Download pdf - mmse

Spring'09 ELE 739 - Channel Equalization 1

Minimum Mean Square Error (MMSE) Equalizer

• Linear equalizer. • Aims at minimizing the variance of the difference between the

transmitted data and the signal at the equalizer output.– This effectively equalizes the freq. selective channel.

• First, consider the infinite length filter case:

• The output of the equalizer is

where the equalized channel IR is


MMSE Equalizer – Infinite Length • The difference between the Tx.ed data and the equalizer output is:

• and the MMSE cost function is:

• This is a quadratic function ⇒ with a unique minimum– Take derivative w.r.t. wj and equate to 0 to find this minimum.

– Using


MMSE Equalizer• Principle of orthogonality:

• The necessary and sufficient condition for the cost function J to attain its minimum value is, for the corresponding value of the estimation error ε[n] to be orthogonal to each input sample t[n] that enters into the estimation of the desired response at time n.

• Error at the minimum is uncorrelated with the filter input!

• In other words, nothing else can be done for the error by just observing the filter inputs.

• A good basis for testing whether the linear filter is operating in its optimum condition.


MMSE Equalizer

• Corollary:

If the filter is operating in optimum conditions (in the MSE sense)

• When the filter operates in its optimum condition, the filter outputz[n] and the corresponding estimation error ε[n] are orthogonal to each other.

z[n]

ε[n]x[n]


MMSE Equalizer• We can calculate the MMSE equalizer by either minimizing J over w:

• or using the principle of orthogonality:

which gives us the Wiener-Hopf Equations

ACF of the WMF output Cross-CF of the Tx.ed data and the WMF output


Optimum Equalizer

• It can easily be shown that

And

Taking the z-transform of the eqn. at the top, we get

Alternatively, incorporating the WMF into the MMSE equalizer, we get


MMSE vs. ZF

• MMSE: ZF:

• MMSE suppresses noise, besides equalizing the channel.– MMSE will not let infinite noise as ZF does when the channel has a spectral

null.

• As noise becomes negligible → N0→0– MMSE and ZF becomes identical.– When N0=0, MMSE cancels ISI completely (ZF cancels for all SNR values)– When N0 ≠0, residual ISI and noise will be observed at the output of the

MMSE equalizer.


MMSE - Performance• What is the value of Jmin?

• Due to the principle of orthogonality, , then

• The summation is a convolution evaluated at shift zero.

=b0


MMSE - Performance

• Then

• No ISI → X(ejωT)=1 →

• Note that,

• Furthermore, output SNR is

• No ISI → → Same as ZF.


MMSE – Performance

• Example 1: The effective channel has two taps,

• Spectrum is

• When we evaluate the integral of b0, Jmin becomes

• When , Jmin and output SNR γ∞ are

• No ISI →

(has a null at ω=π/T when)


MMSE - Performance

• Example 2: Let the equiv. channel have exponentially decaying taps, a<1

• Then,

which is minimum at ω=π/T.• Then the output SNR is

• No ISI →

(fl has a zero at z=0 and a pole at z=a, performance degrades as |a| → 1)


MMSE - Performance• BER Analysis: No straightforward way. • Unlike ZF, residual ISI remains at the output of the MMSE equalizer

and this ISI cannot be modeled as AWG noise.

• Consider PAM signalling with levels 2n-M-1, n=1,2,...,M

where the WMF output/equalizer input is

and the convolution of the equalizer and the equivalent channel IRs is

• Obviously, the variance of noise is

equalizer has2K+1 taps!


MMSE - Performance• The ISI terms are

• For a fixed sequence of information symbols xJ={x[k]}, .• Then, the probability of error for this sequence is

• Average probability is found by averaging over all

• is dominated by the sequence yielding highest which occurs when x[n]= ±(M-1) and the signs of x[n]’s match the corresponding {qn}.


MMSE - Performance

• Then, following

and

• And, the upper bound for PM is found to be


0 0.5 1-20

-10

0

10

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

(a)

0 0.5 1-60

-40

-20

0

20


Mag

nitu

de (

dB)

(b)

0 0.5 1-100

-50

0

50


Mag

nitu

de (

dB)

(c)-2 0 2

-1

0

1

10

Real Part

Imag

inar

y P

art

(a)

-1 0 1-1

0

1

2

Real Part

Imag

inar

y P

art

(b)

-1 0 1-1

0

1

4

Real Part

Imag

inar

y P

art

(c)


MMSE – Finite Length Case

• The MMSE equalizer of length L is

• Then, applying the filter to the WMF output, the equalizer output is

• Express the signal at the WMF output as

• Then, the MMSE equalizer output becomes

Toeplitz Matrix


Cost Function

• ZF equalizer aims at

• MMSE equalizer aims at minimizing

Filter, wEffectiveChannel, f + +

Delay, δ

x[n] t[n]

x[n-δ]

ε[n]z[n]-

η[n]


Cost Function

• Expanding the cost function

• Using the property that data and noise are uncorrelated E{xη*}=0

• This is a quadratic function of w, take derivative wrt. w and equate to 0


Optimum Equalizer• Optimum equalizer coefficients are:

• Substituting back to the MSE term

where we used the matrix inversion lemma in the second line

• Jmin still depends on the delay parameter δ.


MMSE Equalizer - Example

• SNR=20dB.

unit norm


MMSE Equalizer - Example• Signal at the equalizer output:

• Signal power:

• Noise power:

• Interference power:

• SNR at the equalizer output:

• SINR at the equalizer output:

• ZF equalizer – SNR:– SINR: no interference ⇒ same as SNR (6.59 dB)

•


Principle of Orthogonality

• Principle of orthogonality:

•

• Using and


Principle of Orthogonality

• Then, the principle of orthogonality becomes:

• Corollary:

i.e.

• In words, when the equalizer taps are optimum in the MMSE sense, the error sequence, ε [n], is orthogonal to the current filter output z[n] and to the input sequence generating that output t[n].


Canonical Form of the Error-Performance Surface

• The cost function in matrix form

• Next, express J(w) as a perfect square in w

• Then, by substituting

• In other words,



• Observations:– J(w) is quadratic in w,– Minimum is attained at w=wo,– Jmin is bounded below, and is always a positive quantity,– Jmin>0 →



• Transformations may significantly simplify the analysis,• Use Eigendecomposition for R

• Then

• Let

• Substituting back into J

• The transformed vector v is called as the principal axes of the surface.

a vector

Canonical form



w1

w2

woJ(wo)=Jmin

J(w)=c curve

v1(λ1)

v2(λ2)

Jmin

J(v)=c curve

Q

Transformation