Structure and Design of MMSE Channel Equalizers

Cenk Toker

www.ee.hacettepe.edu.tr/~toker/equalizers.pdf

cenk.toker@ieee.org

H

yk=∑n=0

N H−1

hn xk−nvk

+xk yk

vk

transmitted data ISI channel

received signal-no noise

noisy received signal

Intersymbol Interference (ISI)

MMSE FIR Equalizers

h +xk y[k]

v[k]

+w

δ - m b

h +y[k]

v[k]

δ - m

+w

ε[k]

ε[k]

h +y[k]

v[k]

δ - m

+wε[k]

b

● Equalizer:b = 1

● Partial response equalizer:b: fixed (for example, b=1+z-1)● Channel shortening equalizer:b: design parameter

-

-

-

z[k]

z[k]

^

xk

xk

h +y[k]

v[k]

+w

δ - m

Qε[k]

b

MMSE FIR Decision Feedback Equalizer (DFE)

+

● Assumption: Past decisions are correct

h +y[k]

v[k]

δ - m

+w

b

+ε[k]-

-

-

xk

xk

δ - m

MMSE FIR Equalizers

h +y[k]

v[k]

+w

δ - m b

ε[k]-

z[k]

z[k]

^

z k=∑n=0

nw−1

wn* yk−n=[w0

* w1* ⋯ wnw−1

* ][ yk

yk−1

⋮y k−nw−1

]=w H yk

yk=∑m=0

nw−1

wm* ∑

n=0

nh−1

hn xk−n−mvk−m

zk=[w0* w1

* ⋯ wnw−1* ][h0 h1 ⋯ hnh−1 0 ⋯ 0

0 h0 h1 ⋯ hnh−1 ⋯ 00 ⋱ ⋱ ⋱ ⋯ ⋱ 00 ⋯ 0 h0 h1 ⋯ hnh−1

][ xk

xk−1

⋯xk−nhnw−2

][ vk

vk−1

⋮vk−nw−1

]zk=w H H x kw H vk

z k=∑n=0

nb−1

bn−m* xk−n

=[0 ⋯ 0 b0* b1

* ⋯ bnb−1* 0 ⋯ 0][ xk

xk−1

⋮xk−nwnh−2

]=[0m×1

b0 ]

H

x k

z k=bH xk

xk

h +y[k]

v[k]

δ - m

+wε[k]

b

-

MMSE FIR Equalizers

z[k]

z[k]

^

k=z k− zk● Error:

● Mean Square Error (MSE): J=E {k k*}

=[ bH−w H H ] x k−w H vk

=[ bH−w H H ] Rxx [ b−H H w ]w H Rvv w

R xx=E {x k xkH }

=E {[ xk

xk−1

⋯xk−nhnw−2

][ xk* xk−1

* ⋯ xk−nhnw−2*]}=[ r xx [0] r xx [1] ⋯ r xx [nhnw−1]r xx

* [−1] r xx [0] ⋯ r xx [nhnw−2]⋮ ⋱ ⋱ ⋮

r xx* [−nhnw−1] r xx

* [−nhnw−2] ⋯ r xx [0 ]]

xk

h +y[k]

v[k]

δ - m

+wε[k]

b

-

MMSE FIR Equalizers

z[k]

z[k]

^

● J: quadratic function; optimum w is found by differentiating J wrt w and equating to zero[4] (or by orthogonality principle, E{ε [k]yH[k]}=0[1,2]).

w H=bH Rxx H H HRxx H HRvv−1

=bH R xx−1H H Rvv

−1 H −1H H Rvv

−1

● MMSE FIR equalizer, b=1: the optimum receiver is

w H=emH R xx

−1H H Rvv−1 H −1

H H Rvv−1

whiteningmatched filter

(*)

(*): Matrix Inversion Lemma: Rxx−1H H Rvv

−1 H −1=R xx−Rxx H H HR xx H HRvv

−1HR xx

xk

em=[0⋮010⋮0] m

h +y[k]

v[k]

δ - m

+wε[k]

b

-

MMSE FIR Partial Response Equalizers

z[k]

z[k]

^

● MMSE FIR fixed partial response equalizer, b fixed: ● the optimum receiver is

w H=bH R xx−1H H Rvv

−1 H −1H H Rvv

−1whitening

matched filter

xk

h +y[k]

v[k]

δ - m

+wε[k]

b

-

MMSE FIR Channel Shortening Equalizers

z[k]

z[k]

^

● Substitute w into J: J=bH Rxx−1H H Rvv

−1 H −1 b

bH0 0 b

0

0

R=

=bH Rb● Impose a constraint on b in order to avoid the trivial case b=0, J=0

● Orthogonality constraint bHb=1.

● MMSE FIR channel shortening equalizer, b fixed length variable filter.

xk

h +y[k]

v[k]

δ - m

+wε[k]

b

-

MMSE FIR Channel Shortening Equalizers

z[k]

z[k]

^

● Problem becomes : J=bH Rb

● Constrained optimization: Solution for b is the eigenvector corresponding tothe smallest eigenvalue, λmin, of R.

bH b=1subject to

minimize

MMSE=λmin

=bH U U H b

=bH [u1 u2 ⋯ unb][1 0 ⋯ 00 2 ⋯ 0⋮ ⋱ ⋱ ⋮0 0 ⋯ nb

][u1H

u2H

⋮unb

H]b

xk

MMSE FIR Decision Feedback Equalizer (DFE)

h +y[k]

v[k]

δ - m

+w

b-

+ε[k]-z[k]^

z[k]δ - m

y [k ]=∑n=0

nh−1

h [n] x [k−n]v [k ]zk=∑n=0

nw−1

wn yk−n−∑n=1

nb−1

bn xk−n−m

= ∑n=0

nhnw−1

cn xk−n−∑n=1

nb−1

bn xk−n−m∑n=0

nw−1

wn vk−n

= cm xk−minformation

bearing cursor

∑n=0

m−1

cn xk−nresidual precursor ISI

∑n=1

nb−1

cnm−d n xk−n−mmodeled postcursor ISI

∑n=mnb1

nhnw−1

cn xk−nresidual postcursor ISI

∑n=0

nw−1

wn vk−nfiltered noise

c [n]=w [n]∗h [n]

xk

MMSE FIR Decision Feedback Equalizer (DFE)

h +y[k]

v[k]

δ - m

+w

b-

+ε[k]-z[k]^

z[k]δ - m

● Goals: ● Feedforward filter:

● shape cn=hn*wn so that● small residual ISI,● keep noise gain as small as possible,

● Feedback filter:● cancel the remaining ISI by matching dn=cn+m, n=1,...,nb-1

xk

zk= cm xk−minformation

bearing cursor

∑n=0

m−1

cn xk−nresidual precursor ISI

∑n=1

nb−1

cnm−bn xk−n−mmodeled postcursor ISI

∑n=mnb1

nhnw−1

cn xk−nresidual postcursor ISI

∑n=0

nw−1

wn vk−nfiltered noise

cm≈1cn≈0, n≠m

∑k∣ f k∣

2

cn

nm0

cm

bw

MMSE FIR Decision Feedback Equalizer (DFE)

h +y[k]

v[k]

δ - m

+w

b-

+ε[k]-z[k]^

z[k]

k=xk−m−{w H Hx kvk −bH x k}

J=E {[k ]*[k ]}

δ - m

xk

xk , c

mm+1 xk , b

~

E {xk−m xk}=0, E {xk−m vk}=0 , E {x k vkH }=0

= x2−[ x

2 emHbH Rxx ]H H w−w H H [ x

2 emRxx b ]wH [HRxx H HRvv ]wbH Rxx b

x k=Mx k

M=[0nb×m I nb×nb0nb×nhnw−nb−m]

● Assumptions:

MMSE FIR Decision Feedback Equalizer (DFE)

h +y[k]

v[k]

δ - m

+w

b-

+ε[k]-z[k]^

z[k]δ - m

xk

J= x2−[ x

2 emHbH Rxx ]H H w−w H H [ x

2 emR xx b ]w H [HR xx H HRvv ]wbH R xx b

● Optimum feedback filter, : ∇ b J=0

w H= x2 em

HbH MRxxH H HR xx H HRvv−1

● Optimum feedforward filter, : J b ,∇w J=0

bH=w H HM H

w H= x2 em

H H H I−M H Rxx M H HRvv−1

● Equivalent optimum feedforward filter, : ∇w J=0

Conclusions

● MMSE linear equalization is a well-studied field for combatting ISI channel.● All MMSE equalizers share common feedforward filter structure:

● All filters first equalize the channel with , then reshape the IR with either or . This decreases the ISI due to the extra degree of freedom provided by these IRs.●There is an inherent whitening matched filter front end in an equalizer designed for the MMSE criterion:

● Design of the feedback filter can be application specific.

w H=emH Rxx H H HRxx H HRvv

−1

w H=bH Rxx H H HRxx H HRvv−1

w H= x2 em

HbH MR xxH H HRxx H HRvv−1

● MMSE equalizer:● Partial response and channel shortening equalizers:● Decision feedback equalizer:

w H=emH bH R xx

−1H H Rvv−1 H −1

H H Rvv−1

whiteningmatched filter

Rxx H H HRxx H HRvv−1

bH x2 em

HbH MRxx

References

[1]: Simon Haykin, Adaptive Filter Theory, Prentice Hall, 2001,

[2]: Al-Dhahir, “FIR Channel-Shortening Equalizers for MIMO ISI Channels”, IEEE Trans. Commun., vol. 49, Feb. 2001, pp. 213-8,

[3]: R. A. Casas, et al, DFE Tutorial, http://www.ece.osu.edu/~schniter/postscript/dfetutorial.pdf,

[4]: M. Brookes, Matrix Reference Manual, http://www.ee.ic.ac.uk/ hp/staff/dmb/matrix/intro.html.