3F4 Equalisation

1
3F4 Equalisation

• Dr. I. J. Wassell

2
Introduction

• When channels are fixed, we have seen that it is
  possible to design optimum transmit and receive
  filters, subject to zero ISI
• In practice, this is not usually possible,
• Ideal filters cannot be realised
• The channel responses can be unknown and/or time
  varying
• The same transmitter may be used over many
  different channels

3
Introduction

• We can improve the situation by including an
  additional filtering stage at the receiver. This
  is known as an equalisation filter and usually it
  is designed to reduce ISI to a minimum
• Equalisers may be categorised as,
• Fixed- The optimal equalisation filter is
  calculated for a fixed (known) received pulse
  shape
• Adaptive- The filter is adapted continuously to
  the changing characteristics of the channel

4
Introduction

• Equalisation may be implemented using,
• Analogue filters- A traditional technique mainly
  confined to fixed channels. Now superseded by,
• Digital filters- Have all usual advantage of
  digital systems, e.g. flexibility, reliability
  etc. May be either fixed or adaptive. We will
  consider fixed equalisers implemented as digital
  filters

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Digital Filters

• An analogue signal x(t) is sampled at times tnT
  to give a digital signal xn

• The Z-transform of xn is defined analogously to
  the Laplace transform of a continuous signal as,

6
FIR Filter

• A Finite Impulse Response (FIR) filter generates
  a new digital signal yn from xn using delay,
  multiply and addition operations

Where bi are known as the filter coefficients and
delay D is equal to the sample (symbol) period T
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FIR Filter

• Taking the Z transform yields,

Where z-n may be taken to mean a delay of n
sample periods

• Now,

• Hence the transfer function H(z) is,

8
IIR Filters

• A recursive Infinite Impulse Response Filter
  generates a new digital signal yn from the input
  xn as follows,

Where ai are known as the filter coefficients and
delay D is equal to the sample (symbol) period T
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IIR Filters

• Taking Z transform yields,

• Rearranging,

• Now,

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Zero-Forcing Equalisers

• Suppose the received pulse in a PAM system is
  p(t), which suffers ISI
• This signal is sampled at times tnT to give a
  digital signal pnp(nT)
• We wish to design a digital filter HE(z) which
  operates on pn to eliminate ISI
• Zero ISI implies that the filter output is only
  non-zero in response to pulse n at sample instant
  n, i.e. the filter output is the unit pulse dn in
  response to pn

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Zero-Forcing Equalisers

• Note that the Z transform of dn is equal to 1, so,

• Now,

Where pi are the sample values of the isolated
received pulse

• So,

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Zero-Forcing Equalisers

• We see that this expression has the form of an
  IIR filter,

If,
That is we define the amplitude of the isolated
pulse at the optimum sampling point to be unity
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FIR Approximations to ZFE

• IIR filters are difficult to deal with in
  practice
• stability is not guaranteed
• adaptive methods are difficult to derive
• Their recursive nature makes them prone to
  numerical instability
• The simplest solution is to use an FIR
  approximation to the ideal response

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Truncated Impulse Response

• A simple way to create an FIR approximation is
  simply to truncate the ideal impulse response
• However, this can give rise to significant errors
  in the filter response

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Truncated Impulse Response

• The IIR response has the form,

• The FIR response has the form,

• Thus we must perform polynomial division to
  calculate the coefficients of H(z)

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Truncated Impulse Response

• Example
• The unequalised pulse response at the receiver in
  response to a single unit amplitude transmitted
  pulse at sample times k 0, 1 and 2 is, p0 1,
  p1 - 0.4 and p2 - 0.2

Now,
So in this example,
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Truncated Impulse Response

• Performing the polynomial division,

Now,

• Truncating to 5 terms gives FIR filter with the
  coefficients 1, 0.4, 0.36, 0.224, 0.1616

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Direct Zero Forcing

• The FIR filter equaliser output in the time
  domain is,

• In the time domain, the zero forcing constraint
  is yn 1 for one value of n and yn 0 otherwise

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Direct Zero Forcing

• This constraint implies an infinite set of
  simultaneous equations corresponding to,

• However, we only have q1 filter coefficients, so
  we set up q1 equations in q1 unknowns and solve
  for the coefficients

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Direct Zero Forcing-Example

• The sampled received pulse in response to a
  single binary 1 is,

• Design a 3-tap FIR equaliser to make the response
  at n0 equal to 1, and equal to zero for n1 and
  n2

• The FIR equaliser filter output is,

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Direct Zero Forcing-Example

• The zero forcing constraint is,
• y0 1, y1 0, y2 0

• Write out previous equation for n0, 1 and 2,

• Solving these equations gives,

22
Error Rates and Noise

• Equalisation is designed to reduce ISI and hence
  increase the eye opening
• However, channel noise also passes through the
  equaliser and must be handled carefully to
  predict performance
• The frequency response of a digital filter may be
  obtained by substituting,

23
Error Rates and Noise

• The ideal ZFE has a response,

• So in the frequency domain,

• Thus at frequencies where P(ejwT) is small, large
  noise amplification will occur.

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Error Rates and Noise
Received pulse spectrum
Equaliser spectral response

• In this example the low pulse spectrum response
  near zero will give rise to high gain and noise
  enhancement by the equaliser in this region.

25
Error Rates and Noise

• What is the mean-square value (sw)2 of the noise
  at the equaliser output?
• Suppose the equaliser filter has impulse response
  bn, (n0,..,q).
• Consider the response of the equaliser to noise
  alone,

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Error Rates and Noise

• The mean-squared value is,

• Assume that vn has a mean-squared value,

And that vn is uncorrelated white noise. Then all
the terms Evnvm will be zero except when mn,
so,
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Error Rates and Noise

• That is, the mean square noise at the filter
  output is that at the input multiplied by the sum
  squared of the filter impulse response

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Error Rates and Noise

• Hence the worst case BER may be calculated as
  follows,
• Calculate the eye opening h for the equalised
  pulse
• Calculate the mean-squared noise power
• Substitute into the BER expression,

29
Error Rates and Noise- Example

• Returning to the previous example, calculate the
  worst case BER after equalisation if unipolar
  line coding with transmit levels of 1V and 0V is
  employed.

• The sampled received pulse in response to a
  single binary 1 is,

• The direct zero forcing solution is an FIR filter
  with the following coefficient values,

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Error Rates and Noise- Example

• We now need to calculate the worst case eye
  opening for the equalised pulse.
• To do this we need to calculate the residual
  values at the output of the equaliser in response
  to a single received pulse, pn
• From the earlier equations the FIR filter
  (equaliser) output is given by,

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Error Rates and Noise- Example

• In the example, the input sequence xn is the
  single pulse pn and q 2. In this case we have,

• This direct convoulution yields,

• Thus the equalised pulse response is,

1, 0, 0, -0.141, 0.0702
residuals
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Error Rates and Noise- Example

• So, remembering that for a unipolar scheme only
  1s give rise to residuals, the worst case
  received 1 is,
• 1 - 0.141 0.859 i.e., 1 other 1
  contributing
• The worst case 0 is,
• 0 0.0702 0.0702 i.e., 1 other 1
  contributing
• The minimum eye opening h is,
• h 0.859 - 0.0702 0.789

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Error Rates and Noise- Example

• To calculate the rms noise at the output of the
  equaliser we utilise,

Where sv is the rms noise at the equaliser input

• For our example,

Showing that the noise power has been increased
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Error Rates and Noise- Example

• The probability of bit error is given by,

• Substituting for h and sw gives,

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Error Rates and Noise- Example

• Note that instead of performing the convolution
  to give the equaliser output in response to a
  single received pulse (and hence determine the
  residuals), an alternative is to multiply the
  pulse response and equaliser response in the z
  domain, so

36
Error Rates and Noise- Example
Equating this to the expansion for Y(z),
Which yields the same expressions for the output
sample values yn obtained previously by direct
convolution
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Other Equalisation Methods

• We have seen that with ZFEs, noise can be
  amplified leading to poor BER performance
• Alternative design approaches take into account
  noise as well as signal propagation through the
  equaliser
• The Minimum Mean Squared Error (MMSE) equaliser
  is one such approach

38
MMSE Equaliser

• The MMSE explicitly accounts for the presence of
  noise in the system
• Assuming a similar model to that used previously,
  then in Z transform notation,

Where X(z) is the Z transform of the sampled
received signal xn, and V(z) is the Z transform
of the noise vn
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MMSE Equaliser

• Ideally, the equalised output yn depends only on
  the transmitted symbols ak. This is not possible
  owing to the random noise, hence we choose to
  minimise the total expected mean square error
  (MSE) between yn and an with respect to the
  equaliser HE(z), i.e.,

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MMSE Equaliser
MMSE equaliser formulation
minimise
From data source
For a fixed equaliser E(.)2 is minimised by
adjusting the coefficients of HE(z). Effectively
we have a trade off between noise enhancement and
ISI.
41
MMSE Equaliser

• The solution has the form,

Where P(z) is the Z transform of the channel
pulse response and No is the noise PSD

• Note,
• The equaliser needs knowledge of the noise PSD
• If No0, the solution is the same as the ZFE
• When noise is present the ZFE solution is
  modified to make a trade-off between ISI and
  noise amplification

42
Non-Linear Equalisation

• The equalisers considered so far are linear,
  since they simply involve linear filtering
  operations
• An alternative we consider now is non-linear
  equalisation
• An example is the Decision Feedback Equaliser
  (DFE)

43
DFE

• The DFE is a non-linear filter.
• Again, its purpose is to cancel ISI.
• The non-linearity allows some of the noise
  problems associated with linear equalisers to be
  overcome

44
DFE

• The structure is,

Detected symbols
Where ai are known as the filter coefficients
(not to be confused with the transmitted symbols
an) and delay D is equal to the sample (symbol)
period T
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DFE

• The DFE has almost the same structure as the
  standard IIR filter based equaliser
• In the following development this relationship is
  demonstrated
• We see that the only significant difference is
  the position of the data slicer (decision block)
• A minor difference is the subtract function at
  the DFE input. Its only effect is to alter the
  sign of the DFE coefficients compared with those
  in the IIR filter

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DFE Development
yn
IIR Structure
Slicer
xn
D
D
D
D

ap
a1
a2
X
X
X

xn
yn

Slicer
D
D
D
D
ap
a1
a2
IIR
X
X
X

47
DFE Development
xn
yn

Slicer

IIR
ap
a1
a2
X
X
X
D
D
D
D
yn
xn

Slicer
DFE

ap
a1
a2
X
X
X
D
D
D
D
48
DFE

• The DFE is almost a standard IIR filter
• For this structure we know that the ZFE solution
  is,

Where p0, p1, etc. is the sampled response at the
equaliser input received in response to one
transmitted symbol of unity amplitude. Because we
define the amplitude of the isolated pulse at the
optimum sampling point to be unity, then po 1.
Comparing with the previous ZF solution where ai
-pi, this time ai pi owing to the subtract
function at the DFE input.

• The outputs of this filter with no channel noise
  are unit pulses, weighted by the transmitted
  symbol amplitudes, an

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DFE Example

• The sampled response to an isolated received
  pulse pn is given by,
• p0 1, p1 0.5, p2 -0.25
• Design a suitable DFE
• From the earlier work we see that the DFE
  coefficients are given by ai pi, so,
• a1 p1 0.5 and a2 p2 -0.25
• Assuming polar data pulses the effect is to
• add (subtract) 0.25 (if previous but one bit is
  a binary one (zero)) to the current input value
  to remove its effect.
• subtract (add) 0.5 (if previous bit is a binary
  one (zero)) to the current input value to remove
  its effect.
• Thus the effect of the previous pulses is
  eliminated

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DFE Example
p(nT)
1
Sampled isolated pulse
0.5
0
T
2T
3T
nT
-0.5
p0 1, p1 0.5, p2 -0.25
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DFE

• In noise, we have seen that noise amplification
  occurs in the IIR filter approach
• To overcome this, the decision slicer is moved
  inside the filter loop in the DFE
• The slicer outputs the symbol estimate which
  is closest to the value at its input
• With no noise, this change makes no difference
  because the ISI is cancelled perfectly by the IIR
  filter, so the slicer input is ak anyway

52
DFE

• However, in noise, the slicer acts to clean-up
  the signal, giving a noise free decision at its
  output.
• For example, the slicer input becomes akvk,
  where vk is the noise value
• Provided that vk is small enough the slicer still
  outputs the correct decision ak
• So error free cancellation continues without
  problems of noise amplification

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DFE-Problems

• Consider what happens when vk is large enough to
  cause an error in the slicer decision
• The error feeds back around the loop and so the
  ISI is no longer cancelled.
• Often a long run or burst of errors will then
  be experienced- known as error propagation
• The length of the burst is of the order of 2N
  bits, where N is the number of taps in the
  feedback filter

54
Automatic Equalisers

• In practical communication systems the channel is
  often unknown and/or time varying
• To overcome this problem so called Automatic
  Equalisers are employed
• Two approaches are used
• Preset equaliser The channel is measured
  periodically by sending some known data. The
  equaliser coefficients are re-calculated and
  subsequent data is equalised using the new
  coefficients.

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Automatic Equalisers

• Adaptive equaliser The coefficients are adapted
  continuously based on the received data. A simple
  approach uses the Least Mean Squares (LMS)
  algorithm to adjust the coefficient values based
  on an error criterion. This approach requires
  that the equaliser is initially trained, so that
  the coefficients are initialised with
  approximately the correct values
• An alternative adaptation algorithm is known as
  Recursive Least Square (RLS). This has the
  advantage of faster training but at the cost of
  higher complexity

56
Summary

• In this section we have seen
• That in practical systems it is difficult to
  arrange optimum TX and RX filters subject to zero
  ISI
• That additional filters (equalisers) can be added
  to reduce ISI and improve BER performance
• Equalisers can be implemented in an analogue or
  digital manner and may be fixed or adaptive
• Digital implementations are fundamentally of IIR
  structure but may be approximated by a truncated
  FIR filter
• The zero-forcing criterion may lead to noise
  enhancement and poor performance
• MMSE and non-linear (DFE) approaches can reduce
  the problem of noise enhancement