These considerations make the random carrier allocation an interesting academic problem, but of little practical use.
often outperforms MUSIC both in accuracy [86, Chap. 4.7] and compu- tational cost [134] and is therefore the preferred choice.
A comprehensive description of ESPRIT can be found in [86]. Only the most relevant steps are recapitulated here.
Let x(k) = PH−1
h=0 bhej(ωhk+ϕh) +z(k) be a superposition of H sinu- soids with unknown phase, frequency, and amplitude as well as a white Gaussian noise processz(k). If x∈ C1×N is a vector representing the time-domain signal,Rxx∈CN×N is the auto-covariance matrix ofx(k), which can be decomposed into its eigenvalues and -vectors,
Rxx=UΛU∗ , Λ = diag(λ0, λ1, . . . , λN−1). (3.122)
If the eigenvalues are sorted by size (λi < λi+1, i = 0, . . . , N−1), the firstH columns ofU (i.e. the eigenvectors corresponding to the largest eigenvalues) span thesignal subspace ofx(k), and the remainingN−H eigenvectors span the noise subspace,
U= [S G] , S∈CN×H,G∈CN×(N−H) (3.123)
The noise subspace is of no interest for this estimator. The signal sub- space is used to create two new matrices,
S1= [IN−10]S (3.124)
S2= [0IN−1]S (3.125)
These are in turn used to create
Φ= (S∗2S2)−1S∗2S1. (3.126)
It can be shown thatΦ=C−1DC, whereCis a nonsingular matrix and D = diag(e−jω0,...,−jωN1). The eigenvalues of Φ can therefore be used to estimate the frequencies inx(k), ˆωk =−arg [λk].
Estimation of the auto-covariance matrix
ESPRIT requires the auto-covariance matrix Rxx as input, whereas a digital receiver only provides the time-domain signalx. The definition of the auto-covariance matrix is
Rxx=E[x∗x]. (3.127)
To estimate the auto-covariance matrix, Lindependent representations ofx are required. Because the noise component inxis white Gaussian noise (WGN), the maximum likelihood estimate for the auto-covariance matrix is [86]
Rˆxx= 1 M
L−1X
i=0
x∗ixi. (3.128)
Application to OFDM radar
As discussed, there will be two one-dimensional estimation problems to solve. For ESPRIT, this requires two auto-covariance matrices, for range and Doppler, respectively.
Given the radar processing matrixF, this is achieved quite simply. Since the column-wise oscillations correspond to the range of the individual targets, and every row is a representation of the same process, the auto- covariance matrix is estimated by
RˆFF,d= 1
NFF∗∈CN×N. (3.129)
In an analog fashion, the auto-covariance matrix for the row-wise os- cillations (corresponding to the Doppler, or relative velocity) are calcu- lated:
RˆFF,vrel = 1
MF∗F∈CM×M. (3.130) The algorithm to determine Doppler and range of targets from Fusing the ESPRIT algorithm is thus straightforward:
1. CalculateRˆFF,dfromFas shown in (3.129).
2. Using (3.122) through (3.126), calculate the matrixΦd.
3. For every one of the P (this value is discussed in Section 3.4.2) largest eigenvalues ofΦd, the range of the object is calculated by
dˆi=−argh λˆi
i· c0
2π·2U∆f, i= 0, . . . , P−1 (3.131) similarly to the waydis estimated from the periodogram in (3.32).
U is the optional sub-carrier spacing explained in Section 3.3.8.
4. Analogously, calculateRˆFF,vrel from (3.130).
5. Compute a new matrixΦvrel fromRˆFF,vrel.
6. The eigenvalues ˆλi ofΦvrel can then be used to calculate the rela- tive velocities of the objects,
ˆ
vrel,i=−argh λˆi
i· c0
2π·2U∆f. (3.132) Dimensionality reduction of the auto-covariance matrices
The computational bottleneck in the ESPRIT is the eigenvalue decom- position of the auto-covariance matrices. Decreasing the dimension of the auto-covariance matrices will also decrease the computational com- plexity [3]. For the range estimation, this is easily achieved by splitting Fvertically into K sub-matrices,
F=
F0
F1
... FK−1
, (3.133)
where each sub-matrix has the dimension Fi ∈ CNK×M. The auto- covariance for the range estimation is then the average of the auto- covariance matrices of the sub-matrices,
RˆFF,d= 1 N
K−1X
i=0
F∗iFi∈CNK×NK. (3.134)
Analogously, the auto-covariance matrix for the Doppler estimation is obtained by splitting F horizontally into sub-matrices, and averaging
their auto-covariance matrices RˆFF,d= 1
N
K−1X
i=0
F∗iFi∈CNK×KN. (3.135)
These auto-covariance matrices with reduced dimension can be used in exactly the same fashion as before.
3.4.2 Comparison to periodogram-based processing
The advantage of the ESPRIT algorithm is its simplicity: To acquire the estimates from the matrix F, only a few algebraic operations are required. No target detection algorithms are necessary. Also, ESPRIT does not suffer from quantization issues and therefore does not need the interpolation algorithms described in Section 3.3.5.
However, it has three major disadvantages over the periodogram: First, the estimation orderP is required as an input to the estimator, although the number of targets might not be known a-priori. It is possible to over-estimateP, but that results in additional estimates, which do not correspond to existing targets.15 Second, the list of values for the range and Doppler estimates are not linked–it is not clear which range estimate corresponds to which Doppler estimate.
Gansman et al. [96] have described a method to perform a coupled es- timation using two-dimensional ESPRIT algorithm, but that makes as- sumptions towards the matrix F which are not generally fulfilled for radar, e.g. that no two objects have the same distance or velocity.
If there are only a few targets with large difference in backscattered power, there is a high probability that the order of the target’s eigenval- ues is most likely the same for both estimates, and sorting the eigenvalues by amplitude can solve this problem (i.e. the estimate for range corre- sponding to the largest eigenvalue ofRˆFF,dand the estimate for velocity corresponding to the largest eigenvalue of RˆFF,vrel belong to the same target). However, this is not a reliable method. If ESPRIT is to be used
15Preliminary research was done to combine the periodogram and ESPRIT algo- rithms [136], but very little benefit was found of adding the ESPRIT algorithm after the periodogram, see also 5.2.
for two-dimensional estimation in the same way as the periodogram, ad- ditional heuristics must be implemented to create a useful radar estimate.
In any case, this method only works for a very small number of targets.
The third disadvantage is the required SNR. ESPRIT-based estimators require a better SNR than the periodogram (cf. [6] and the results in Section 5.2.1). This can be explained with the matched filter analogy of the periodogram (Section 3.3.3), which suggests that the periodogram is the optimal estimator with regard to SNR.
For practical use, the disadvantages outweigh the advantages. Unless the application is very specific, such as the tracking of a single object in an otherwise uncluttered environment, the robustness and versatility of the periodogram make it the better choice for OFDM radar signal processing.