Structure and Design of MMSE Channel Equalizerstoker/equalizers.pdf · Structure and Design of MMSE...
Transcript of Structure and Design of MMSE Channel Equalizerstoker/equalizers.pdf · Structure and Design of MMSE...
Structure and Design of MMSE Channel Equalizers
Cenk Toker
www.ee.hacettepe.edu.tr/~toker/equalizers.pdf
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.