## 1 Introduction

The first step of time-delay (TD) estimation is an estimation of the multipath components of the underlying communication channel. Since the impulse response of multipath radio channels is considered to be not bandlimited, it is challenging to achieve high resolution TD estimation from bandlimited observations of communications signals. Traditional channel modeling is mainly suited for communication system design, where it is more important to estimate the effects of the channel on the signal to perform equalization, rather than estimating the parameters of the underlying multipath channel. Therefore, radio channels are typically modeled as FIR filters where the time resolution of the channel is inversely proportional to the bandwidth of the signal used for channel probing [23].

Therefore, high resolution channel estimation requires modeling assumptions. Modeling the channel impulse response (CIR) as a sparse sequence of Diracs, time-delay estimation becomes a problem of parametric spectral inference from observed bandlimited signals. Under this assumption, theoretically, it is possible to obtain perfect estimates of the channel parameters from an equally finite number of samples taken in the frequency domain

[20].Many algorithms for TD estimation from frequency domain samples have been proposed in the past. These algorithms are usually based on (i) subspace estimation [19, 8, 14, 5], (ii) finite rate of innovation [20, 9, 10], or (iii) compressed sampling [1, 12, 11, 24, 2, 3] methods. Few of the previous works [10, 12, 11, 3] discuss issues related to frequency domain sampling. However, these methods are typically complex for the implementation or not robust to noise.

The resolution of TD estimation is proportional to the frequency aperture of the samples taken in the frequency domain. To improve the resolution of TD estimation, without arriving at unrealistic sampling rates, multiband channel sampling has been proposed in [22]. In [7] a practical multibranch receiver for multiband channel sampling has been proposed. Due to hardware impairments of analog electronics components such as low-noise and power amplifiers, the noise level is typically varying across the receiver branches.

In this paper, we are interested in a generalized algorithm for high resolution TD estimation from an arbitrary number of sampled bands with nonuniform noise. Following the shift-invariance structure of Hankel matrices formed from the acquired samples, we propose an algorithm for TD estimation based on weighted subspace fitting [21, 16]. We formulate weighted subspace fitting as a separable nonlinear least squares problem and solve it using the variable projection method [4]. To initialize the variable projection method, we use the TD estimate obtained via the multiresolution TOA algorithm [7]

. With this initialization, the iteration of the variable projection method converges very fast, typically within three steps for moderate or high signal-to-noise ratios (SNR).

The resulting algorithm is benchmarked through simulations by comparing its performance with the algorithms proposed in [15, 17, 7] and the Cramér Rao Lower Bound (CRLB). The results show that for low SNR, the proposed algorithm provides better performance than previously proposed algorithms, and it almost attains the CRLB.

## 2 Problem formulation

Consider a multipath radio channel model with propagation paths defined by a continuous-time impulse and frequency response as

(1) |

where and are collecting unknown gains and time-delays of the MPCs, respectively [13]. We are interested in estimating and by probing the channel using the known wideband OFDM probing signal transmitted over , separate frequency bands (cf. Fig. (a)a). The probed frequency bands are , where is the bandwidth, and is the central angular frequency of the th band. We consider that measurements are collected using nonideal multibranch transceivers with nonuniform noise across the receiver branches. Our objective is to estimate from an arbitrary number of sampled bands while considering the difference in noise levels of the acquired samples.

## 3 Data model

Continous-time signal model: We consider a baseband signal model and assume ideal conversion to and from the passband without phase and synchronization errors. The response of RF chains at the probed frequency bands are modeled using equivalent linear and time-invariant low-pass filters , where the corresponding CTFT has passband (cf. Fig. (b)b). We assume that the frequency responses of the RF chains are characterized during calibration and known. The algorithm for time-delay estimation in the case when , , are unknown is proposed in [6]. The impulse response of the th channel band is , and its CTFT is . Assume that is time-limited to the duration of the OFDM symbol cyclic prefix, that is for . Therefore, there is no inter-symbol interference, allowing us to consider the model for a single OFDM symbol only.

Consider that the same OFDM probing signal is transmitted in all bands, that is, and for , and is given by

where are the known pilot symbols, is the sub-carrier spacing, and is the duration of one OFDM symbol.The signal received at the th band after conversion to the baseband and low-pass filtering is

(2) |

where

is low-pass filtered Gaussian white noise. The corresponding CTFT of the signal

is(3) |

where and are the CTFTs of and , respectively.

Discrete-time signal model: The receiver samples with period , performs packet detection, symbol synchronization, and removes the cyclic prefix. During the period of single OFDM symbol, complex samples are collected, where is the number of sub-carriers and . Next, -point DFT is applied on the collected samples, and they are stacked in increasing order of the DFT frequencies in . Then, the discrete data model of the received signal (3) can be written as

(4) |

where is the Khatri-Rao product, collects the known pilot symbols, , and , collect samples of , and at the subcarrier frequencies, respectively. Likewise, collects samples of as

(5) |

where , and we assume that is an even number. We consider that bands are laying on the discrete frequency grid , where , and denotes the lowest frequency considered during channel probing. Inserting the channel model (1) into (5) gives

(6) |

where we absorbed in

. Now, the channel vector

satisfies the model(7) |

where is a Vandermonde matrix

(8) |

and , where . Likewise, the band dependent phase shifts of MPCs are collected in , where .

Next, we assume that none of the entries of or are zero or close to zero, and estimate by applying deconvolution on the data vector (4) as

which satisfies the model

(9) |

where . The pilot symbols have the constant magnitude, and we assume that frequency responses of receiver chains are almost flat. Therefore

is a zero-mean white Gaussian distributed noise with covariance

, where is the identity matrix. In the case that the frequency responses of the RF chains are not perfectly flat, will be colored noise. However, its coloring is known and can be taken into account.## 4 Multiband time-delay estimation

Our next objective is to estimate from the channel estimates , . We begin by stacking the channel estimates in the multiband channel vector . From (9), satisfies the model

(10) |

Since has a multiple shift-invariance structure and (10) resembles the data model of Multiple Invariance ESPRIT [16], can be estimated using a subspace fitting methods.

### 4.1 Multiband estimation algorithm

From the vectors , , we construct Hankel matrices of size as

(11) |

Here, , is a design parameter and we require and . From (9), and using the shift invariance of the Vandermonde matrix (8), the constructed Hankel matrices satisfy

(12) |

where is an submatrix of ,

is a noise matrix, and .

The column subspaces of , , are spanned by the same dimensional basis. Therefore, a good initial estimate of the orthonormal basis that spans the column subspace of , , can be obtained from the low-rank approximation of the block Hankel matrix

(13) |

The matrix satisfies the model

where , and likewise . Let be a dimensional orthonormal basis for the column span of , then is the corresponding projection matrix. To perform noise reduction, we project , , onto the column subspace of and form the block Hankel matrix

where is the Kronecker product. Now, satisfies the model

(14) |

Note that has multiple shift-invariance structures introduced by the phase shifts of on the (i) subcarrier frequencies of the pilots, , and (ii) carrier frequencies of the bands, , , as shown in Fig.(a)a. The phase shifts can be estimated from the low-rank approximation of and its shift-invariance properties using subspace fitting methods. From the estimate of , immediately follows.

Let be a -dimensional orthonormal basis for the column span of

, obtained using the singular value decomposition

[18], then we can write , where is a nonsingular matrix. Next, let us define selection matrices(15) | ||||

where is transpose, is a zero vector of size , is a vector of size , with th element equal to and zero otherwise. To estimate we select submatrices consisting of the first and, respectively, the last row of each block matrix stacked in , that is and , . In the view of shift-invariance structure of , we have

(16) |

where is a submatrix of . Next, we form block matrices

(17) |

Finally, can be estimated by solving the following weighted subspace fitting problem

(18) | ||||

where , is the pseudoinverse of a matrix, and for the case when noise is uniform accross the receiver chains branches. When the noise power is nonuniform accross the receiver branches, the weighting martix is given by , where and we assume that , , are known.

The subspace fitting problem (18) can be formulated as a separable nonlinear least squares problem, which can be solved efficiently using several iterative optimization methods (e.g., variable projection, Gauss-Newton or Levenberg-Marquardt). We use the variable projection method [4] to find a solution, where a good initialization is obtained by the multiresolution TOA algorithm [7]. With this initialization, the variable projection method converges very fast, typically within three steps for moderate and high signal-to-noise ratios (SNR).

## 5 Numerical Experiments

This section evaluates the performance of the proposed algorithm via numerical simulations. We consider a scenario where the multipath channel has nine dominant MPCs, i.e., , with gain of line-of-sight (LOS) MPC distributed according to a Rician distribution. The continuous-time channel is modeled using a GHz grid, with channel tap delays spaced at ps. We consider that the receiver estimates the channel frequency response in four frequency bands, i.e., , using probing signal with subcarriers and bandwidth of MHz. The central frequencies of the bands are MHz, respectively. To evaluate the performance of TD estimation, we use the root mean square error (RMSE) of the LOS multipath component TD estimate. The RMSEs are computed using independent Monte-Carlo trials and compared with the CRLB and RMSEs of the algorithms proposed in [7, 15, 17] which are shortly denoted with MR-ESPRIT, ESPRIT, and MI-MUSIC, respectively.

Fig. (a)a shows the performance of the proposed, MR-ESPRIT, ESPRIT, and MI-MUSIC algorithms in different signal-to-noise ratio (SNR) regimes for the case when noise power is uniform across the receiver branches. The number of channel snapshots is set to and kept fixed during trials. From Fig. (a)a, we observe that the RMSE of TD estimation decreases with SNR. The MR-ESPRIT and ESPRIT algorithms utilize only samples available from the first and fourth band. Therefore, as expected, their performance is worse, compared to the performance of the proposed and MI-MUSIC algorithms. The MR-ESPRIT, MI-MUSIC and the proposed algorithms are all almost attaining the CRLB for sufficiently high SNR, while ESPRIT due to inefficient use of available data is not able to resolve closely spaced MPCs even for high SNR. The proposed algorithm attains the CRLB for lower SNR than any of the algorithms used for comparison.

In the second scenario, we fixed the signal-to-noise ratio to dB and evaluated the performance of algorithms for the different number of snapshots. From Fig. (b)b, it is seen that the RMSE of TD estimation decreases with the number of snapshots. It can be observed that the number of snapshots needs to be sufficiently high, i.e., equal or higher than , for the algorithms to perform well, which is the consequence of the errors introduced by signal subspace estimation.

The same simulation scenarios are repeated for the case when the noise power is nonuniform across the receiver branches, and the corresponding RMSEs are shown in Fig. (c)c and Fig. (d)d. The signal-to-noise ratios in the third and fourth band are set to dB and dB compared to the SNR of the x-axis on the Fig.(c)c. Likewise, in the fourth scenario, the signal-to-noise ratios for the third and fourth bands are set to dB and dB, respectively. Due to the appropriate weighting of the cost function (18), the proposed algorithm is still close to the CRLB also in the case of nonuniform noise.

## 6 Conclusions

In this paper, we proposed an algorithm for time-delay estimation from multiband channel measurements. Considering the channel impulse response as a sparse signal in the time domain, we have formulated time-delay estimation as a problem of parametric spectral inference from observed multiband measurements. The acquired measurements exhibit multiple shift-invariance structures, and we estimate time-delays by solving the subspace fitting problem. The solution to the problem is found efficiently using the variable projection method. Future directions aim towards evaluating the proposed algorithm with real channel measurements and solving the problem of joint time-delay estimation and calibration of RF chains.

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