Estimation of Multipath Delays

Consider a group of L_gsuccessive OFDM symbols and let them be indexed j = 0, . . . , L_g− 1. Assume that, within the time span of these L_g symbols (i.e., L_gT ), the complex multipath gains may vary due to fading, but the path delays remain the same. This assumption is appropriate because the path delays usually change much more slowly than

the path gains [39]. In our earlier notations, W stays constant over this period but h may change. For convenience, we attach an index to h and let h^j denote the channel response in the jth OFDM symbol period in the group. Likewise, we also use superscript j to index other quantities that may change with symbols, such as ˆg^j and S^j.

Let there be Q candidate delay values between 0 and τ_maxfrom which we will identify L for the delay subspace. One reasonable choice of these Q values is τ_maxk/Q, k = 0, . . . , Q − 1. We can define an N × Q dictionary matrix as V = [v₀, . . . , v_Q−1] where its kth column is given by

v_k= [1, e^{−j2πτmaxkTQ}, . . . , e−j2π(N −1)τmaxkT

Q]^T. (5.7)

Define V^j , S^jV .

Estimation Based on the MUSIC Algorithm

The MUltiple SIgnal Classification algorithm (MUSIC) [36] algorithm has been widely used in array signal processing for direction of signal arrival (DOA) estimation. With the assumption of uncorrelated sources, MUSIC algorithm generally is capable of high resolution identification. We mentioned that the MUSIC has been proposed for use in multipath delay estimation for OFDM transmission, with the assumption that the pilot locations be fixed and equal-spaced [32]. Below we outline the algorithm without giving all the details. It is written in a form applicable to the case with fixed but not necessarily equal-spaced pilots.

The fundamental idea of the MUSIC technique is to first find the null (noise) subspace based on the received signal and then project all candidate basis vectors of the delay subspace (i.e., columns of V^j or V ) into the null subspace. Since the actual basis vectors of the delay subspace (which correspond to signal) do not lie in the null subspace, the reciprocals of the projections should show peak at these basis vectors. From this we can identify the delay subspace. Procedure-wise, the steps are as follows:

1. For each OFDM symbol group, collect the Lg estimated channel frequency response vectors ˆg^j for pilot locations. Solve for the projection matrix P_N of the noise subspace with rank D − L.

2. Project all the columns in V with PN. Find the L columns with the smallest projection magnitudes. These L columns define the desired delays.

3. Follow the procedure in Sec. 5.1.2 to complete the channel estimation.

Note in the second step we omit the index j for V because V^j is identical for all j.

Indeed, having fixed pilot locations is a requirement of the MUSIC technique. Besides the limitation of fixed pilot locations, a property of the MUSIC technique is that, if some

path coefficients do not change significantly over the L_g OFDM symbols, then there may be a rank-deficiency problem. The result is that these paths may not be identified and resolved properly. This property appears quite undesirable, because it seems to imply the unpalatable conclusion that, in order to achieve good multipath delay estimation, we should make the OFDM symbol period a significant fraction of the channel coherence time.

In the area of direction-of-arrivals estimation, this effect has been known as the problem of correlated signal sources, and it may heavily degrade the estimation performance even in high SNR [38]. A technique called spatial smoothing [32], [38] can solve the problem, but the remedy itself also requires equal-spaced pilots. Moreover, it would divide the pilots into several groups, which is an unaffordable solution when the pilots are very few.

Estimation Based on Orthogonal Matching Pursuit Employing One Single OFDM Symbol

In preparation for the description of the proposed GMP technique, we describe how con-ventional orthogonal matching pursuit (OMP) can be applied to multipath delay estima-tion with a single OFDM symbol [45].

Ideally, to choose L delays out of Q candidate values, we should try all ^Q_L
possible combinations. For each combination, ˆg may be projected into the corresponding delay subspace. We then choose the combination with the largest projection magnitude as the estimation result. But the above exhaustive search approach is obviously impractical even with a moderate number of candidate delays Q. One suboptimal but much more efficient approach is the OMP technique [1], which employs a kind of greedy search method to determine the chosen candidates in a sequential fashion.

In applying OMP to multipath delay estimation for OFDM, we determine one path delay at a time. At each iteration, say iteration p, let U_p be the matrix containing the columns from V that define the (partial) delay subspace found so far. We project ˆg to the subspace and find the residual. Then from all columns of V that have not entered or covered by U_p, we choose the one that has the maximum inner product with the residual and add it to U_p. At this, we go to the next iteration until the required number of paths is found. The concept of above-mentioned orthogonal matching pursuit is shown in Fig. 5.1.

Mathematically, let dp be the index of the column from V that is chosen in iteration p. Let k_p denote this vector, that is, k_p = v_dp. Let P_Up denote the matrix that, when premultiplied to a vector, projects the vector onto the range space of U_p. And let r_p be the residual after the pth iteration. Then the OMP algorithm, in iteration p, works as follows:

d_p = arg max

i |r^H_p−1v_i|, 0 ≤ i ≤ Q − 1, (5.8) k_p = v_dp, U_p = [U_p−1, k_p], (5.9)

r_p = (I − P_Up)ˆg, (5.10)

Fig. 5.1: In the pth iteration,projection and selection based on projection residual where r₀ , ˆg, U₀ = ∅, and UL−1 gives the desired estimate of W (See Sec.5.1.2).

The Group Matching Pursuit Algorithm for Multipath Delay Estimation Now we turn to the proposed GMP algorithm for estimation of multipath delays based on observation of one OFDM symbol group. While the multipath delays and the delay subspace characterized by W are (assumed to be) fixed within one OFDM symbol group, the changing pilot locations result in different W^j and different V^j. One approach, based on OMP, to address this condition is to perform Lg OMP operations, one for each OFDM symbol, and combine the results. But how the results can be combined poses a problem, because the estimated delays may be different for different OFDM symbols.

The idea of GMP is to make use of the whole set of ˆg^j, j = 0, . . . , L_g− 1, and obtain a jointly optimal delay estimation in some sense. This results in the following steps for iteration p of the algorithm:

d_p = arg max

i Lg−1

X

j=0

|(r^j_p−1)^Hv^j_i|, 0 ≤ i ≤ Q − 1, (5.11) k^j_p = v^j_dp, U^j_p = [U^j_p−1, k^j_p], 0 ≤ j ≤ Lg− 1, (5.12) r^j_p = (I − P_U^j

p)ˆg, 0 ≤ j ≤ L_g− 1. (5.13)

(See the beginning paragraph of Sec. 5.1.3 for the meaning of the superscript j.) As in OMP, U^j_p, j = 0, . . . , L_g− 1, give the the desired estimates of W^j, j = 0, . . . , L_g− 1, that define the delay subspace and can be used as described in Sec. 5.1.2 to obtain a channel estimate for each OFDM symbol in the group. Note that the channel estimates may vary

for different symbols, because the channel is subject to fading, but the delay subspace is the same.

On the Number of Path Delays to Estimate

Throughout the chapter, we have assumed that the number of path delays to be estimated is known. This information can be obtained through other means of channel analysis [46]

or empirical data. Even if the number of estimated delays is different from the actual number, in many cases it should not be critical. For example, if we have estimated less path delays than the actual but have captured the most significant paths, then the loss may be acceptable. Conversely, if we have estimated several more path delays than the actual, the resulting enhancement in noise may have little implication as long as its correlation with the actual delay subspace is small [45]. In any case, the number of multipath delays that can be estimated with the proposed technique is upper bounded by D, for otherwise we would have an under-determined set of equations for ˆh (see, e.g., (5.5)).