11

Techniques to control noise and fadingTechniques to control noise and fadingTechniques to control noise and fadingTechniques to control noise and fading NoiseNoise and and fadingfading are the primary sources of are the primary sources of

distortion in communication channelsdistortion in communication channels Techniques to reduce noise and fading are Techniques to reduce noise and fading are

usually implemented at the receiverusually implemented at the receiver The most common mechanism is to have a The most common mechanism is to have a

receiver filter that can cancel the effects of receiver filter that can cancel the effects of noise and fading, at least partiallynoise and fading, at least partially

Digital technology has made it possible to Digital technology has made it possible to have have adaptive filtersadaptive filters

NoiseNoise and and fadingfading are the primary sources of are the primary sources of distortion in communication channelsdistortion in communication channels

Techniques to reduce noise and fading are Techniques to reduce noise and fading are usually implemented at the receiverusually implemented at the receiver

The most common mechanism is to have a The most common mechanism is to have a receiver filter that can cancel the effects of receiver filter that can cancel the effects of noise and fading, at least partiallynoise and fading, at least partially

Digital technology has made it possible to Digital technology has made it possible to have have adaptive filtersadaptive filters

22

Principle of EqualizationPrinciple of EqualizationPrinciple of EqualizationPrinciple of Equalization Equalization is the process of compensation Equalization is the process of compensation

at the receiver, to reduce noise effectsat the receiver, to reduce noise effects The channel is treated as a filter with transfer The channel is treated as a filter with transfer

functionfunction Equalization is the process of creating a filter Equalization is the process of creating a filter

with an inverse transfer function of the with an inverse transfer function of the channelchannel

Since the channel is a varying filter, equalizer Since the channel is a varying filter, equalizer filter also has to change accordingly, hence filter also has to change accordingly, hence the term the term adaptiveadaptive..

Equalization is the process of compensation Equalization is the process of compensation at the receiver, to reduce noise effectsat the receiver, to reduce noise effects

The channel is treated as a filter with transfer The channel is treated as a filter with transfer functionfunction

Equalization is the process of creating a filter Equalization is the process of creating a filter with an inverse transfer function of the with an inverse transfer function of the channelchannel

Since the channel is a varying filter, equalizer Since the channel is a varying filter, equalizer filter also has to change accordingly, hence filter also has to change accordingly, hence the term the term adaptiveadaptive..

33

Equalization Model-Signal detectionEqualization Model-Signal detectionEqualization Model-Signal detectionEqualization Model-Signal detection

TransmitterTransmitter Receiver Receiver Front EndFront End

ChannelChannel

IF StageIF Stage

DetectorDetector

CarrierCarrier

Message signal x(t)Message signal x(t)

Detected signal y(t)Detected signal y(t)

44

Equalization model-CorrectionEqualization model-CorrectionEqualization model-CorrectionEqualization model-Correction

d(t)d(t)

+EqualizerEqualizerDecision Decision MakerMaker

ReconstructedReconstructedSignalSignal

eqh (t)

y(t)

Equivalent Equivalent NoiseNoise

nb(t)

55

Equalizer System EquationsEqualizer System Equations

Detected signalDetected signal

y(t) = x(t) * f(t) + ny(t) = x(t) * f(t) + nbb(t)(t)

=> Y(f) = X(f) F(f) + N=> Y(f) = X(f) F(f) + Nbb(f)(f)

Output of the Equalizer Output of the Equalizer ^ ^ d(t) = y(t) * h d(t) = y(t) * heqeq(t) (t)

Equalizer System EquationsEqualizer System Equations

Detected signalDetected signal

y(t) = x(t) * f(t) + ny(t) = x(t) * f(t) + nbb(t)(t)

=> Y(f) = X(f) F(f) + N=> Y(f) = X(f) F(f) + Nbb(f)(f)

Output of the Equalizer Output of the Equalizer ^ ^ d(t) = y(t) * h d(t) = y(t) * heqeq(t) (t)

66

Equalizer System EquationsEqualizer System Equations

Desired outputDesired output

^ ^ D(f) = Y(f) H D(f) = Y(f) Heqeq(f) = X(f)(f) = X(f)

=> H => Heqeq(f) X(f) F(f) = X(f)(f) X(f) F(f) = X(f)

=> H=> Heqeq(f) F(f) = 1(f) F(f) = 1

H Heqeq(f) = 1/ F(f) => Inverse filter(f) = 1/ F(f) => Inverse filter

Equalizer System EquationsEqualizer System Equations

Desired outputDesired output

^ ^ D(f) = Y(f) H D(f) = Y(f) Heqeq(f) = X(f)(f) = X(f)

=> H => Heqeq(f) X(f) F(f) = X(f)(f) X(f) F(f) = X(f)

=> H=> Heqeq(f) F(f) = 1(f) F(f) = 1

H Heqeq(f) = 1/ F(f) => Inverse filter(f) = 1/ F(f) => Inverse filter

77

System EquationsSystem EquationsSystem EquationsSystem Equations

ErrorError

MSE Error =MSE Error =

Aim of equalizer: To minimize MSE errorAim of equalizer: To minimize MSE error

ErrorError

MSE Error =MSE Error =

Aim of equalizer: To minimize MSE errorAim of equalizer: To minimize MSE error

e(t) d(t) d(t)

2E[| e(t) | ]

88

Equalizer Operating Modes Equalizer Operating Modes Equalizer Operating Modes Equalizer Operating Modes

TrainingTraining TrackingTracking

TrainingTraining TrackingTracking

99

Training and Tracking functionsTraining and Tracking functionsTraining and Tracking functionsTraining and Tracking functions

The Training sequence is a known pseudo-The Training sequence is a known pseudo-random signal or a fixed bit pattern sent by the random signal or a fixed bit pattern sent by the transmitter. The user data is sent immediately transmitter. The user data is sent immediately after the training sequence after the training sequence

The equalizer uses training sequence to adjust The equalizer uses training sequence to adjust its frequency response Hits frequency response Heqeq (f) and is optimally (f) and is optimally ready for data sequenceready for data sequence

Adjustment goes on dynamically, it is adaptable Adjustment goes on dynamically, it is adaptable equalizerequalizer

The Training sequence is a known pseudo-The Training sequence is a known pseudo-random signal or a fixed bit pattern sent by the random signal or a fixed bit pattern sent by the transmitter. The user data is sent immediately transmitter. The user data is sent immediately after the training sequence after the training sequence

The equalizer uses training sequence to adjust The equalizer uses training sequence to adjust its frequency response Hits frequency response Heqeq (f) and is optimally (f) and is optimally ready for data sequenceready for data sequence

Adjustment goes on dynamically, it is adaptable Adjustment goes on dynamically, it is adaptable equalizerequalizer

1010

Block Diagram of Digital EqualizerBlock Diagram of Digital EqualizerBlock Diagram of Digital EqualizerBlock Diagram of Digital Equalizer

y(k) y(k 1) y(k 2)

Z-1 Z-1 Z-1

∑

y(k N)

d(k)

d(k)

e(k)

w0k w1kw2k wNk

Adaptive Algorithm ∑+

-

1111

Digital Equalizer equationsDigital Equalizer equationsDigital Equalizer equationsDigital Equalizer equations• In discrete form, we sample signals at In discrete form, we sample signals at interval of ‘T’ seconds : t = k T;interval of ‘T’ seconds : t = k T;

•The output of Equalizer is:The output of Equalizer is:

• In discrete form, we sample signals at In discrete form, we sample signals at interval of ‘T’ seconds : t = k T;interval of ‘T’ seconds : t = k T;

•The output of Equalizer is:The output of Equalizer is:

eqd(k) y(k) * h (k)

e(k) d(k) d(k) eqd(k) y(k) * h (k)

N

nkn 0

W y(k n)

ok 1k NkW y(k) W y(k 1)...W y(k N)

1212

Error minimizationError minimizationError minimizationError minimization The adaptive algorithm is controlled by the The adaptive algorithm is controlled by the

error signal,error signal,

The adaptive algorithm is controlled by the The adaptive algorithm is controlled by the error signal,error signal,

e(k) d(k) d(k)

2Min E [e(k) e(k)*] Min E [| e(k) | ]

nk 1 nk k 1 nW W Ke * y

The equalizer weights are varied until convergence is reached.

1313

Types of equalizersTypes of equalizersTypes of equalizersTypes of equalizers

Linear Equalizers.Linear Equalizers. Non Linear Equalizers.Non Linear Equalizers.

Linear Equalizers.Linear Equalizers. Non Linear Equalizers.Non Linear Equalizers.

1414

Diversity techniquesDiversity techniquesDiversity techniquesDiversity techniques Powerful communications receiver Powerful communications receiver

technique that provides wireless link technique that provides wireless link improvement at relatively low cost.improvement at relatively low cost.

Unlike equalization, diversity requires no Unlike equalization, diversity requires no training overhead.training overhead.

Powerful communications receiver Powerful communications receiver technique that provides wireless link technique that provides wireless link improvement at relatively low cost.improvement at relatively low cost.

Unlike equalization, diversity requires no Unlike equalization, diversity requires no training overhead.training overhead.

1515

Principle of diversity Principle of diversity Principle of diversity Principle of diversity Small Scale fading causes deep and rapid Small Scale fading causes deep and rapid

amplitude fluctuations as mobile moves amplitude fluctuations as mobile moves over a very small distances.over a very small distances.

Small Scale fading causes deep and rapid Small Scale fading causes deep and rapid amplitude fluctuations as mobile moves amplitude fluctuations as mobile moves over a very small distances.over a very small distances.

1616

……Principle of diversityPrinciple of diversity……Principle of diversityPrinciple of diversity If we space 2 antennas at 0.5 m, one may If we space 2 antennas at 0.5 m, one may

receive a null while the other receives a receive a null while the other receives a strong signal. By selecting the best signal at strong signal. By selecting the best signal at all times, a receiver can mitigate or reduce all times, a receiver can mitigate or reduce small-scale fading. This concept is small-scale fading. This concept is Antenna Antenna Diversity.Diversity.

If we space 2 antennas at 0.5 m, one may If we space 2 antennas at 0.5 m, one may receive a null while the other receives a receive a null while the other receives a strong signal. By selecting the best signal at strong signal. By selecting the best signal at all times, a receiver can mitigate or reduce all times, a receiver can mitigate or reduce small-scale fading. This concept is small-scale fading. This concept is Antenna Antenna Diversity.Diversity.

1717

Diversity ImprovementDiversity ImprovementDiversity ImprovementDiversity Improvement• Consider a fading channel (Rayleigh)Consider a fading channel (Rayleigh)

Input s(t) Output r(t)Input s(t) Output r(t)• Input-output relationInput-output relation

r (t) = r (t) = (t) e (t) e -j -j (t)(t) s (t) + n (t) s (t) + n (t) Average value of signal to noise ratio Average value of signal to noise ratio

___ ___

SNR = SNR = = (E = (Ebb / N / Noo) ) 22 (t) (t)

• Consider a fading channel (Rayleigh)Consider a fading channel (Rayleigh)

Input s(t) Output r(t)Input s(t) Output r(t)• Input-output relationInput-output relation

r (t) = r (t) = (t) e (t) e -j -j (t)(t) s (t) + n (t) s (t) + n (t) Average value of signal to noise ratio Average value of signal to noise ratio

___ ___

SNR = SNR = = (E = (Ebb / N / Noo) ) 22 (t) (t)

Channel

1818

Average SNR Improvement Average SNR Improvement Using DiversityUsing Diversity Average SNR Improvement Average SNR Improvement Using DiversityUsing Diversity

p.d.f., p(p.d.f., p(γγii) = (1 / ) = (1 / ) e ) e – – γγii / /

where (where (γγii 0 ) and 0 ) and γγii = instantaneous = instantaneous SNRSNR

Probability [Probability [γγii γγ] ]

M diversity branches,M diversity branches,

Probability [Probability [γγii > > γγ] ]

p.d.f., p(p.d.f., p(γγii) = (1 / ) = (1 / ) e ) e – – γγii / /

where (where (γγii 0 ) and 0 ) and γγii = instantaneous = instantaneous SNRSNR

Probability [Probability [γγii γγ] ]

M diversity branches,M diversity branches,

Probability [Probability [γγii > > γγ] ]

0

p( i)d i

/ M1 (1 e )

1919

Average Snr Improvement Average Snr Improvement Using DiversityUsing DiversityAverage Snr Improvement Average Snr Improvement Using DiversityUsing Diversity Average SNR improvement using Average SNR improvement using

selection Diversity,selection Diversity, Average SNR improvement using Average SNR improvement using

selection Diversity,selection Diversity,M

k 1

/ 1/ k

2020

ExampleExample : Assume that 5 antennas : Assume that 5 antennas are used to provide space diversity. are used to provide space diversity. If average SNR is 20 dB, determine If average SNR is 20 dB, determine the probability that the SNR will be the probability that the SNR will be 10 dB. 10 dB. Compare this with the case of a single Compare this with the case of a single receiver.receiver.

SolutionSolution : : = 20 dB => 100. = 20 dB => 100.

Threshold Threshold γγ = 10 dB = 10. = 10 dB = 10.

ExampleExample : Assume that 5 antennas : Assume that 5 antennas are used to provide space diversity. are used to provide space diversity. If average SNR is 20 dB, determine If average SNR is 20 dB, determine the probability that the SNR will be the probability that the SNR will be 10 dB. 10 dB. Compare this with the case of a single Compare this with the case of a single receiver.receiver.

SolutionSolution : : = 20 dB => 100. = 20 dB => 100.

Threshold Threshold γγ = 10 dB = 10. = 10 dB = 10.

2121

……Example Example ……Example Example

Prob Prob [[γγii > > γγ] = 1 – (1 – e ] = 1 – (1 – e – – γγ/ / ))MM

For M = 5, For M = 5,

Prob Prob = 1 – (1 – e = 1 – (1 – e – – 0.10.1 ))5 5 = 0.9999= 0.9999

For M = 1(No Diversity), For M = 1(No Diversity),

Prob Prob = 1 – (1 – e = 1 – (1 – e – – 0.10.1 )) = 0.905 = 0.905

Prob Prob [[γγii > > γγ] = 1 – (1 – e ] = 1 – (1 – e – – γγ/ / ))MM

For M = 5, For M = 5,

Prob Prob = 1 – (1 – e = 1 – (1 – e – – 0.10.1 ))5 5 = 0.9999= 0.9999

For M = 1(No Diversity), For M = 1(No Diversity),

Prob Prob = 1 – (1 – e = 1 – (1 – e – – 0.10.1 )) = 0.905 = 0.905

2222

Maximal Ratio Combining (MRC)Maximal Ratio Combining (MRC)Maximal Ratio Combining (MRC)Maximal Ratio Combining (MRC) MRC uses each of the M branches in MRC uses each of the M branches in

co-phased and weighted manner such co-phased and weighted manner such that highest achievable SNR is that highest achievable SNR is available. If each branch has gain Gavailable. If each branch has gain G ii, ,

rrMM = total signal envelope = total signal envelope

= =

MRC uses each of the M branches in MRC uses each of the M branches in co-phased and weighted manner such co-phased and weighted manner such that highest achievable SNR is that highest achievable SNR is available. If each branch has gain Gavailable. If each branch has gain G ii, ,

rrMM = total signal envelope = total signal envelope

= = M

i ii 1

G r

2323

……Maximal Ratio Combining (MRC)Maximal Ratio Combining (MRC)……Maximal Ratio Combining (MRC)Maximal Ratio Combining (MRC) … … assuming each branch has some assuming each branch has some

average noise power N, total noise average noise power N, total noise power Npower NTT applied to the detector is, applied to the detector is,

… … assuming each branch has some assuming each branch has some average noise power N, total noise average noise power N, total noise power Npower NTT applied to the detector is, applied to the detector is,

M2

T ii 1

N N G

2

M M TSNR r / 2N M

2M i i i i

i 1

1Max (r )N r whenG r /N2

2424

Average SNR ImprovementAverage SNR ImprovementAverage SNR ImprovementAverage SNR Improvement

MAverage SNR M

k 1M

/M

k 1

( / )Pr obability ( ) e

(k 1)!

2525

EXAMPLEEXAMPLE : Repeat earlier problem for : Repeat earlier problem for MRC caseMRC case EXAMPLEEXAMPLE : Repeat earlier problem for : Repeat earlier problem for MRC caseMRC case

k 1M

/M

k 1

( / )Pr obability ( ) e

(k 1)!

= 10, = 100, M = 5

k 150.1

Mk 1

(0.5)Pr obability ( 10) e

(k 1)!

2626

……ExampleExample……ExampleExample

= 0.905 [ 1.1051708 ]

= e - 0.1 [ 1 + 0.1 / 1 + 0.12 / 2 ! + 0.13 / 3 ! + 0.14 / 4 ! ]

= 0.9999998

e-0.1

2727

Types of diversity Types of diversity Types of diversity Types of diversity Space Diversity Space Diversity

Either at the mobile or base station.Either at the mobile or base station. At base station, separation on order of At base station, separation on order of

several tens of wavelength are several tens of wavelength are required.required.

Polarization DiversityPolarization Diversity Orthogonal Polarization to exploit Orthogonal Polarization to exploit

diversitydiversity

Space Diversity Space Diversity Either at the mobile or base station.Either at the mobile or base station. At base station, separation on order of At base station, separation on order of

several tens of wavelength are several tens of wavelength are required.required.

Polarization DiversityPolarization Diversity Orthogonal Polarization to exploit Orthogonal Polarization to exploit

diversitydiversity

2828

……Types of diversityTypes of diversity……Types of diversityTypes of diversity Frequency Diversity : Frequency Diversity :

More than one carrier frequency is More than one carrier frequency is usedused

Time Diversity : Time Diversity : Information is sent at time spacings Information is sent at time spacings Greater than the coherence time of Greater than the coherence time of

Channel, so multiple repetitions can be Channel, so multiple repetitions can be resolvedresolved

Frequency Diversity : Frequency Diversity : More than one carrier frequency is More than one carrier frequency is

usedused Time Diversity : Time Diversity :

Information is sent at time spacings Information is sent at time spacings Greater than the coherence time of Greater than the coherence time of

Channel, so multiple repetitions can be Channel, so multiple repetitions can be resolvedresolved

2929

Practical diversity receiver Practical diversity receiver – rake receiver– rake receiverPractical diversity receiver Practical diversity receiver – rake receiver– rake receiver CDMA system uses RAKE Receiver to CDMA system uses RAKE Receiver to

improve the signal to noise ratio at the improve the signal to noise ratio at the receiver.receiver.

Generally CDMA systems don’t require Generally CDMA systems don’t require equalization due to multi-path resolution.equalization due to multi-path resolution.

CDMA system uses RAKE Receiver to CDMA system uses RAKE Receiver to improve the signal to noise ratio at the improve the signal to noise ratio at the receiver.receiver.

Generally CDMA systems don’t require Generally CDMA systems don’t require equalization due to multi-path resolution.equalization due to multi-path resolution.

3030

Block Diagram Of Rake ReceiverBlock Diagram Of Rake ReceiverBlock Diagram Of Rake ReceiverBlock Diagram Of Rake Receiver

αα11

M1 M2 M3M1 M2 M3 αα22

r(t)r(t) ααMM Z’ Z’ Z Z

αα11

M1 M2 M3M1 M2 M3 αα22

r(t)r(t) ααMM Z’ Z’ Z Z

Correlator 1

Correlator 2

Correlator M

Σ ()dt

><

m’(t)

3131

Principle Of Operation Principle Of Operation Principle Of Operation Principle Of Operation M Correlators – Correlator 1 is synchronized M Correlators – Correlator 1 is synchronized

to strongest multi-path Mto strongest multi-path M11. The correlator 2 . The correlator 2 is synchronized to next strongest multipath is synchronized to next strongest multipath MM22 and so on. and so on.

The weights The weights 11 , , 22 ,……, ,……,MM are based on are based on SNR from each correlator output. (SNR from each correlator output. ( is is proportional to SNR of correlator.)proportional to SNR of correlator.)

M M Z’ = Z’ = M M Z ZM

m =1 m =1

M Correlators – Correlator 1 is synchronized M Correlators – Correlator 1 is synchronized to strongest multi-path Mto strongest multi-path M11. The correlator 2 . The correlator 2 is synchronized to next strongest multipath is synchronized to next strongest multipath MM22 and so on. and so on.

The weights The weights 11 , , 22 ,……, ,……,MM are based on are based on SNR from each correlator output. (SNR from each correlator output. ( is is proportional to SNR of correlator.)proportional to SNR of correlator.)

M M Z’ = Z’ = M M Z ZM

m =1 m =1

3232

……Principle Of OperationPrinciple Of Operation……Principle Of OperationPrinciple Of Operation Demodulation and bit decisions are then Demodulation and bit decisions are then

based on the weighted Outputs of M based on the weighted Outputs of M Correlators.Correlators.

Demodulation and bit decisions are then Demodulation and bit decisions are then based on the weighted Outputs of M based on the weighted Outputs of M Correlators.Correlators.