Lecture 12 1

Lecture 12

Equalization

Lecture 12 2

Intersymbol Interference

– With any practical channel the inevitable filtering effect will cause a spreading (or smearing out) of individual data symbols passing through a channel.

Lecture 12 3

• For consecutive symbols this spreading causes part of the symbol energy to overlap with neighbouring symbols causing intersymbol interference (ISI).

Lecture 12 4

ISI can significantly degrade the ability of the data detector to differentiate a current symbol from the diffused energy of the adjacent symbols.

With no noise present in the channel this leads to the detection of errors known as the irreducible error rate.

It will degrade the bit and symbol error rate performance in the presence of noise

Lecture 12 5

Pulse Shape for Zero ISI

Careful choice of the overall channel characteristic makes it possible to control the intersymbol interference such that it does not degrade the bit error rate performance of the link.

Achieved by ensuring the overall channel filter transfer function has what is termed a Nyquist frequency response.

Lecture 12 6

Nyquist Channel Responsetransfer function has a transition band

between passband and stopband that is symmetrical about a frequency equal to 0.5 x 1/ Ts.

Lecture 12 7

For such a channel the the data signals are still smeared but the waveform passes through zero at multiples of the symbol period.

Lecture 12 8

If we sample the symbol stream at the precise point where the ISI goes through zero, spread energy from adjacent symbols will not affect the value of the current symbol at that point.

This demands accurate sample point timing - a major challenge in modem / data receiver design.

Inaccuracy in symbol timing is referred to as timing jitter.

Lecture 12 9

Achieving the Nyquist Channel

Very unlikely that a communications channel will inherently exhibit a Nyquist transfer response.

Modern systems use adaptive channel equalisers to flatten the channel transfer function.

Adaptive equalisers use a training sequence.

Lecture 12 10

Eye Diagrams

Visual method of diagnosing problems with data systems.

Generated using a conventional oscilloscope connected to the demodulated filtered symbol stream.

Oscilloscope is re-triggered at every symbol period or multiple of symbol periods using a timing recovery signal.

Lecture 12 11

Lecture 12 12

Example eye diagrams for different distortions, each has a distinctive effect on the appearance of the ‘eye opening’:

Lecture 12 13

– Example of complex ‘eye’ for M-ary signalling:

Lecture 12 14

Raised Cosine Filtering• Commonly used realisation of a Nyquist filter. The

transition band (zone between pass- and stopband) is shaped like a cosine wave.

Lecture 12 15

The sharpness of the filter is controlled by the parameter β, the filter roll-off factor.

When β = 0 this conforms to the ideal brick-wall filter.

The bandwidth B occupied by a raised cosine filtered data signal is thus increased from its minimum value, Bmin = 0.5 x 1/Ts, to:

Actual bandwidth B = Bmin ( 1 + β)

Lecture 12 16

Impulse Response of Filter

– The impulse response of the raised cosine filter is a measure of its spreading effect.

– The amount of ‘ringing’ depends on the choice of β.

– The smaller the value of a (nearer to a ‘brick wall’ filter), the more pronounced the ringing.

Lecture 12 17

Choice of Filter Roll-off β

• Benefits of small β– maximum bandwidth efficiency is achieved

• Benefits of large β– simpler filter with fewer stages hence easier to

implement– less signal overshoot– less sensitivity to symbol timing accuracy

Lecture 12 18

Symbol Timing Recovery

– Most symbol timing recovery systems obtain their information from the incoming message data using ‘zero crossing’ information in the baseband signal.

Lecture 12 19

Three kinds of systems

mT T

Narrowband system:Flat fading channel, single-tap channel model.

bit or symbolbit or symbol

Tm = delay spread of multipath channel

T = bit or symbol duration

System bandwidth

T W

Lecture 12 20

No intersymbol interference (ISI)

Adjacent symbols (bits) do not affect the decision process (in other words there is no intersymbol interference).

However: Fading (destructive interference)

is still possible

No intersymbol interference at

decision time instant

Received replicas of same symbol overlap in multipath channel

Narrowband system:

Lecture 12 21

Decision circuit

In the binary case, the decision circuit compares the received signal with a threshold at specific time instants (usually somewhere in the middle of each bit period):

Decision time instant

Decision circuitDecision circuitDecision threshold

Noisy and distorted symbols “Clean” symbols

Lecture 12 22

Three kinds of systems, Cont.

...m mT T T T

Wideband system (TDM, TDMA):Selective fading channel, transversal filter channel model, good performance possible through adaptive equalization

T = bit or symbol duration

Intersymbol interference causes signal distortion

Tm = delay spread of multipath channel

Lecture 12 23

Receiver structure

The intersymbol interference of received symbols (bits) must be removed before decision making (the case is illustrated below for a binary signal, where symbol = bit):

Decision circuit

Decision circuit

Adaptiveequalizer

Adaptiveequalizer

Symbols with ISI

Symbols with ISI removed

“Clean” symbols

Decision time instantDecision threshold

Lecture 12 24

Three kinds of systems: BER performance

S/N

BER

Frequency-selective channel (no equalization)

Flat fading channel

AWGN channel

(no fading)

Frequency-selective channel (with equalization)

“BER floor”

Lecture 12 25

Three kinds of systems, Cont.

m cT T T

Wideband system (DS-CDMA):Selective fading channel, transversal filter channel model, good performance possible through use of Rake receiver (this lecture).

Bit (or symbol)Bit (or symbol)

Tm = delay spread of multipath channel

T = bit (or symbol) duration

Tc = Chip duration ...

cT W

Lecture 12 26

Three basic equalization methods

1)- Linear equalization (LE):Performance is not very good when the frequency response of the frequency selective channel contains deep fades.

Zero-forcing algorithm aims to eliminate the intersymbol interference (ISI) at decision time instants (i.e. at the centre of the bit/symbol interval).

Least-mean-square (LMS) algorithm will be investigated in greater detail in this presentation.

Recursive least-squares (RLS) algorithm offers faster convergence, but is computationally more complex than LMS (since matrix inversion is required).

Lecture 12 27

Three basic equalization methods

2)-Decision feedback equalization (DFE):Performance better than LE, due to ISI cancellation of tails of previously received symbols.

Decision feedback equalizer structure:

Feed-forward filter (FFF)

Feed-forward filter (FFF)

Feed-back filter (FBF)

Feed-back filter (FBF)

Adjustment of filter coefficients

Input Output

++

Symbol decision

Lecture 12 28

Three basic equalization methods, Cont.

3)- Maximum Likelihood Sequence Estimation using the Viterbi Algorithm (MLSE-VA):Best performance. Operation of the Viterbi algorithm can be visualized by means of a trellis diagram with m

K-1 states, where m is the symbol alphabet size and K is the length of the overall channel impulse response (in samples).

State trellis diagram

Sample time instants

State

Allowed transition between states

Lecture 12 29

Linear equalization, zero-forcing algorithm

Raised cosine

spectrum

Raised cosine

spectrum

Transmitted symbol

spectrum

Transmitted symbol

spectrum

Channel frequencyresponse

(incl. T & R filters)

Channel frequencyresponse

(incl. T & R filters)

Equalizer frequency response

Equalizer frequency response

=

Z f B f H f E f

0 ffs = 1/T

B f

H f

E f

Z f

Basic idea:

Lecture 12 30

Zero-forcing equalizer

Communication channel

Equalizer

FIR filter contains 2N+1 coefficients

r k z kTransmitted impulse sequence Input to

decision circuit

N

nn N

h k h k n

Channel impulse response Equalizer impulse response

M

mm M

c k c k m

Coefficients of equivalent FIR filter

( )M

k m k mm M

f c h M k M

(in fact the equivalent FIR filter consists of 2M+1+2N coefficients, but the equalizer can only “handle” 2M+1 equations)

FIR filter contains 2M+1 coefficients

Overall channel

Lecture 12 31

Zero-forcing equalizer

We want overall filter response to be non-zero at decision time k = 0 and zero at all other sampling times k 0 :

1, 0

0, 0

M

k m k mm M

kf c h

k

0 1 1 2

1 0 1 2 1

1 1

2 1 2 2 1 1

2 2 1 1 0

... 0

... 0

:

... 1

:

... 0

... 0

M M M M

M M M M

M M M M M M

M M M M M

M M M M M

h c h c h c

h c h c h c

h c h c h c

h c h c h c

h c h c h c

This leads to a set of 2M+1 equations:

(k = –M)

(k = 0)

(k = M)

Lecture 12

32 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Equalization: Removing Residual ISI Consider a tapped delay line equalizer with

Search for the tap gains cN such that the output equals zero at sample intervals D except at the decision instant when it should be unity. The output is (think for instance paths c-N, cN or c0)

that is sampled at yielding

( ) ( ) N

eq nn N

p t c p t nD ND

( ) ( ) ( )N N

eq n nn N n N

p kD ND c p kD nD c p D k n

k

t kD ND

( ) ( 2 )N N

p t c p t ND

( ) ( )N N

p t c p t

0( ) ( )

Np t c p t ND

Lecture 12

33 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Example of Equalization

Read the distorted pulse values into matrix from fig. (a)

and the solution is

1

0

1

1.0 0.1 0.0 0

0.2 1.0 0.1 1

0.1 0.2 1.0 0

c

c

c

1

0

1

0.096

0.96

0.2

c

c

c

Zero forced values

0p

1p

2p

1p

2p

Lecture 12 34

Minimum Mean Square Error (MMSE)

The aim is to minimize:2

kJ E e

ˆ ˆk k k k ke z b b z (or depending on the source)

EqualizerEqualizerChannelChannelkz kb

ke

r k s k

+

Estimate of k:th symbol

Input to decision circuit

z k b k

Error

Lecture 12 35

MSE vs. equalizer coefficients

2

kJ E e

J

1c

2c

quadratic multi-dimensional function of equalizer coefficient values

MMSE aim: find minimum value directly (Wiener solution), or use an algorithm that recursively changes the equalizer coefficients in the correct direction (towards the minimum value of J)!

Illustration of case for two real-valued equalizer coefficients (or one complex-valued coefficient)

Lecture 12 36

Wiener solution

opt Rc p

R = correlation matrix (M x M) of received (sampled) signal values

p = vector (of length M) indicating cross-correlation between received signal values and estimate of received symbol

copt = vector (of length M) consisting of the optimal equalizer coefficient values

(We assume here that the equalizer contains M taps, not 2M+1 taps like in other parts of this presentation)

We start with the Wiener-Hopf equations in matrix form:

kb

kr

kr

Lecture 12 37

Correlation matrix R & vector p

Before we can perform the stochastical expectation operation, we must know the stochastical properties of the transmitted signal (and of the channel if it is changing). Usually we do not have this information => some non-stochastical algorithm like Least-mean-square (LMS) must be used.

*TE k k R r r

*kE k b p r

1 1, ,...,T

k k k Mk r r r rwhere

M samples

Lecture 12 38

Algorithms

Stochastical information (R and p) is available:

1. Direct solution of the Wiener-Hopf equations:

2. Newton’s algorithm (fast iterative algorithm)

3. Method of steepest descent (this iterative algorithm is slow but easier to implement)

R and p are not available:

Use an algorithm that is based on the received signal sequence directly. One such algorithm is Least-Mean-Square (LMS).

opt Rc p 1opt

c R p Inverting a large matrix is difficult!

Lecture 12 39

Conventional linear equalizer of LMS type

k Mr k Mr TT TT TT

LMS algorithm for adjustment of tap coefficients

Transversal FIR filter with 2M+1 filter taps

Estimate of kth symbol after symbol decision

Complex-valued tap coefficients of equalizer filter

ke

+

kbkz

Mc 1 Mc 1Mc Mc

WidrowReceived complex

signal samples

Lecture 12 40

Joint optimization of coefficients and phase

Equalizer filterEqualizer filter

Coefficient updating

Coefficient updating

Phase synchronization

Phase synchronization

ke

je

+

r kkz

kb

ˆ ˆexpM

k k k m k m km M

e z b c r j b

Minimize:2

kJ E e

Lecture 12 41

Least-mean-square (LMS) algorithm(derived from “method of steepest descent”) for convergence towards minimum mean square error (MMSE)

2

Re 1 ReRe

kn n

n

ec i c i

c

Real part of n:th coefficient:

2

Im 1 ImIm

kn n

n

ec i c i

c

Imaginary part of n:th coefficient:

2

1 kei i

Phase:

2 2 1 1M equations Iteration index Step size of iteration

2

k k ke e e

Lecture 12 42

LMS algorithm (cont.)

After some calculation, the recursion equations are obtained in the form

ˆRe 1 Re 2 ReM

j jn n m k m k k n

m M

c i c i e c r b r e

ˆIm 1 Im 2 ImM

j jn n m k m k k n

m M

c i c i e c r b r e

ˆ1 2 ImM

jk m k m

m M

i i b e c r

ke

Lecture 12 43

Effect of iteration step size

Slow acquisitionSlow acquisition

smaller larger

Poor tracking performance

Poor tracking performance

Poor stabilityPoor stability

Large variation around optimum

value

Large variation around optimum

value

Lecture 12 44

Decision feedback equalizer

TT TT TT

LMS algorithm

for tap coefficient adjustment

TT TT

FFFFFF

FBFFBF +

+ ke

kb

kz

k Qb 1kb

k Mr k Mr

Mc 1 Mc 1Mc Mc

1Qq Qq1q?

Lecture 12 45

The purpose is again to minimize

Decision feedback equalizer (cont.)

1

ˆ ˆ ˆQM

k k k m k m n k n km M n

e z b c r q b b

2 2

k kJ E e e

Feedforward filter (FFF) is similar to filter in linear equalizer tap spacing smaller than symbol interval is allowed => fractionally spaced equalizer => oversampling by a factor of 2 or 4 is common

Feedback filter (FBF) is used for either reducing or canceling samples of previous symbols at decision time instants

Tap spacing must be equal to symbol interval

where

Lecture 12 46

The coefficients of the feedback filter (FBF) can be obtained in either of two ways:

Recursively (using the LMS algorithm) in a similar fashion as FFF coefficients

By calculation from FFF coefficients and channel coefficients (we achieve exact ISI cancellation in this way, but channel estimation is necessary):

Decision feedback equalizer (cont.)

1, 2, ,M

n m n mm M

q c h n Q

Lecture 12 47

Lecture 12 48

Channel estimation circuit

TT TT TT

LMS algorithm

Estimated symbols

+

Mc1Mc 1c0c

k Mb kb

krkr

ˆm mh c

k:th sample of received signal Estimated channel coefficients

Filter length = CIR length

Lecture 12 49

Channel estimation circuit (cont.)

1. Acquisition phase Uses “training sequence”Symbols are known at receiver, .

2. Tracking phase Uses estimated symbols (decision directed mode) Symbol estimates are obtained from the decision circuit (note the delay in the feedback loop!) Since the estimation circuit is adaptive, time-varying channel coefficients can be tracked to some extent.

k kb b

Alternatively: blind estimation (no training sequence)

Lecture 12 50

Channel estimation circuit in receiver

Channel estimation

circuit

Channel estimation

circuit

Equalizer & decision

circuit

Equalizer & decision

circuit

h m b k

b k

r k

Estimated channel coefficients

“Clean” output symbolsReceived signal samples

Symbol estimates (with errors)

Training symbols

(no errors)

Mandatory for MLSE-VA, optional for DFE

Lecture 12 51

MLSE-VA receiver structure

Matched filter

Matched filter

MLSE(VA)

MLSE(VA)

Channel estimation circuit

Channel estimation circuit

NW filter

NW filter

r t

f k

y k b k

f k

MLSE-VA circuit causes delay of estimated symbol sequence before it is available for channel estimation

=> channel estimates may be out-of-date (in a fast time-varying channel)

Lecture 12 52

MLSE-VA receiver structure (cont.)

The probability of receiving sample sequence y (note: vector form) of length N, conditioned on a certain symbol sequence estimate and overall channel estimate:

21

2 21 01

1 1 ˆ ˆˆ ˆˆ ˆ, , exp22

N N K

k k n k nN Nk nk

p p y y f b

y b f b f

Metric to be minimized

(select best .. using VA)

Objective: find symbol

sequence that maximizes this

probability

This is allowed since noise samples are

uncorrelated due to NW (= noise whitening) filter

Length of f (k)Since we have AWGN

b

Lecture 12 53

MLSE-VA receiver structure (cont.)

We want to choose that symbol sequence estimate and overall channel estimate which maximizes the conditional probability.

Since product of exponentials <=> sum of exponents, the metric to be minimized is a sum expression.

If the length of the overall channel impulse response in samples (or channel coefficients) is K, in other words the time span of the channel is (K-1)T, the next step is to construct a state trellis where a state is defined as a certain combination of K-1 previous symbols causing ISI on the k:th symbol.

K-10 k

f k Note: this is overall CIR, including response of matched

filter and NW filter

Lecture 12 54

MLSE-VA receiver structure (cont.)

At adjacent time instants, the symbol sequences causing ISI are correlated. As an example (m=2, K=5):

1 0 0 1 0

1 0 0 1 0

0 0 1 0 0

0

1

At time k-3

At time k-2

At time k-1

:

:

Bits causing ISI not causing ISI at time instant

At time k

1

0 1 0 0 11 0 1

Bit detected at time instant16 states

Lecture 12 55

MLSE-VA receiver structure (cont.)

State trellis diagram

k-2 k-1 kk-3

1Km

Number of states The ”best” state

sequence is estimated by

means of Viterbi algorithm (VA)

k+1

Of the transitions terminating in a certain state at a certain time instant, the VA selects the transition associated with highest accumulated probability (up to that time instant) for further processing.

Alphabet size

Lecture 12 56

Rake receiver structure and operation

Rake receiver <=> a signal processing example that illustrates some important concepts

Rake receiver is used in DS-CDMA (Direct Sequence Code Division Multiple Access) systems

Rake “fingers” synchronize to signal components that are received via a wideband multipath channel

Important task of Rake receiver is channel estimation

Output signals from Rake fingers are combined, for instance calculation of Maximum Ratio Combining (MRC)

Lecture 12 57

Principle of RAKE Receiver

Lecture 12 58

To start with: multipath channel

in which case the received (equivalent low-pass) signal is of the form

Suppose a signal s (t) is transmitted. A multipath channel with M physical paths can be presented (in equivalent low-pass signal domain) in form of its Channel Impulse Response (CIR)

1

0

( ) ( ) ( ) m

Mj

m mm

r t s t h t a e s t

1

0

( ) m

Mj

m mm

h t a e t

.

Lecture 12 59

Sampled channel impulse response

Delay ( )

1

0

( ) n

Nj

nn

h t a e t n

Sampled Channel Impulse Response (CIR)

The CIR can also be presented in sampled form using N complex-valued samples uniformly spaced at most 1/W apart, where W is the RF system bandwidth:

CIR sampling rate = for instance sampling rate used in receiver during A/D conversion.

Uniformly spaced channel samples

1 W

Lecture 12 60

Rake finger selection

Delay ( )

1

( ) i

Lj

rake i ii

h t a e t

Channel estimation circuit of Rake receiver selects L strongest samples (components) to be processed in L Rake fingers:

In the Rake receiver example to follow, we assume L = 3.

1 2 3

Only one sample chosen, since adjacent samples may be correlated

Only these samples are constructively utilized in

Rake fingers

Lecture 12 61

Received multipath signal

Received signal consists of a sum of delayed (and weighted) replicas of transmitted signal.

All signal replicas are contained in received signal

:

Signal replicas: same signal at different delays, with different amplitudes and phases

Summation in channel <=> “smeared” end result

Blue samples indicate signal replicas processed in Rake fingersGreen samples only cause interference

Lecture 12 62

Rake receiver

Finger 1Finger 1

Finger 2Finger 2

Channel estimationChannel estimation

Received baseband multipath signal

Finger 3Finger 3

Output signal

(to decision

circuit)

Rake receiver Path combining

(Generic structure, assuming 3 fingers)

WeightingWeighting

Lecture 12 63

Channel estimation

Channel estimationChannel estimation

AB

C

A

B

C

Amplitude, phase and delay of signal components detected in Rake fingers must be estimated.

ia i i

Each Rake finger requires delay (and often also phase) information of the signal component it is processing.

iai

Maximum Ratio Combining (MRC) requires amplitude (and phase if this is utilized in Rake fingers) of components processed in Rake fingers.

( )i ( )i

Lecture 12 64

Rake finger processing

Case 1: same code in I and Q branches

Case 2: different codes in I and Q branches

- for purpose of easy demonstration only

- the real case e.g. in IS-95 and WCDMA

- no phase synchronization in Rake fingers

- phase synchronization in Rake fingers

Lecture 12 65

DelayDelay

Rake finger processing

Tdt

Received signal

To MRC

Tdt if

Stored code sequenceStored code sequence

(Case 1: same code in I and Q branches)

I branch

Q branch

I/QI/Q

Output of finger: a complex signal value for each detected bit

Lecture 12 66

Correlation vs. matched filtering

TdtReceived

code sequence

Received code sequence

Stored code sequenceStored code sequence

Basic idea of correlation:

Same result through matched filtering and sampling:

Received code sequence

Received code sequence

Matched filter

Matched filter

Sampling at t = T

Sampling at t = T

Sam

e e

nd

resu

lt (in th

eory

)

Lecture 12 67

Rake finger processing

1

i n

Nj j

i i n nnn i

r t z t v t w t

a e s t a e s t w t

Correlation with stored code sequence has different impact on different parts of the received signal

= desired signal component detected in i:th Rake finger

= other signal components causing interference

= other codes causing interference (+ noise ... )

z t

v t

w t

Lecture 12 68

Rake finger processing

Illustration of correlation (in one quadrature branch) with desired signal component (i.e. correctly aligned code sequence)

Desired component

Stored sequence

After multiplication

Strong positive/negative “correlation result” after integration

“1” bit “0” bit “0” bit

Lecture 12 69

Rake finger processing

Illustration of correlation (in one quadrature branch) with some other signal component (i.e. non-aligned code sequence)

Other component

Stored sequence

After multiplication

Weak “correlation result” after integration

“1” bit “0” bit

Lecture 12 70

Rake finger processing

Mathematically:

0

2

0

1 0 0

i

n

T

i

Tj

i

T TNj

n n inn i

C z t v t w t s t dt

a e s t dt

a e s t s t dt w t s t dt

Correlation result for bit between

Interference from same signal

Interference from other signals

Desired signal

0,T

Lecture 12 71

Rake finger processing

Set of codes must have both: - good autocorrelation properties (same code sequence) - good cross-correlation properties (different sequences)

2

0

1 0 0

i

n

Tj

i i

T TNj

n n inn i

C a e s t dt

a e s t s t dt w t s t dt

Large

Small Small

Lecture 12 72

DelayDelay

Rake finger processing

Tdt

Received signal

Tdt

Stored I code sequenceStored I code sequence

(Case 2: different codes in I and Q branches)

I branch

Q branch

I/QI/Q

Stored Q code sequenceStored Q code sequence

i

To MRC for I signal

To MRC for Q signal

Required: phase synchronization

if

Lecture 12 73

Phase synchronization

I/QI/Q

i

When different codes are used in the quadrature branches (as in practical systems such as IS-95 or WCDMA), phase synchronization is necessary.

Phase synchronization is based on information within received signal (pilot signal or pilot channel).

Signal in I-branch

Pilot signalPilot signal

Signal in Q-branch

I

Q

Note: phase synchronization must

be done for each finger separately!

Lecture 12 74

Weighting

Maximum Ratio Combining (MRC) means weighting each Rake finger output with a complex number after which the weighted components are summed “on the real axis”:

3

1

i ij ji i

i

Z a e a e

Component is weighted

Phase is aligned

Rake finger output is complex-valued

real-valued

(Case 1: same code in I and Q branches)

Instead of phase alignment: take

absolute value of finger outputs ...

Lecture 12 75

Phase alignment

The complex-valued Rake finger outputs are phase-aligned using the following simple operation:

1i ij je e

Before phase alignment:

ije

ije

1

After phase alignment:

Phasors representing complex-valued Rake

finger outputs

Lecture 12 76

Maximum Ratio Combining

The idea of MRC: strong signal components are given more weight than weak signal components.

The signal value after Maximum Ratio Combining is:

2 2 21 2 3Z a a a

(Case 1: same code in I and Q branches)

Lecture 12 77

Maximum Ratio Combining

Output signals from the Rake fingers are already phase aligned (this is a benefit of finger-wise phase synchronization).

Consequently, I and Q outputs are fed via separate MRC circuits to the decision circuit (e.g. QPSK demodulator).

(Case 2: different codes in I and Q branches)

Quaternarydecisioncircuit

Quaternarydecisioncircuit

Finger 1Finger 1

Finger 2Finger 2

MRC

MRC

MRC

MRC

:

I

Q

I

Q

I

Q