if t

he system transfer characteristic consists of a raised-cosine spectrum wit

h 100% excess bandwidth (r=1)

(b) The same 4-ary PAM sequence is modulated onto a carrier wave, so that

the baseband spectrum is shifted and centered at frequency f0. Find the

minimum required DSB bandwidth for transmitting the modulated PAM

sequence

12Dept. of EE, NDHU

Nyquist Pulse

13Dept. of EE, NDHU

Square-root Nyquist Pulse and Raised-cosine Pulse

14Dept. of EE, NDHU

Equalization

• Maximum-likelihood sequence estimation (MLSE)

– Make measurement of channel response and adjust the receiver to the transmission environment

– Enable the detector to make good estimates from the distorted pulse sequence (ex. Viterbi equaliza

tion)

• Equalization with filtering

– Use filter to compensate the distorted pulse

– Linear filter contains only feedforward elements (ex. transversal equalizers)

– Non-linear filter contains both feedforward and feedback elements (ex. decision feedback equalize

rs)

– Preset or adaptive filter design

– Filter’s resolution and update rate

15Dept. of EE, NDHU

Receiving / Equalizing Filter

• The overall transfer function

• System design goal

then Ht(f) and Hr(f) each have frequency transfer functions that are the square r

oot of the raised cosine.

• Equalizing filter sometimes not only compensates the channel effect but compen

sates the ISI brought by the transmitter and receiver (ex. Gaussian filter)

)()()()()( fHfHfHfHfH erctRC

)()()(

)(

1

)(

1)( )(

fHfHfH

efHfH

fH

rtRC

fj

cce

c

16Dept. of EE, NDHU

Eye Pattern

• Eye pattern is a filtering effect

17Dept. of EE, NDHU

Distorted Pulse Response

18Dept. of EE, NDHU

Transversal Equalizer

• A training sequence (like PN sequence) is needed to estimate the channel freque

ncy response

• A transversal filter is the most popular form of an easily adjustable equal

izing filter consisting of a delay line with T-second tapes

• The main contribution is from a central tap of a transversal filter

• In practice, a finite-length transversal filter is realized to approximate the ideal fi

lter (infinite-length transversal filter)

• Consider there are (2N+1) taps with weights c-N, c-N+1, …,cN, the equalizer output

samples {z(k)}

NNnNNkcnkxkzN

Nnn , 2,2 , )()(

19Dept. of EE, NDHU

Transversal Filter

20Dept. of EE, NDHU

Zero-Forcing Solution

• Relationship among {z(k)}, {x(k)}, and {cn} for the transversal filter

• Disposing the top N the bottom N rows of the matrix X into a square matrix with dimension

of 2N+1 and transform Z vector into a vector of 2N+1

• Rewrite the relationship

• Select the weights {cn} so that the equalizer output is

)(0000

)1()(000

)()1()2()1()(

0)()1(

0000)(

and

)2(

)0(

)2(

0

Nx

NxNx

NxNxNxNxNx

NxNx

Nx

X

c

c

c

c

Nz

z

Nz

z

N

N

zXccXz 1

Nkfor

kforkz

,,2,1 0

0 1)(

21Dept. of EE, NDHU

Example: A Zero-Forcing Equalizer

• Consider a three-taps transversal filter, the given received data {x(k)} are 0.0, 0.2,

0.9, -0.3,0.1. Using the zero-forcing solution to find the weights {c-1, c0, c1}

– For the relationship

1

0

1

1

0

1

9.03.01.0

2.09.03.0

02.09.0

)0()1()2(

)1()0()1(

)2()1()0(

0

1

0

c

c

c

c

c

c

xxx

xxx

xxx

Xcz

3448.0

9631.0

2140.0

1

0

1

c

c

c

22Dept. of EE, NDHU

Minimum MSE Solution

• Minimize the mean-square error (MSE) of all the ISI terms plus the

noise power at the output of the equalizer

• MSE is defined as the expected value of the squared difference between

the desired data symbol and the estimated data symbol

• MSE solution

• Minimum MSE solution is superior to zero-forcing solution

• Minimum MSE is more robust in the presence of noise and large ISI

xzxxxxxz

TT

RRccRR

XcXzX

1

23Dept. of EE, NDHU

Decision Feedback Equalizer

• Limitation of a linear equalizer is that it performs poor on channel

having spectral nulls

• Decision feedback equalizer (DFE) is a non-linear equalizer and uses

previous detector decisions to eliminate the ISI on pulse

• Basic idea is that if the values of the symbols previously detected are

known, then the ISI contributed by these symbols can be cancelled out

• Forward filter and feedback filter are used in the DFE

• The advantage of DFE is that the feedback filter operates on noiseless

quantized levels, and thus its output is free of channel noise

24Dept. of EE, NDHU

Decision Feedback Equalizer

25Dept. of EE, NDHU

Preset and Adaptive Equalization

• The equalizer weights remain fixed during transmission of data, then the

equalization is called preset equalization

• Preset equalization sets the tap weights according to some average

knowledge of the channel (Ex. Voice-grade telephone)

• Adaptive equalization can be implemented to perform tap-weight

adjustments periodically or continually

• Periodic adjustments are accomplished by periodically transmitting a

preamble sequence

• Continually adjustment are performed by the decision directed procedure

26Dept. of EE, NDHU

Preset and Adaptive Equalization

• Disadvantages of preset equalization

– Require an initial training period

– A time-varying channel can degrade system performance

• If the probability of error exceeds one percent (rule of thumb), decision-directed adaptive

equalizer might not converge

• Common solution to the adaptive equalization

– Initialize the equalizer with a preamble to provide good channel-error performance

– Then switch to the decision-directed mode

– Blind equalization algorithm can be used to form initial channel estimates without a

preamble