mmse
TRANSCRIPT
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
Spring'09 ELE 739 - Channel Equalization 2
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
Spring'09 ELE 739 - Channel Equalization 3
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.
Spring'09 ELE 739 - Channel Equalization 4
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]
Spring'09 ELE 739 - Channel Equalization 5
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
Spring'09 ELE 739 - Channel Equalization 6
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
Spring'09 ELE 739 - Channel Equalization 7
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.
Spring'09 ELE 739 - Channel Equalization 8
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
Spring'09 ELE 739 - Channel Equalization 9
MMSE - Performance
• Then
• No ISI → X(ejωT)=1 →
• Note that,
• Furthermore, output SNR is
• No ISI → → Same as ZF.
Spring'09 ELE 739 - Channel Equalization 10
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)
Spring'09 ELE 739 - Channel Equalization 11
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)
Spring'09 ELE 739 - Channel Equalization 12
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!
Spring'09 ELE 739 - Channel Equalization 13
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}.
Spring'09 ELE 739 - Channel Equalization 14
MMSE - Performance
• Then, following
and
• And, the upper bound for PM is found to be
Spring'09 ELE 739 - Channel Equalization 15
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
Normalized Frequency (×π rad/sample)
Mag
nitu
de (
dB)
(b)
0 0.5 1-100
-50
0
50
Normalized Frequency (×π rad/sample)
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)
Spring'09 ELE 739 - Channel Equalization 16
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
Spring'09 ELE 739 - Channel Equalization 17
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]
Spring'09 ELE 739 - Channel Equalization 18
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
Spring'09 ELE 739 - Channel Equalization 19
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.
Spring'09 ELE 739 - Channel Equalization 20
MMSE Equalizer - Example
• SNR=20dB.
unit norm
Spring'09 ELE 739 - Channel Equalization 21
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)
•
Spring'09 ELE 739 - Channel Equalization 22
Principle of Orthogonality
• Principle of orthogonality:
•
• Using and
Spring'09 ELE 739 - Channel Equalization 23
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].
Spring'09 ELE 739 - Channel Equalization 24
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,
Spring'09 ELE 739 - Channel Equalization 25
Canonical Form of the Error-Performance Surface
• 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 →
Spring'09 ELE 739 - Channel Equalization 26
Canonical Form of the Error-Performance Surface
• 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
Spring'09 ELE 739 - Channel Equalization 27
Canonical Form of the Error-Performance Surface
w1
w2
woJ(wo)=Jmin
J(w)=c curve
v1(λ1)
v2(λ2)
Jmin
J(v)=c curve
Q
Transformation