Proofs in Section 2.5.2

= W ΣΣ^†W^T, (2.278)

where the second equality follows from the facts that W^TW = I_N and Σ^TΣ^†T = V^TV = V V^T = I_K. Since the N × N diagonal matrix ΣΣ^†= diag (I_K, 0), it follows from (2.278) that HG is symmetric. Similarly GH is also symmetric. Next we verify condition (b).

HGH = 

W ΣV^TAV Σ^TW^T

W Σ^†TV^T|A|⁻²V Σ^†W^T 

W ΣV^TAV Σ^TW^T

= W ΣΣ^†ΣV^TAV Σ^TW^T

= W ΣV^TAV Σ^TW^T

= SAS^T = H, (2.279)

where in the second equality, the following facts are used: W^TW = IN, Σ^TΣ^†T = IK and V^TV = V V^T = I_K; the third equality follows from the fact that ΣΣ^†Σ = Σ. Condition (c) can be similarly verified, i.e., GHG = G. Therefore, we have

U Λ^†₀U^T = H^† = G = W Σ^†TV^T|A|⁻²V Σ^†W^T. (2.280) Now (2.107) follows immediately from (2.280) and the fact U^TU = U U^T = I_N. 2

2.8.3 Proofs in Section 2.5.2

Some Useful Lemmas

We first list some lemmas which will be used in proving the results in Section 2.5.2. A random matrix is said to be Gaussian distributed, if the joint distribution of all its elements

is Gaussian. First we have the following vector form of the central limit theorem.

Lemma 2.4 (Theorem 1.9.1B in [434]) Let {xi} be i.i.d. random vectors with mean µ and covariance matrix Σ. Then

Next we establish that the sample auto-correlation matrix ˆC_r given by (2.122) is asymptot-ically Gaussian distributed as the sample size M → ∞.

Lemma 2.5 Denote

M ∆Cr converges in probability towards a Gaussian matrix with mean 0 and an (N²× N²) covariance matrix whose elements are specified by

M · cov {[∆Cr]i,j, [∆Cr]m,n} by Lemma 2.4, it is asymptotically Gaussian, with an (N²× N²) covariance matrix whose elements are given by the covariance of the zero-mean random matrix

r[i]r[i]^T . To calculate this covariance, note that (for notational convenience, in what follows we drop the time index i.)

We have where the last equality follows from the fact that

[C_r]_i,j = Note that the last term of (2.284) is due to the non-normality of the received signal r[i].

If the signal had been Gaussian, the result would have been the first two terms of (2.284)

only (compare this result with Theorem 3.4.4 in [17]). Using a different modulation scheme (other than BPSK) will result in a different form for the last term in (2.284).

In what follows, we will make frequent use of the differential of a matrix function (cf.

[412], Chapter 14). Consider a function f :Rⁿ→ R^m. Recall that the differential of f at a point x₀ is a linear function L_f(·; x0) :Rⁿ → R^m, such that

∀ > 0, ∃δ > 0 : x − x0 < δ ⇒ f(x) − f(x0)− Lf(x− x0; x₀) < . (2.288) If the differential exists, it is given by L_f(x; x₀) = T (x₀)x, where T (x₀) =
^∂f

∂x |x⁼x0. Let y = f (x) and consider its differential at x₀. Denote ∆x = x
− x0 and ∆y = L
_f(∆x; x₀).

Hence for fixed x₀, ∆y is a function of ∆x; and for fixed x₀, if x is random, so is ∆y. We have the following lemma regarding the asymptotic distribution of a function of a sequence of asymptotically Gaussian vectors.

Lemma 2.6 (Theorem 3.3A in [434]) Suppose that x(M )∈ Rⁿ is asymptotically Gaussian, i.e.,

√M [x(M )− x0] → N (0, Cx), in distribution, as M → ∞.

Let f : Rⁿ → R^m be a function. Denote y(M ) = f [x(M )]. Suppose that f has a nonzero differential L_f(x; x₀) = T (x₀)x at x₀. Denote ∆x(M ) = x(M )
− x0, and ∆y(M ) = T (x₀)∆x(M ). Then

√M [y(M )− f(x0)] → N (0, Cy), in distribution, as M → ∞, (2.289)

where

C_y = T (x₀)C_xT (x₀)^T =
lim

M→∞T (x₀)E{∆x(M)∆x(M)} T (x0)^T (2.290)

= lim

M→∞E



∆y(M )∆y(M )^T



. (2.291)

To calculate Cy we can use either (2.290) or (2.291). When dealing with functions of matrices, however, it is usually easier to use (2.291). In what follows, we will make use of the following identities of matrix differentials.

C = f (X) = M X =
⇒ ∆C = M∆X, (2.292) C = f (X, Y ) = XY
=⇒ ∆C = X∆Y + ∆XY , (2.293) C = f (X) = X
⁻¹ =⇒ ∆C = −X⁻¹∆XX⁻¹. (2.294)

Finally, we have the following lemma regarding the differentials of the eigencomponents of a symmetric matrix. It is a generalization of Theorem 13.5.1 in [17]. Its proof can be found in [193].

Lemma 2.7 Let the N × N symmetric matrix C0 have an eigendecomposition C₀ = U0Λ0U^T₀, where the eigenvalues satisfy λ⁰₁ > λ⁰₂ > · · · > λ⁰K > λ⁰_K+1 = λ⁰_K+2 = · · · = λ⁰N. Let ∆C be a symmetric variation of C₀ and denote C = C
₀ + ∆C. Let T be a unitary transformation of C as

T (C) = U
^T₀CU₀. (2.295) Denote the eigendecomposition of T as

T = W ΛW^T. (2.296)

(Note that if C = C0, then W = IN and Λ = Λ0.) The differential of Λ at Λ0, and the differential of W at I_N, as a function of ∆T = U₀∆CU^T₀, are given respectively by

∆λ_k = [∆T ]_k,k, 1≤ k ≤ K, (2.297)

[∆W ]_i,k =





0, i = k

1 λ⁰_k−λ⁰i

[∆T ]_i,k, i= k , 1≤ i ≤ N, 1 ≤ k ≤ K. (2.298)

Proof of Theorem 2.1

DMI Blind Detector - Consider the function ˆC_r→ ˆw₁ = ˆC⁻¹_r s₁. The differential of ˆw₁ at C_r is given by

∆w₁ = −C⁻¹r ∆C_rC⁻¹_r s₁, (2.299) where ∆Cr

= ˆ
Cr− Cr. Then according to Lemma 2.6, √

M ( ˆw1− w1) is asymptotically Gaussian as M → ∞, with zero-mean and covariance matrix given by (2.291)³

C_w = M
· E

∆w₁∆w^T₁

= M· E

C⁻¹_r ∆C_rC⁻¹_r s₁s^T₁C⁻¹_r ∆C_rC⁻¹_r 

= M· C⁻¹r E

∆C_rw₁w^T₁∆C_r

C⁻¹_r . (2.300)

3We do not need the limit here, since the covariance matrix of (√

M∆C_r) is independent ofM.

Now, by Lemma 2.5, we have

Writing (2.301) in a matrix form, we have M · E The eigendecomposition of C_r is

C_r = U_sΛ_sU^T_s + σ²U_nU^T_n. (2.303) Substituting (2.302) and (2.303) into (2.300), we get

M · Cw = (w^T₁C_rw₁)C⁻¹_r + w₁w^T₁ − 2C⁻¹r SDS^TC⁻¹_r

where the last equality follows from the fact that U^T_nS = 0. 2 Subspace Blind Detector - We will prove the following more general proposition, which will be used in later proofs. The part of Theorem 2.1 for the subspace blind detector follows with v = s₁.

Proposition 2.6 Let w₁ = U_sΛ⁻¹_s U^T_sv be the weight vector of a detector, v ∈ range(S), and let ˆw₁ = ˆU_sΛˆ⁻¹_s Uˆ^T_sv be the weight vector of the corresponding estimated detector. Then

√M ( ˆw₁− w1) → N (0, Cw), in distribution, as M → ∞,

with

C_w =

w^T₁v₁ U_sΛ⁻¹_s U^T_s + w₁w^T₁ − 2UsΛ⁻¹_s U^T_sSDS^TU_sΛ⁻¹_s U^T_s + τ U_nU^T_n, (2.304) where

D = diag

 A⁴₁

s^T₁w₁ ², A⁴₂

s^T₂w₁ ²,· · · , A⁴K

s^T_Kw₁ ²



, (2.305)

and τ = σ
²v^TU_sΛ⁻¹_s 

Λ_s− σ²I_K ⁻²U^T_sv. (2.306) Proof: Consider the function ( ˆU_s, ˆΛ_s) → ˆw₁ = Uˆ_sΛˆ⁻¹_s Uˆ^T_sv. By Lemma 2.6,

√M ( ˆw₁− w1) is asymptotically Gaussian as M → ∞, with zero-mean and covariance matrix given by C_w = M
· E

∆w₁∆w^T₁

, where ∆w₁ is the differential of ˆw₁ at (U_s, Λ_s).

Denote U = [U_sU_n]. Define

T = U
^TCˆ_rU = U^T

Uˆ_sΛˆ_sUˆ^T_s + ˆU_nΛˆ_nUˆ^T_n 

U . (2.307)

Since T is a unitary transformation of ˆC_r, its eigenvalues are the same as those of ˆC_r. Hence its eigendecomposition can be written as

T = W_sΛˆ_sW^T_s + W_nΛˆ_nW^T_n, (2.308) where W = [W_s W_n]= U
^T[ ˆU_sUˆ_n]U are eigenvectors of T . From (2.307) and (2.308) we have

Uˆ_sΛˆ⁻¹_s Uˆ^T_s = U W_sΛˆ⁻¹_s W^T_sU^T. (2.309)

Thus we have where E_s is composed of the first K columns of I_N. Using Lemma 2.7, after some manipu-lations, we have

Using (2.312) and (2.315), we have

where (2.316) follows from the fact that U^Tv =

since it is assumed that v ∈ range(S); a similar relationship holds for U^Ts_α. Writing (2.316) in matrix form, we obtain

M · E

Finally by (2.310), M · E

∆w₁∆w^T₁

= M · UE zz^T

U^T. Substituting (2.318) into this

expansion, we obtain (2.304). 2

Proof of Corollary 2.1

First we compute the term given by (2.120). Using (2.303) and (2.128), and the fact that U^T_sU_n= 0, we have Next note that the linear MMSE detector can also be written in terms of R, as [511]

W = [w1 · · · wK] = C
⁻¹S = S

By (2.130), for the DMI blind detector, we have τ σ² = s^T₁w₁; and for the subspace blind

where we have used the fact that the decorrelating detector can be written as [540]

D = U_s

Λ_s− σ²I_K ⁻¹U^T_sS = SR⁻¹A⁻². (2.332) Finally substituting (2.327)-(2.331) into (2.119), we obtain (2.132). 2

SINR for Equicorrelated Signals

In this case, R is given by

R = S
^TS = ρ11^T + (1− ρ)IK, (2.333) where 1 is an all-1 K-vector. It is straightforward to verify the following eigen-structure of R,

= 1 Substituting (2.337)-(2.339) into (2.132)-(2.135), and by defining

α =

we obtain expression (2.143) for the average output SINR’s of the DMI blind detector and

the subspace blind detector. 2

In document 20471280 Wireless Communication Systems (Page 121-134)