S-38.211 Signal Processing in Communications Stefan Werner

S-38.211 Signal Processing in Communications Stefan Werner
S-38.211 Signal Processing in Communications
Stefan Werner
Solution exercise 4, October 29, 1997
phone: 451 2437 room: SG224
email: [email protected]
Solution
a) The optimum coefficient vector is given by (LM 11.16)
−1
copt
− 0.5  0.5 
 1 0.5  0.5 
1  1
=Φ a=
=



2 
1  0.25
0.5 1  0.25 1 − 0.5 − 0.5
1  0.5 − 0.5 ⋅ 0.25  0.5
=
=
0.75 − 0.5 ⋅ 0.5 + 0.25  0 
−1
b) The minimum mean-square error (MMSE) is given by (LM 11.17)
0.5
ξmin = E a k2 − a T Φ −1a = σ a2 − a T c opt = σ a2 − 0.5 0.25   = σ a2 − 0.25
0
[ ]
[
]
c) The MSEG algorithm is given by (LM 11-27)
c j +1 = c j −
[ ]= c
β
∇ c E ek
2 j
2
j
 0.5 
 1 0.5
+ β ( a − Φc j ) = c j + β 
− β

c j
0.25
0.5 1 
d) The maximum step size that can be used is given by (LM 11-39) 0 < β <
2
λ max
where λmax is the
maximum eigenvalue to the autocorrelation matrix Φ .
 λ 0   1 0.5
 λ − 1 − 0.5
2
2
 = det 
det( λI − Φ) = 0 ⇔ det 
−


 = ( λ − 1) − 0.5 = 0
0
0
.
5
1
0
.
5
1
λ
−
λ
−


 


. , λ 2 = 0.5
⇒ λ = 1 ± 0.5 ⇒ λ1 = 15
We get the maximum step size as
2
4
βmax =
= .
λmax 3
Note that in order to be sure that the MSEG algorithm converges we choose a smaller step size than
the above!
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