EE6110 Adaptive Signal Processing Problem Set 3 Solutions

EE6110 Adaptive Signal Processing Problem Set 3 Solutions

1.

Xˆ =K0Y, whereK0 is any solution to K0RY =RXY

We want to minimize E[( ˜X^′)^HWX˜^′] for some W ≥0 ( ˜X^′ =X−Xˆ^′, Xˆ^′ =KY)

E[( ˜X^′)^HWX˜^′] =E[(X−KY)^HW(X−KY)]

=E[(X−K₀Y+K₀Y−KY)^HW(X−K₀Y+K₀Y−KY)]

=E[(X−K0Y)^HW(X−K0Y)] +E[(X−K0Y)^HW(K0Y−KY)]

+E[(K0Y−KY)^HW(X−K₀Y)] +E[(K0Y−KY)^HW(K0Y−KY)]→eqn

We know that K₀Y is the linear MMSE estimate ofX from Y

Therefore, we know (X−K₀Y) is perpendicular to any linear function ofY

=⇒ second term in the eqn

E[(X−K₀Y)^HW(K0Y−KY)] =E[(X−K₀Y)^H(W(K0−K))Y] = 0 Also,E[(K0Y−KY)^HW(X−K₀Y)] = 0

=⇒ E[( ˜X^′)^HWX˜^′] =E[(X−K₀Y)^HW(X−K₀Y)] +E[(K0Y−KY)^HW(K0Y−KY)]

Both the terms in the right hand side of the above equation are ≥0, (since W ≥0 =⇒ a^HWa≥0,∀a)

=⇒ E[( ˜X^′)^HWX˜^′]≥E[(X−K₀Y)^HW(X−K₀Y)]

This lower bound is achieved by K =K₀ i.e the linear MMSE estimate K₀Y also minimizes E[( ˜X^′)^HWX˜^′] for anyW ≥0

2.

J(x) = (x−c)^HA(x−c)

Since A is Hermitian nonnegative-definite, J(x)≥0 1

=⇒ minimum possible value of J(x) is 0.

Forx=c+d where Ad=0,

J(x) = (c+d−c)^HA(c+d−c) = d^HAd=d^H0= 0

ThusJ(x) = 0 is acheived at x=c+d for any d satisfying Ad=0

3.

Y1 =X1+N1, X1 is zero mean, variance 1

Y2 =X2+N2 , N1, N2 i.i.d zero mean ,variance σ², independent of X1

X₂ =ρX₁+p

1−ρ²Z,

Z is zero mean, variance 1, independent of X₁, X₂, N₁, N₂ E[X₂²] =ρ² + 1−ρ² = 1

(a) EstimateX1 fromY1

Xˆ1 =k0Y1, k0(1 +σ²) = 1

=⇒ Xˆ1 = _(1+σ^Y12)

(b) EstimateX2 fromY1

Xˆ21 =ρXˆ1 = _(1+σ^ρY12)

(c) EstimateY2 fromY1

Yˆ2 =kY1, k(1 +σ²) =RY1Y2 =E[X1X2] =ρ Yˆ2 = ˆX2+ ˆN2 =ρXˆ1 = _(1+σ^ρY12)

=⇒ Yˆ₂ = _(1+σ^ρY12)

(d) LetE₂ =Y₂−Yˆ₂ =Y₂− ^ρY1

(1+σ²)

EstimateX₂ fromE₂

Xˆ_2e=keE₂, keRE₂ =RE₂X₂

2

R_E2X2 =E[(Y2− _(1+σ^ρY1₂₎)X2] = 1− _(1+σ^ρ2₂₎

RE2 =E[(Y2−Yˆ2)(Y2−Yˆ2)]

=E[(Y2)(Y2−Yˆ₂)] →(1)

=E[Y₂²]−E[Y2Yˆ2]

= (1 +σ²)− ρ

(1 +σ²)E[Y2Y₁]

= 1 +σ²− ρ

(1 +σ²)E[X2X1]

= 1 +σ²− ρ² (1 +σ²)

(1) comes from the fact that (Y2−Yˆ₂)⊥AY₁ =⇒ (Y2−Yˆ₂)⊥Yˆ₂

So, ke = ¹⁻

ρ2 (1+σ2)

1+σ²− ^ρ

2 (1+σ2)

=⇒ Xˆ_2e= ¹⁻

ρ2 (1+σ2)

1+σ²− ^ρ

2 (1+σ2)

(Y₂− ^ρY1

(1+σ²)) (e)

Xˆ₂ = ˆX₂₁+ ˆX_2e = ρY₁

(1 +σ²) + 1−_(1+σ^ρ2₂₎ 1 +σ²− ^ρ2

(1+σ²)

(Y2− ρY₁ (1 +σ²))

= σ²

1 +σ²− ^ρ2

(1+σ²)

! ρY1

(1 +σ²)

+ 1− ^ρ2

(1+σ²)

1 +σ²− ^ρ2

(1+σ²)

! Y2

=

ρσ² (1 +σ²)²−ρ²

Y₁+

1 +σ²−ρ² (1 +σ²)²−ρ²

Y₂

This can also be done directly by using ˆX2 =h k₀i

"

Y₁ Y₂

#

h k0

i

=R_X2YR⁻¹_Y

=h ρ 1i

"

(1 +σ²) ρ ρ (1 +σ²)

#−1

=h ρ 1i

"

(1 +σ²) −ρ

−ρ (1 +σ²)

#

1 (1 +σ²)²−ρ²

=⇒ Xˆ₂ =

ρσ² (1+σ²)²−ρ²

Y₁+

1+σ²−ρ² (1+σ²)²−ρ²

Y₂

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