equalization - islamic university of...

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© Ammar Abu-Hudrouss Islamic University Gaza

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EqualizationEqualization

Slide 2Wireless Communications

ISI can cause an irreducible error floor when the modulation symbol time is on the same order as the channel delay spread.

In a broad sense, equalization defines any signal processing technique used at the receiver to alleviate the ISI problem caused by delay spread.

Signal processing can also be used at the transmitter to make the signal less susceptible to delay spread. These are such as spread spectrum and multicarrier modulation.

Equalizer design must typically balance ISI mitigation with noise enhancement, since both the signal and the noise pass through the equalizer

Introduction

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goal of equalization is to mitigate the effects of ISI. However, this goal must be balanced so that in the process of removing ISI, the noise power in the received signal is not enhanced.

At the receiver front end white Gaussian noise n(t) is added to the signal, so the signal input to the receiver is

Y (f) = S(f)H(f)+N(f),

Equalizer Noise Enhancement


Suppose we wish to equalize the received signal so as to completely remove the ISI introduced by the channel. This is easily done by introducing an analog equalizer in the receiver defined by

And the output of the equalizer become

where N(f) is colored Gaussian noise with power spectral density N0 /|H (f )|2.

For small values of H (f) the noise is enhanced.

fHfH eq /1

fNfSfHfNfSfY '/

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Equalizer Types

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Equalization techniques fall into two broad categories: linear and nonlinear. The linear techniques are generally the simplest to implement and to understand conceptually. However, linear equalization techniques typically suffer from more noise enhancement than nonlinear equalizers.

Among nonlinear equalization techniques, decision-feedback equalization (DFE) is the most common, since it is fairly simple to implement and generally performs well.

However, on channels with low SNR, the DFE suffers from error propagation when bits are decoded in error, leading to poor performance.


The optimal equalization technique is maximum likelihood sequence estimation (MLSE).

Unfortunately, the complexity of this technique grows exponentially with the length of the delay spread, and is therefore impractical on most channels of interest.

Equalizers can also be categorized as symbol-by-symbol (SBS) or sequence estimators (SE).

Linear and nonlinear equalizers are typically implemented using a transversal or lattice structure.

The transversal structure is a filter with N − 1 delay elements and N taps with tunable complex weights.

The lattice filter uses a more complex recursive structure

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Folded Spectrum and ISI-free Trans


The symbol dk is passed through pulse shape filter g (t ) and then transmitted over the ISI channel with impulse response c (t).

We define the equivalent channel impulse response h (t) = g (t) ∗c (t), and the transmitted signal is thus given by

d (t) is the train of information symbols and given by

tctgtd **

k sk KTtdtd

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Let f (t) denote the combined baseband impulse response of the transmitter, channel, and matched filter:

Then the matched filter output is given by

Then sampling y (t) every Ts seconds yields

tgtctgtf m ***

tnkTtfdtgtntftdty gskm ***

nvknfdnyk

k

nvknfdfdnynk

kn

0


We notice that we get zero ISI if f [n − k] = 0 for k ≠ n, i.e. f [k] = δ [k]f[0].

In this y [n] = dnf [0] + ν [n].

We now show that the condition for ISI-free transmission, f [k]= δ [k]f [0 ]

The function FΣ(f) is often called the folded spectrum, and FΣ (f ) = f [0] implies that the folded spectrum is flat.

01 fTnfF

TfF

n ss

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To show this equivalence, first note that

We first show that a flat folded spectrum implies that f [k] = δ [k]f [0 ].

If we assume the folded spectrum equal f (0 )


If we assume f [k] = δ [k]f [0 ]

So f [k] is the inverse Fourier transform of FΣ(f). Therefore, if f [k] = δ [k]f [0 ] and FΣ (f) = f [0].

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Example: Consider a channel with combined baseband impulse response f (t) = sinc (t/Ts). Find the folded spectrum and determine if this channel exhibits ISI.

The Fourier transform of f (t ) is

Thus

No ISI . This can be seen from f (nTs) = 1 only for n = 0;

TfTfTTfT

fTrectTfF s

s

ss

/5.00/5.05.0/5.0

11

n ss TnfF

TfF


If FΣ (f ) is not flat, we can use the equalizer Heq(z) to reduce ISI. In this section we assume a linear equalizer implemented via an N = 2L + 1 tap transversal filter:

The length of the equalizer N is typically dictated by implementation considerations, since a large N usually entails higher complexity.

Causal linear equalizers have wi = 0, i < 0.

For a given equalizer size N the equalizer design must specify and update the tap weights {wi} for a given varying channel frequency response.

Linear Equalizer

L

Li

iieq zwzH

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The input to the equalizer can be represented as

where Ng (z) is the power spectrum of the white noise after passing through the matched filter G∗ (1/z∗) and

This is the discrete-time equivalent to the analog equalizer , and it suffers from the same noise enhancement properties. Specifically, the power spectrum N(z) is given by

Zero Forcing (ZF) Equalizer

zNzFzDzY g

**** /11

/11

)(1

zGzHzGzCzGzFzH ZF

20

2

zHNzZFHzNzN g


The ZF equalizer defined by HZF (z) = 1/F(z ) may not be implementable as a finite impulse response (FIR) filter. Specifically, it may not be possible to find a finite set of coefficients w−L, . . ., wL such that

IIR approximation can be done or the tap weights can be set to minimize the peak distortion by convex optimization.

zFzw

L

Li

ii

1

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Example: Consider an channel with impulse response

The channel also has AWGN with power spectral density N0. Find a two-tap ZF equalizer for this channel.

Solution: We have

So

The two tap ZF equalizer therefore has tap weight coefficients w0 = 1 and w1 = -e −(Ts/τ) .

000/

tte

nht

1/11

zezH

sT

1/1 zezH sTeq


In MMSE equalization the goal of the equalizer design is find the filter coefficients wi that minimize

Since the MMSE is a linear equalizer, its output is a linear combination of the input samples y [k]:

Minimum Mean Square Error (MMSE) Equalizer

2ˆkk ddE

L

Liik ikywd

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Consider a linear filter with N = 2L + 1 taps:

Define v = (v [k + L], v [k + L − 1 ] . . . , v [k − L]) = (vk+L, vk+L−1, . . . , vk−L) as a vector of inputs to the filter then

Thus, we want to minimize the mean square error

L

Li

iieq zwzH

wvvwd TTk ˆ

22*2*ˆ

kkHHT

kk ddwvwvvwEddEJ

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Define Mv = E[vvH] and vd = E[vHdk]. The matrix Mv is an N × N Hermitian matrix and vd is a length N row vector. Assume E|dk|2 = 1. Then the MSE J is

Finding the gradient of J with respect to w and solving for

Which leads to

12 ** wvwMwJ dvT

022,,

dvT

LLw vMw

wJ

wJJ

Td

Tvopt vMw

1


Substituting in these optimal tap weights we obtain the minimum mean square error as

For an infinite length equalizer, v = (vn+∞, . . . , vn, vn−∞) and w = (w−∞, . . ., w0, . . ., w∞). Then wTMv = vd can be written as [ref]

Taking z transforms

Ref: G.L. St¨uber, Principles of Mobile Communications, 2nd Ed. Kluwer Academic Publishers, 2001. (Ch 7.4)

Hdvd vMvJ 11

jjgijNijfw mi

i*

0

0

**

)(/1ˆNzFzGzH m

eq

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The full MMSE equalizer when it is not restricted to finite length

1) The ideal infinite length MMSE equalizer cancels out the noise whitening filter.

2) This infinite length equalizer is identical to the ZF filter except for the noise term N0, so in the absence of noise the two equalizers are equivalent.

3) This ideal equalizer design clearly shows a balance between inverting the channel and noise enhancement

0

** )(1

/1

ˆ

NzFzGzH

zHm

eqeq


It can be shown that the minimum MSE can be expressed in term of the folded spectrum FΣ(f )

Example: Find Jmin when the folded spectrum FΣ(f ) is flat, FΣ(f )= f [0], in the asymptotic limit of high and low SNR.

For high SNR, f0 >> N0 so Jmin ≈ N0/f0 = N0/Es =1/snr

For low SNR, N0 >> f0, so Jmin = 1

s

s

T

Ts dfNff

NTJ/5.0

/5.00

0min

00

0/5.0

/5.000

0min Nf

NdfNf

NTJ s

s

T

Ts

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Given the channel response h(t), the MLSE algorithm chooses the input sequence {dk} that maximizes the likelihood of the received signal w(t).

MLSE does not encode each symbol dk by itself but it estimated the most likely data sequence based on Veterbi algorithm.

The algorithm is very efficient but it has a very high computation complexity.

Maximum Likelihood Seq. Estimation


The DFE consists of a feedforward filter B (z) with the received sequence as input (similar to the linear equalizer) followed by a feedback filter D (z) with the previously detected sequence as input.

Decision Feedback Equalization

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The basic idea behind decision feedback equalization is that once the information symbol has been detected, its ISI that it induces on future symbols can be estimated and subtracted.


The feedback approximate the baseband channel F(z), the resultant feedback is subtracted from the incoming signal.

The feedback filter must be strictly causal.

Assuming W (z) has N1 taps and V (z) has N2 taps, we can write the DFE output as

The typical criteria for selecting the coefficients for W(z) and V (z) are either zero-forcing (remove all ISI) or MMSE (minimize the expected MSE between the DFE output and the original symbol).

2

1 1

0ˆ

N

iiki

Niik dvknywd

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When both W(z) and V (z) have infinite duration, it was shown by Price that the

optimal feedforward filter for a zero-forcing DFE is1/G∗m(1/z∗).

For the MMSE criterion, we wish to minimize

Let fn = f[n] denote the samples of f(t). Then this minimization implies that the coefficients of the feedforward filter must satisfy the following set of linear equations:

For

2ˆ

kk ddE

*0

1

lNi

ili fwq

0,,, 10

0 * NililNffqlj iljjli


The coefficients of the feedback filter are then determined from the feedforward coefficients by

It was shown that the resulting minimum MSE is

0

1Niikik fwv

s

s

T

Ts dfNff

NTJ/5.0

/5.00

0min lnexp

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Although MLSE is the optimal form of equalization, its complexity precludes its widespread use.

Most of these techniques either reduce the number of surviving sequences in the Viterbi algorithm or reduce the number of symbols spanned by the ISI through preprocessing or decision-feedback in the Viterbi detector.

These reduced complexity equalizers have better performance versus complexity tradeoffs than the other equalization techniques, and achieve performance close to that of the optimal MLSE with significantly less complexity.

Other Equalization Methods


All of the equalizers described so far are designed based on a known value of the composite channel response h(t) = g(t) ∗ c(t).

Since in wireless channels c(t) = c(τ, t) will change over time, the system must periodically estimate the channel c(t) and update the equalizer coefficients accordingly.

This process is called equalizer training or adaptive equalization . The equalizer can also use the detected data to adjust the equalizer coefficients. This process is called equalizer tracking.

Blind equalizers do not use training: they learn the channel response via the detected data only

Adaptive Equalizers: training and tracking

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During training, the coefficients of the equalizer are updated at time k based on a known training sequence [dk−M, . . . , dk] that has been sent over the channel.

The length M of the training sequence depends on the number of equalizer coefficients that must be determined and the convergence speed of the training algorithm.

If the training algorithm is slow relative to the channel coherence time then the channel may change before the equalizer can learn the channel.

we will choose the updated coefficient {w−L(k + 1), . . ., wL(k + 1)} as the coefficients that minimize the MSE between dk and d^k


Recall that

The w coefficients can be found by

Where

L

Liik ikykwd

pRkwkwkw LL11,,11

2*

*1

21

*1

**1

2

LkLkLk

LkLkLkLkLk

LkLkLkLkLk

yyy

yyyyyyyyyy

R

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And

Note that the optimal tap updates in this case requires a matrix inversion, which requires N 2 to N 3 multiply operations on each iteration (each symbol time Ts).

However, the convergence of this algorithm is very fast: it typically converges in around N symbol times for N the number

of equalizer tap weights

TLkLkk yydp


If complexity is an issue then the large number of multiply operations needed to do MMSE training can be prohibitive. A simpler technique is the least mean square (LMS) algorithm .

In this algorithm the tap weight vector w(k + 1) is updated linearly as

where k = dk − ˆ dk is the error between the bit decisions and the training sequence and Δ is the step size of the algorithm, which is a parameter that can be chosen.

The choice of Δ dictates the convergence speed and stability of the algorithm.

**1 LkLkk yykwkw

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However, the LMS algorithm exhibits significantly reduced complexity compared to the MMSE algorithm since the tap updates only require approximately 2N +1 multiply operations per iteration.


Consider a 5 tap equalizer that must retrain every .5Tc, where Tc is the coherence time of the channel. Assume the transmitted signal is BPSK with a rate of 1 Mbps for both data and training sequence transmission.

Compare the length of training sequence required for the LMS equalizer versus the Fast Kalman DFE. For an 80 Hz Doppler, by how much is the data rate reduced in order to do periodic training for each of these equalizers.

How many operations does each require for this training

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The equalizers must retrain every .5Tc = .5/Bd = .5/80 = 6.25 msec. From the table, for a data

rate of Rb = 1/Tb = 1 Mbps, the LMS algorithm requires 10NTb = 50 × 10−6 seconds to train, and the Fast

Kalman DFE requires NTb = 50 × 10−5 seconds to train. If training occurs every 6.25 msec, the fraction of time

the LMS algorithm uses for training is 50 × 10−6/6.25 × 10−3 = .008. Thus, the effective data rate becomes

(1 − .008)Rb=.992 Mbps. The fraction of time used by the Fast Kalman DFE for training is 50 × 10−5/6.25 ×

10−3 = .0008, resulting in an effective data rate of (1 − .0008)Rb=.9992 Mbps. The LMS algorithm requires

approximately 2N + 1 = 11 operations for training per training period, whereas the Fast Kalman DFE requires

20N + 5 = 105 operations, an order of magnitude more than the LMS algorithm. With processor technology

today, this is not a significant difference in terms of processor requiremen

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