Detection and channel equalization

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Ad hoc Sensor Networks

Detection and channel equalization

Érzékelő mobilhálózatok

Detekció és csatornakiegyenlítés

Dr. Oláh András

Lecture 4 review

• Advantage of digital modulation

• Bandwidth of a signal

• ISI-free system requirements

• IQ modulator

• Constellation and eye diagrams

• Tradeoff between spectral efficiency and power efficiency

• Linear and constant envelope modulation scheme

• Spread Spectrum Modulation

Outline

• Signal space representation

• Optimal detection of signal in AWGN (Bayesian decision)

• Probability of error (BER and SER)

• Demodulation and detection for modulation schemes

• BER in fading channel

• Equalization

Structure of a wireless communications link

Receiver tasks

• Demodulation and sampling:

– Waveform recovery and preparing the received signal for detection – Improving the signal-to-noise ratio (SNR) using matched filter – Reducing ISI by using equalizer

– Sampling the recovered waveform

• Detection:

– Estimate the transmitted symbol based on the received samples

Receiver stucture

• Note: the bandpass model of detection process is equivalent to the baseband model because the received bandpass waveform is first transformed into a baseband waveform.

• Received signal in fading channel:

Model of the received signal

• Received signal in AWGN:

Simplifying the model

Ideal channel:

h_c(τ) = δ(τ)

Signal space representation

• What is a signal space?

– Vector representations of signals in an N-dimensional space

• Why do we need a signal space?

– It is a mean to convert signals to vectors and vice versa.

– It is a mean to calculate signals energy and Euclidean distances between signals.

• Why are we interested in Euclidean distances between signals?

– For detection purposes: The received signal is transformed to a

received vectors. The signal which has the minimum distance to the received signal is estimated as the transmitted signal (in the case of Gaussian noise Euclidean metric provides the best – minimum error probability – decision).

Signal space representation (cont’)

• The inner product between two signals (functions):

• The distance in signal space is measured by calculating the norm:

– Norm of a signal:

– The norm between two signals:

( ), ( ) ( ) *( ) x t y t x t y t dt

∞

−∞

< >=

∫

( ) ( ), ( ) ( ) 2 _x

x t x t x t ^∞ x t dt E

= < > =

∫

, ( ) ( )

dx y = x t − y t

( It is the “length” of x(t))

(We refer to it as the Euclidean distance.)

Example of signal space representation

1 11 1 12 2 1 11 12

2 21 1 22 2 2 21 22

3 31 1 32 2 3 31 32

1 1 2 2 1 2

( ) ( ) ( ) ( , )

( ) ( ) ( ) ( , )

( ) ( ) ( ) ( , )

( ) ( ) ( ) ( , )

s t a t a t a a

s t a t a t a a

s t a t a t a a

r t r t r t r r

ψ ψ

ψ ψ

ψ ψ

ψ ψ

= + ⇔ =

= + ⇔ =

= + ⇔ =

= + ⇔ =

s s s r

Transmitted signal Received signal at matched filter output The Euclidean distance between signals r(t) and s(t)

2 2

, ( ) ( ) ( 1 1) ( 2 2) 1, 2,3

s ri i i i

d = s t −r t = a −r + a −r i =

Orthogonal signal space

• N-dimensional orthogonal signal space is characterized by N linearly independent functions { Ψ

j

(t)}

called basis functions. The basis functions must satisfy the orthogonality condition

where 0 ≤ t ≤ T, and i,j=1,…,N.

• If all K

=1 , the signal space is orthonormal.

* 0

( ), ( ) ( ) ( )

0

T

i

i j i j i ji

K i j

t t t t dt K

i j

ψ ψ ψ ψ δ ^{ =}

< >= = = 

≠

∫ 

Orthogonal signal space (cont’)

• Example: 1 dimensional orthonormal signal space

• Example: 2 dimensional orthonormal signal space

Signal space representation

• Any arbitrary finite set of waveforms {s_i(t)}_i=1,..,M, where each member of the set is of duration T, can be expressed as a linear combination of N orthonogal waveforms {Ψ

j(t)}_j=1,..,N where N ≤ M : where

Waveform to vector conversion

Vector to waveform conversion

1

( ) ( )

N

i ij j

j

s t a ψ t

=

=

∑

* 0

1 1

( ), ( ) ( ) ( )

T

ij i j i j

j j

a s t t s t t dt

K ψ K ψ

= < >=

∫

0 ≤ ≤t T

2 1

N

i j ij

j

E K a

=

=

∑

Waveform energy

1 2

( , ,..., )

i = a ai i aiN

Vector rep. s

of waveform

1,..., i = M 1,...,

j = N

Gram-Schmidt procedure

• To find an orthonormal basis functions for a given set of signals, the Gram- Schmidt procedure can be used:

• given a signal set {s_i(t)}_i=1,..,M , compute an orthonormal basis {Ψ

j(t)}_j=1,..,N

1. Define

2. For i=1,…,M compute

If b_i(t)≠0 let

If b_i(t)=0, do not assign any basis function.

3. Renumber the basis functions such that basis is {Ψ

1(t), Ψ

2(t),…, Ψ

N(t)}

This is only necessary if b_i(t)=0 for any i in step 2. Note that N≤M.

1( )t s t1( ) / E1 s t1( ) / s t1( )

ψ = =

1

1

( ) ( ) ( ), ( ) ( )

i

i i i j j

j

b t s t ⁻ s t ψ t ψ t

=

= −

∑

( ) ( ) / ( )

i t b ti b ti

ψ =

• Find the basis functions and plot the signal space for the following transmitted signals:

• Using Gram-Schmidt procedure:

2 2

1 1

0

1 1 1 1

2 1 2 1

0

2 2 1

( )

( ) ( ) / ( ) /

( ), ( ) ( ) ( )

( ) ( ) ( ) ( ) 0

T

T

E s t dt A

t s t E s t A

s t t s t t dt A

b t s t A t

ψ

ψ ψ

ψ

= =

= =

< >= = −

= − − =

∫

∫

Gram-Schmidt procedure (cont’)

White noise in the orthonormal signal space

• AWGN, n(t), can be expressed as

• Vector representation of noise:

n=(n₁,n₂,…,n_N)

• {Ψ

j(t)}_j=1,..,N independent zero-mean Gaussain random variables with variance

σ(n_i)=N₀/2

( ) ˆ( ) ( ) n t = n t + n tɶ

Noise projected on the signal space

which has an impact on the detection process.

Noise outside of the signal space

( ), ( )

j j

n =< n t¹ ψ t > < n tɶ( ),ψ _j( )t >= 0

ˆ( ) ( )

N

j j j

n t n ψ t

=

=

∑

1,..., j = N

Detection of signals at the output of AWGN

• Detection problem: given the observation vector r, perform a mapping from r to an estimate m^ˆ of the transmitted symbol, m_i , such that the average probability of error in the decision is minimized.

– Signal vector s_i is deterministic.

– Elements of noise vector n are i.i.d Gaussian random variables with zero-mean and variance N₀/2. The noise vector pdf is

– The elements of observed vector r are independent Gaussian random variables.

Its pdf is

( )

2 / 2

0 0

( ) 1 _N exp

p πN N

 

= − 

 

n

n n

( )

2 / 2

( | )_i 1 _N exp ⁱ

p πN N

 − 

= − 

 

r

r s r s

Detection of signal at the output of AWGN (cont’)

• Optimum decision rule (maximum a posteriori probability):

– Applying Bayes’ rule gives:

– For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to:

– or equivalently:

which is known as maximum likelihood.

1,...,

: arg max Pr( | )_i

i M

k s

= = r

( ) ( )

( )

1,...,

: arg max _i ⁱ

i M

p s

k p s

p

= = ^r

r

r r

1,...,

( )

: arg max _i

i M

k p s

= = _r r

( ( ) )

1,...,

: arg max ln _i

i M

k p s

= = _r r

ˆ _k

m = m

ˆ _k

m = m

ˆ _k

m = m

ˆ _k

m = m

Detection of signal at the output of AWGN (cont’)

• Partition the signal space into M decision regions, D₁,D₂,…,D_M.

• Restate the maximum likelihood decision rule as follows:

vector r lies inside region D_k if

for all k≠i, that means m^~=m_k.

( ( ) ) _{( )}

( ) { }

2

1,..., 1,..., 0 / 2 0

2

2 2

1,..., 0 / 2 0 1,..., 1,...,

: arg max ln arg max ln 1 exp

arg max ln 1 arg max arg min

i

i N

i M i M

i

i i

i M N i M i M

k p s

N N

N N

π π

= =

= = =

  − 

= =  − 

   − 

 

=   −  = − − = −

 

   

 

r

r r s

r s r s r s

ˆ _k

m = m

k i

− < − r s r s

Detection of signal at the output of AWGN (cont’)

• The optimal decision (cont’):

( ( ) ) {

} {

}

1,..., 1,..., 1,...,

1,..., 1

: arg max ln arg max arg max 2

arg max

2

i i i i

i M i M i M

N

i j ij

i M j

k p s

r a E

= = =

= =

= = − − = − + ⋅ − =

 

=  − 



∑

r r r s r r s s

The probability of symbol error

• Erroneous decision: for the transmitted symbol m_i or equivalently signal vector s_i, an error in decision occurs if the observation vector r does not fall inside region D_i.

– Probability of erroneous decision for a transmitted symbol:

or equivalently

– Probability of correct decision for a transmitted symbol:

thus the probability of erroneous decision:

e( _i) Pr( ˆ _i and _i sent) P m = m ≠ m m

Pr(mˆ ≠ m_i) = Pr(m_i sent)Pr( does not lie inside r D m_{i i} sent)

( )

( )

c Pr( ˆ ) Pr( sent)Pr( lies inside sent) ( )

i

i i i i i

i i

D

P m m m m D m

p m p m d

∈

= = = =

= ⋅

∫

r

r

r r

( )

e( _i) ( _i) _i 1 c( _i)

P m = p m ⋅

∫

The probability of symbol error (cont’)

• Average probability of symbol error :

• Example for binary PAM:

( )

E e c

1 1

1

( ) 1 ( )

1 1 ( | )

i

M M

i i

i i

M

i

i D

P M P m P m

p m d M

= =

= ∈

= = − =

= −

∑ ∑

∑ ∫

r

r r

b

B E

0

(2) 2E

P P Q

N

 

= =  

 

1 2

e 1 e 2

0

( ) ( ) / 2

P m P m Q / 2

N

 − 

= =  

 

s s

( )

2

1 2

2

u

x

Q x e du

π

∞ −

=

∫

Recall:

Example of Symbol error prob. for PAM signals

Demodulation and detection

• Demodulation: The receiver signal is converted into a baseband signal, and then filtered and sampled.

• Detection: Sampled values are used for detection and we use a decision rule such as the ML detection rule.

• The matched filters output (observation vector r) is the detector input and the decision variable is a function of r. We know that for calculating the probability of symbol error, we need to determine

Pr( lies inside r D m_{i i} sent)

M-ary Pulse Amplitude modulation

• One dimensional modulation, demodulation and detection

2

( )

( ) cos

i i c

s t t

α T ω

=

( )

( )

1

1

b

2 2

b 2

s b

( ) ( ) 1, , ( ) 2 cos

(2 1 )

2 1

( 1)

3

i i

c

i

i i

s t a t i M

t t

T

a i M E

E E i M

E M E

ψ

ψ ω

= =

=

= − −

= = − −

= − s

…

b 2

E 2

0

2( 1) 6log

( )

1 E

M M

P M Q

M M N

 

= −  

 − 

M-ary Phase Shift Keying (M-PSK)

• Two dimensional modulation, demodulation and detection

2 s 2

( ) cos

i c

E i

s t t

T M

ω π

 

=  + 

 

( ) ( )

1 1 2 2

1 2

1 s 2 s

2 s

( ) ( ) ( ) 1, ,

2 2

( ) cos ( ) sin

2 2

cos sin

i i i

c c

i i

i i

s t a t a t i M

t t t t

T T

i i

a E a E

M M

E E

ψ ψ

ψ ω ψ ω

π π

= + =

= = −

   

=   =  

   

= = s

…

(

)

0

2 log

( ) 2 M E sin

P M Q

N M

  π 

≈   

M-ary Quadrature Amplitude Mod. (M-QAM)

• Two dimensional modulation, demodulation and detection

( ) ( )

1 1 2 2

1 2

1 2 s

( ) ( ) ( ) 1, ,

2 2

( ) cos ( ) sin

2( 1)

where and are PAM symbols and

3

i i i

c c

i i

s t a t a t i M

t t t t

T T

a a E M

ψ ψ

ψ ω ψ ω

= + =

= =

= −

…

(

)

( 1, 1) ( 3, 1) ( 1, 1)

( 1, 3) ( 3, 3) ( 1, 3)

,

( 1, 1) ( 3, 1) ( 1, 1)

i i

M M M M M M

M M M M M M

a a

M M M M M M

 − + − − + − − − 

 

− + − − + − − −

 

=  

 

 − + − + − + − + − − + 

 

⋯

⋯

⋮ ⋮ ⋮ ⋮

⋯

( )

( ) 2 ⁱ cos

i c i

s t E t

T ω ϕ

= +

M-QAM (cont’)

b 2

E

0

1 3log

( ) 4 1

1 E

P M Q M

M N

M

 

 

=  −   

−

   

( )

1

2 s

( ) ( ) 1, ,

2

( ) cos

0

M

i ij j

j

s

i i ij

i i

s t a t i M

E i j

t t a

T i j

E E

ψ

ψ ω

=

= =

 =

= = 

 ≠

= =

∑

s

…

M-ary Frequency Shift keying (M-FSK)

• Multi-dimensional modulation, demodulation and detection

( ) ( )

s s

2 2

( ) cos cos ( 1)

1

2 2

i i c

E E

s t t t i t

T T

f T

ω ω ω

ω π

= = + − ∆

∆ = ∆ =

( )

E

0

( ) 1 E

P M M Q

N

 

≤ −  

 

Coherent vs. non-coherent detection

Coherent detection:

• It requires carrier phase recovery at the receiver and hence, circuits to perform phase estimation.

• The sources of carrier-phase mismatch at the receiver:

– Propagation delay causes carrier-phase offset in the received signal.

– The oscillators at the receiver which generate the carrier signal, are not usually phased locked to the transmitted carrier.

Non-coherent detection:

• No need for a reference in phase with the received carrier.

• Less complexity compared to coherent detection at the price of

higher error rate.

Bit error probability vs. symbol error probability

• Number of bits per symbol:

k=log

M

• For orthogonal M-ary signaling (M-FSK) :

• For M-PSK, M-PAM and M-QAM:

1 Bit

E Bit

E

2 / 2

2 1 1

lim 1

2

k k

k

P M

P M

P P

−

→∞

= =

− −

=

E

Bit P for E 1

P P

≈ k <<

Designing a Wireless Communication System

• Goals:

– Maximizing the transmission bit rate (R) – Minimizing probability of bit error (P_Bit) – Minimizing the required power (P_TX or PE) – Minimizing required system bandwidth (B) – Maximizing system utilization (SE)

– Minimizing system complexity

• Limitations:

– The Nyquist theoretical minimum bandwidth requirement

– The Shannon-Hartley capacity theorem (and the Shannon limit) – Government regulations

– Technological limitations

– Other system requirements (e.g wireless sensor networks)

• Most famous result in communication theory.

• Channel capacity of AWGN channel:

– B : bandwidth

– C : channel capacity (bps) of real data (not retransmissions or errors). It is the maximum data rate at which error-free communication over the channel is performed.

• SE_max is fundamental limit that cannot be achieved in practice.

Maximum SE: Shannon’s theorem (1948)

Recall from Chapter 4

Claude Elwood Shannon (1916-2001)

S b

max 2 2

N 0

log 1 P log 1 E

C R

SE B P N B

   

= =  +  =  + 

   

Given a modulation scheme and a targeted BER, the communication system designer can determine the SE (spectral efficiency) and the PE (E_b/N₀ required to maintain the average BER target).

Recall from Chapter 4

Power and bandwidth limited systems

• Two major communication resources:

– P_TX transmit power – B channel bandwidth

• In many communication systems, one of these resources is more precious than the other. Hence, systems can be classified as:

– Power-limited systems: save power at the expense of bandwidth (for example by using coding schemes)

– Bandwidth-limited systems: save bandwidth at the expense of power (for example by using spectrally efficient modulation schemes)

M-ary signaling

• Bandwidth efficiency:

– Assuming Nyquist (ideal rectangular) filtering at baseband, the required passband bandwidth is:

• M-PSK and M-QAM (bandwidth-limited systems)

– SE increases as M increases.

• MFSK (power-limited systems)

– SE decreases as M increases

s s

1/ [Hz]

B = T = R

2 s

log 1

[bits/s/Hz]

M R

B = BT = BT

/ log2 [bits/s/Hz]

R B = M

/ log2 / [bits/s/Hz]

R B = M M

Design example of uncoded systems

• Design goals:

– The bit error probability at the modulator output must meet the system error requirement.

– The transmission bandwidth must not exceed the available channel bandwidth.

Design example for uncoded systems (cont’)

• Task: choose a modulation scheme that meets the system requirements.

• Solution:

RX 5

Bit 0

AWGN channel with 4000 [Hz]

53 [dBHz] 9600 [bits/s] 10 BC

P R P

N

−

=

= = ≤

s 2

s b RX

2 2

0 0 0

5

E s 0

6 5

E Bit

2

Band-limited channel M-PSK modulation

8 / log 9600 / 3 3200 [sym/s] 4000 [Hz]

(log ) (log ) 1 62.67

( 8) 2 2 / sin( / ) 2.2 10

( )

7.3 10 10 log

c

c

R B

M R R M B

E E P

M M

N N N R

P M Q E N M

P P M

M

π ⁻

− −

> ⇒ ⇒

= ⇒ = = = < =

= = =

 

= ≈   = ×

≈ = × <

b RX

0 0

c b 0

s 2 c

s b RX

2 2

0 0 0

E

1 6.61 8.2 [dB]

and relatively small / power-limited channel M-FSK

16 /(log ) 16 9600 / 4 38.4 [ksym/s] 45 [kHz]

(log ) (log ) 1 26.44

( 16)

E P

N N R

R B E N

M B MR MR M B

E E P

M M

N N N R

P M M

= = =

< ⇒ ⇒

= ⇒ = = = × = < =

= = =

= ≤ − ^{s 5} _Bit ¹ _E ^{6 5}

0

1 2

exp 1.4 10 ( ) 7.3 10 10

2 2 2 1

k k

E P P M

N

− − − −

 

− = × ⇒ ≈ = × <

 

  −

Design example for uncoded systems (cont’)

• Task: choose a modulation scheme that meets the system requirements.

• Solution:

RX 5

Bit 0

AWGN channel with 45 [kHz]

48 [dBHz] 9600 [bits/s] 10 BC

P R P

N

−

=

= = ≤

BER in fading channels

• Wireless channel impairments:

– system generated interference – Shadowing, large-scale path loss

– Multipath Fading, rapid small-scale signal variations (ISI) – Doppler Spread due to motion of mobile unit

• All can lead to significant distortion or attenuation of the received signal (SNR) which degrade Bit Error Rate (BER) of digitally modulated signal.

BER in fading channels (cont’)

• We have BER expressions for non-fading AWGN channels.

• The pdf of SNR in Rayleigh fading channel:

where is the average SNR.

• The BER in case of Rayleigh fading can be computed as:

( )

pdf SNR

SNR

= −

SNR

( ) ^{( ) ( )}

Rayleigh AWGN

0

BER SNR BER SNR pdf SNR dSNR

= ∞

∫

This is a serious problem!

BER in fading channels (cont’)

• Four techniques are used to improve received signal quality and lower BER:

– Equalization

– Diversity (expensive and resource consuming) – Channel Coding

– Interleaving

• They are used independently or together

• We will consider equalization

Inter-Symbol Interference (ISI)

• ISI in the detection process due to the filtering effects of the system

• Overall equivalent system transfer function G(f)=G_T(f )H(f )G_R(f ) – creates echoes and hence time dispersion

– causes ISI at sampling time: _{k k k i k i}

i k

x y n h y ₋

≠

= + +

∑

Equalization

Equalization (cont’)

• MLSE (Maximum likelihood sequence estimation)

• Filtering

– Transversal filtering: A weighted tap delayed line that reduces the effect of ISI by proper adjustment of the filter taps

• Zero-forcing equalizer (Peak-Distortion Criterion)

• Minimum mean square error (MMSE) equalizer (MSE criterion)

– Decision feedback: using the past decisions to remove the ISI contributed by them

– Adaptive equalizer

Linear channel equalization

• The simplified model and notations:

0

0 0 0 0 0 0

L

k k n n k

n

J J L L J J

k j k j j k j n n k j j k j n n j k j

j j n n j j

x h y

y w x w h y w h y w

ν

ν ν

= −

− − − − − − −

= = = = = =

= +

 

 

= =  +  =   +

   

∑

∑ ∑ ∑ ∑ ∑ ∑

ɶ

0

k k n n k

n

k k n n k

n k

y q y

q y q y

η

η

−

≠ −

= + =

= + +

∑

∑

ɶ

ISI

Equivalent filter

{ }

E η = N w

(

)

k N N

η ^{∼ w}

Zero forcing equalizer

• The zero forcing equalizer minimizes the peak-distortion:

• The ISI can be completely eliminated if:

• In the case of finite-length equalizer:

( )^ZF

opt

, 0

: arg min ( ) arg min _n

n n

PD q

≠

= =

∑

w w

w w

0

1 0

0 0

n n

q n

δ ^ n ⁼

= = 

 ≠

0 0

1 0

0 1,...,

J

j n j n

j

w h n

n J

− δ

=

=

= = 

 =

∑

0 0

1 0 1

2 1 0 2

0 0 0 1

0 0 0

0 0

0

0 0 0 0

h w

h h w

h h h w

h w

    

    

    

    =

    

    

 

  

 

  

…

…

…

⋮ ⋮ ⋮ ⋱ ⋮ ⋮

… ( )^ZF

(J+1) (x J+1) ⋅ opt =

H w δ

( )^ZF 1 opt

= − ⋅

w H δ

Eye open condition:

1 0

1

L n n

h h

= <

∑

Zero forcing equalizer (cont’)

Problems:

1. It eliminates ISI, but enhances the effect of additive noise (SNR ↓) 2. The h_n is unknown

( ) ( ) ( )

0

1

J

l l k j k j k l

j

w k w k y w k x ₋ y ₋

=

 

+ = − ∆ − − 



∑

with learning set τ^(K)

MMSE equalizer

• The filter taps are adjusted such that the MSE of ISI and noise power at the equalizer output is minimized:

• Wiener filtering problem! (see DSP course) – Autocorrelation matrix

– Crosscorrelation vector:

• The linear equations in matrix form:

(^MMSE)

(J+1) (x J+1) ⋅ opt =

R w r ⁽opt^{MMSE) 1}

= − ⋅

w R r

(^MMSE)

opt : arg min _{k j k j}

j

E y w x ₋ 

=  − 



∑

w

w

{ }

ij k i k j

R ₌ E x ₋ x ₋

{ }

i k k i

r = E y x ₋

, 0,..., i j = J

0,..., i = J

MMSE equalizer (cont’)

Problems:

1. The h_n is unknown

with learning set τ^(K)

( ) ( ) ( )

0

1

J

l l k j k j k l

j

w k w k y w k x ₋ x ₋

=

 

+ = − ∆ − − 



∑

Summary

• In the signal space: calculate signals energy and Euclidean distances between signals.

• The receiver signal is converted into baseband, filtered and sampled (demodulation).

• Sampled values are used for detection using a decision rule such as the ML detection rule (detection).

• A communication system designer can determine the modulation scheme according to the SE and the PE.

• All of these can lead to significant distortion or attenuation of received signal (SNR) which degrade (BER) of digitally modulated signal.

• A weighted tap delay-line that reduces the effect of ISI by proper adjustment of the weights of the filter taps.

• Next lecture: Multiple channel access