2.6 Parametric FS-CIR Estimation
2.6.1 Projection Approximation Subspace Tracking
LetH[n] ∈ CKbe the vector of the subcarrier-related CTF coefficients associated with the channel model of Equation (1.14). As described in Section 1.7.1, the CIR associated with the CTF coefficient vectorH[n]
is constituted by a relatively low number of L ≪ K statistically-independent Rayleigh fading paths. The corresponding CIR components are related to the CTF coefficientsH[n,k]by means of Equation (1.7). The motivation for employing the so-called subspace technique [132] here is that usually we have L ≪ Kand thus it is more efficient to estimate a low number of CIR-related taps in the low-dimensional signal subspace than estimating all theKFD-CTF coefficients.
Letλl andulbe the eigenvalues and the corresponding eigenvectors of the CTF’s covariance matrixCH,
which is defined as follows
CH =E n
H[n]HH[n] . (2.42)
Then, we haveCH =UΣUH, whereΣ=diag(λl)andU = [u1· · ·uK].
The eigenvalues aligned in a descending order may be expressed as
λ1 ≥ · · · ≥λL >λL+1 =· · ·= λK =σw2, (2.43)
where the first L dominant eigenvaluesλ1,· · · ,λL in conjunction with the Lcorresponding eigenvectors
u1,· · · ,uL may be termed as the signal eigenvalues and eigenvectors, respectively [117]. The remaining
eigenvalues λL+1,· · · ,λK and eigenvectors uL+1,· · · ,uK are termed the noise eigenvalues and eigenvec-
tors. The resultant sets of signal and noise eigenvectors, which are column vectors, span the mutually orthogonal signal and noise subspacesUSandUN, such that we have
US= [u1,· · · ,uL] and UN = [uL+1,· · · ,uK]. (2.44)
2.6.1. Projection Approximation Subspace Tracking 52
lows
ˆ α=UH
S[n]H˜[n]. (2.45)
Furthermore, the reduced-noise estimate of the CTF vectorH[n]may reconstructed using
ˆ
H[n] =US[n]αˆ[n]. (2.46)
For the sake of evaluating and tracking the potentially time-variant signal subspaceUS[n]we employ sub-
space tracking method developed by Yang [117]. More specifically, we consider the following real-valued scalar objective function having the matrix argument ofW ∈CK×L
J(W) =EkH−WWHHk2 =tr(CH)−2tr WHCHW +tr WHC HW·WHW (2.47)
As demonstrated by Yang in [117], the objective function J(W)of Equation (2.47) exhibits the following important properties
1. W is a stationary point of J(W) if and only if we haveW = ULQ, whereUL ∈ CK×Lcontains
anyLdistinct eigenvectors ofCHandQ∈CL×Lis an arbitrary unitary matrix. Furthermore, at each
stationary point, J(W)equals the sum of these particular eigenvalues, whose eigenvectors are not involved inUL[117, Theorem 1].
2. All stationary points of J(W) are local saddle points except, when UL contains the L dominant
eigenvectors ofCH. In this case,J(W)attains the global minimum [117, Theorem 2].
3. The global convergence ofW is guaranteed by using iterative minimization ofJ(W)and the columns of the resultant value ofW will span the signal subspace ofCH.
4. The use of an iterative algorithm to minimizeJ(W)will always converge to an orthonormal basis of the signal subspace ofCHwithout invoking any orthonormalization operations during the iterations.
5. The global minimum of J(W),W does not necessarily contain the signal eigenvectors, but an arbi- trary orthogonal basis of the signal subspace ofCH as indicated by the unitary matrixQintroduced
in Property 1. In other words, we haveW =argminJ(W)if and only ifW =USQ, whereQis an
arbitrary unitary matrix.
6. For the simple scalar case of L = 1, the solution minimizing J(W)is given by the most dominant normalized eigenvector ofCH.
2.6.1. Projection Approximation Subspace Tracking 53
Subsequently, Yang [117] proposes an iterative RLS algorithm for tracking of the signal subspace of the channel’s covariance matrixCH. Specifically, upon replacing the expectation value in Equation (2.47) by
the exponentially weighted sum of the RLS algorithm, we arrive at the following new objective function
J(W[n]) = n
∑
i=1 ηn−ikH[i]−W[n]WH[n]H[i]k2 =tr(CH)−2tr WH[n]CH[n]W[n] +tr WH[n]CH[n]W[n]·WH[n]W[n], (2.48)whereη∈(0, 1)is the so-called forgetting factor, which accounts for possible deviations of the actual chan- nel statistics encountered from the WSS assumption. Observe that the sole difference between the objective functions of Equations (2.47) and (2.48) is the introduction of the time-variant exponentially weighted sam- ple covariance matrix [117], which may be expressed as
CH[n] = n
∑
m=1 ηn−mH[m]HH[m] =ηC H[n−1] +H[n]HH[n] (2.49)instead of the time-invariant matrixCH =E
H HH of Equation (2.42).
The Projection Approximation Subspace Tracking (PAST) algorithm may be derived by approximating the expressionWH[n]H[m]in Equation (2.48), which may be interpreted as a projection of the vectorH[m] onto the column space of the matrixW[n], by the readily available a posteriori vectorα[m] =WH[m]H[m]. The resultant modified cost function may be formulated as
J′(W[n]) =
n
∑
m=1
ηn−mkH[m]−W[n]α[m]k2. (2.50) As is argued in [117], for stationary or slowly varying signals, the aforementioned projection approximation, hence the name PAST, does not substantially change the error surface associated with the corresponding cost function of Equation (2.50) and therefore does not significantly affect the convergence properties of the derived algorithm.
Similarly to other RLS estimation schemes [64, 101], the cost function J′(W[n])is minimized if
W = CHα[n]C−αα1[n], (2.51) where we have CHα[n] = n
∑
i=1 ηn−iH[i]αH[i] =ηCHα[n−1] +H[n]αH[n] (2.52) and Cαα[n] = n∑
i=1 ηn−iα[i]αH[i] =ηCαα[n−1] +α[n]αH[n]. (2.53)2.6.2. Deflation PAST 54