EURASIP Journal on Wireless Communications and Networking
Volume 2008 (2008), Article ID 892193, 13 pages
doi:10.1155/2008/892193

Copyright © 2008 Jian (Andrew) Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate reduced-rank shift-invariant technique and its application for synchronization and channel identification in UWB systems. Shift-invariant techniques, such as ESPRIT and the matrix pencil method, have high resolution ability, but the associated high complexity makes them less attractive in real-time implementations. Aiming at reducing the complexity, we developed novel reduced-rank identification of principal components (RIPC) algorithms. These RIPC algorithms can automatically track the principal components and reduce the computational complexity significantly by transforming the generalized eigen-problem in an original high-dimensional space to a lower-dimensional space depending on the number of desired principal signals. We then investigate the application of the proposed RIPC algorithms for joint synchronization and channel estimation in UWB systems, where general correlator-based algorithms confront many limitations. Technical details, including sampling and the capture of synchronization delay, are provided. Experimental results show that the performance of the RIPC algorithms is only slightly inferior to the general full-rank algorithms.

1. Introduction

Ultra-wideband (UWB) signals have very high temporal resolution ability. This implies a frequency-selective channel with rich multipath in practice. Identifying and utilizing this multipath is a must for achieving satisfactory performance in a UWB receiver. To estimate the numerous and closely spaced multipath signals in a UWB channel, high temporal resolution channel identification algorithms with low complexity are required for practical implementations.

Some related UWB research based on the traditional correlator techniques have been reported [1, 2]. The correlator-based techniques are simple, but they might confront many limitations in UWB systems. For example, they usually have limited resolution ability which largely depends on the number of samples, and to improve resolution, higher sampling rates are required; they are ineffective in coping with overlapping multipath signals; they are susceptible to interchip interference (ICI) and narrowband interference (they lack flexibility for removing narrowband interference); and with the number of multipaths increasing, the complexity of these algorithms increases rapidly. In [3], a frequency domain approach is introduced based on subspace methods. Although this scheme is derived from the authors' preceding work on the “sampling signals with finite rate of innovation,” it is in essence the same as those in [4, 5] based on the well-known shift-invariant techniques [6, 7].

Shift-Invariant techniques, such as ESPRIT and its variants [8, 9], matrix pencil methods [10], and state space methods [6], are a class of signal subspace approaches with high resolution ability but relatively high computational complexity associated with the singular value decomposition (SVD) and generalized eigenvalue decomposition (GED). This associated high complexity makes these techniques less attractive in online implementations. To make the algorithms noise-stable, truncated data matrices are generally formed using the SVD, and the original GED in a larger space is transformed into that in a relatively smaller space. This is an application of rank reduction techniques.

Rank reduction is a general principle for finding the right tradeoff between model bias and model variance when reconstructing signals from noisy data. Abundant research has been reported, for example, in [11–14]. Based on some linear models, these rank reduction techniques usually try to find a low-rank approximation of the original data matrix following some optimization criteria such as least squares or minimum variance. In the SVD-based reduced-rank methods, the low-rank approximation matrix is a result of keeping dominant singular values while setting insignificant ones to zero.

Although rank reduction is inherent in shift invariant techniques, in the literature, the rank reduction is only limited to separating the signal subspace and noise subspace, and the reduced rank is constrained to the number of signal sources, , which is usually required to be known a priori or estimated online. Further reduction of the rank generally becomes a problem of signal space approximation by excluding weak signal subspaces. Then we ask, is it possible to reduce the rank to any using shift-invariant techniques supposing only out of signals (parameters) need to be estimated?

This reduction finds practical applications such as in the synchronization and channel identification of UWB signals. The UWB multipath channel is dense with as large as [15]. The general -rank algorithms will have a high computational complexity in the order of multiplications for . Although all multipath parameters can be determined, it is usually sufficient to know () multipath with largest energy for the following reasons: (1) for the purposes of synchronization and detection, several multipath components are usually enough; (2) in the presence of noise, estimates cannot be accurate, and the estimates of multipath signals with lower energy contain relatively larger errors according to the Cramer-Rao bounds [16].

In this paper, we present some novel -rank shift-invariant algorithms, and investigate their applications in joint synchronization and channel identification for UWB signals. These -rank algorithms will be referred to as reduced-rank identification of principal components (RIPC) algorithms. Unlike general subspace methods, our schemes remove the constraint on and multipath signals with largest energy can be automatically tracked and identified, while the complexity can be significantly reduced by a factor related to . The word “automatically” means that no further processing is needed to pick up principal ones among more estimates. Actually, only signals are estimated and they are supposed to be the principal ones. The value of can be adjusted freely to meet different performance requirements of synchronization and specific multiple-finger receivers like RAKE.

The rest of this paper is organized as follows. In Section 2, the shift-invariant techniques are introduced. In Section 3, our new RIPC algorithms are derived using the harmonic retrieval model. In Section 4, the application of RIPC algorithms in the joint synchronization and channel estimation is presented. Technical details are given including sampling, deconvolution, FFT, and the capture of synchronization delay. Simulation results are given in Section 5. Finally, conclusions are given in Section 6.

The following notation is used. Matrices and vectors are denoted by boldface upper-case and lower-case letters, respectively. The conjugate transpose of a vector or matrix is denoted by the superscript , the transpose is denoted by , and the pseudoinverse of a matrix is denoted by . Finally, denotes the identity matrix and denotes a diagonal matrix.

2. Formulation of Shift-Invariant Techniques

Typical harmonic retrieval problems can be addressed as the identification of unknown variables from the following equation: (1) where is the imaginary unit, are the measured samples, are the noise samples, is the number of samples, and are the unknown amplitudes and frequencies, to be determined.

Organize these measured samples into an Hankel matrix where the entries along the antidiagonals are constant, we get (2) where and . The used samples usually start from . In order to make the notations in (4) applicable to subsequent equations, for example, (19), we start from here. Without loss of generality, we assume . In the noise-free case, can be factorized as (3) where (4)

The Vandermonde matrix exhibits the so called shift-invariant property, that is, (5) where and denote the operations of omitting the first and omitting the last rows of a matrix, respectively, and contains the desired frequencies. This property facilitates the development of various shift-invariant techniques. By constructing two rank matrices and with the inherent shift-invariant property, the diagonal elements of can be obtained by solving the generalized eigenvalues of the matrix pencil . These two matrices and can be constructed directly from using and , or from the correlation matrices of , or from the singular vectors of . The use of can improve resolution ability and result in smaller variance of estimates, but must be chosen to ensure in order to avoid phase ambiguities, and maintain . In the presence of noise, the above solutions hold as approximations while the criterion of least squares or total least squares is applied [7].

Substituting estimated frequencies into (1), the amplitudes can be obtained by solving a Vandermonde system using least squares type algorithms [13, 17]. The energy of harmonics can also be solved according to the generalized eigenvectors (GVs) [8]. In either method, the accuracy of amplitude estimates is inferior to frequency estimates whose accuracy is guaranteed by the stability of the singular values in the presence of a perturbation matrix. The accuracy of amplitude estimates will sometimes contribute to the overall performance of estimation. For example, when we need to pick out several harmonics with largest energy among all estimates, the errors in amplitude estimates will influence the correctness of the selected harmonics significantly.

3. Reduced-Rank Identification of Principal Components (RIPC)

3.1. Generalization of the Shift-Invariant Methods

The shift-invariant techniques can be interpreted from various angles, such as the subspace viewpoint [8, 9], the state space viewpoint [6], and the matrix pencil viewpoint [10]. We generalize a result in the viewpoint of matrix pencil below, which will be used in the subsequent development of the paper.

Proposition 1. For any two matrices and , if both matrices have rank , and can be factorized as (6) where is an matrix, is an matrix, and (as well as ) is an diagonal matrix with each diagonal element mapping to one of the desired parameters uniquely, then the desired parameters can be uniquely determined by the generalized eigenvalues of the matrix pencil , for example, the desired parameters are the frequencies in the harmonic retrieval problem.

Proof. According to the property that the rank of the product of matrices is smaller than the rank of any factor matrix, both and have rank

For the pencil , if is a generalized eigenvalue of the pencil, the matrix will have rank . This implicitly requires the matrix to be rank deficient [18, page 48]. Thus, equals the reciprocal of one of the diagonal elements of , and the desired parameter can be determined accordingly.

This theory removes the normal constraints on the structures of the basic factor matrices (e.g., Vandermande matrix) and the data matrices (e.g., Hankel or Toeplitz matrix). Any problem can be solved applying this theory if it can be formulated likewise. An example is if the parameters in are independent of those in and , they can still be determined no matter how many unknown parameters are contained in and

3.2. Principal Subspace and Frequency Estimation

Suppose that the formed and are noise-free matrices. Since has rank , the compact SVD of has the form (7) where the diagonal matrix contains singular values in descending order, the matrix and matrix consist of left and right singular vectors, respectively. and are the left and right submatrices of , associated with the principal and the remaining smaller singular values, respectively.

Multiplying the matrix pencil by from the left and by from the right, we get a new matrix pencil (8) where we have utilized the orthogonality between the columns of and , and and

For the new matrix pencil, we have the following results.

Proposition 2. For the two matrices and defined in Proposition 1, when the generalized eigenvalues of the matrix pencil exist, the matrix pencil has distinct generalized eigenvalues , and, specific to a harmonic retrieval problem, the angles of equal to the frequencies up to a known scalar, corresponding to harmonics with largest energy.

Proof. As defined in Proposition 1, and can be factorized as (9) where is an matrix with rank , and is an matrix with rank

Let () denote the matrix containing dominant left (right) singular vectors of , and the corresponding diagonal singular values matrix. According to (10) we know , where we used the property that the rank of a product matrix could not be larger than the rank of every factor matrix.

Similarly, we can get

Then for the matrix (11) if is the generalized eigenvalue of the pencil (we will discuss the possibility of its existence later), it is also a rank-reducing number of the matrix . This implies is rank deficient. Otherwise Rank. Therefore is also a rank reducing number of the matrix and the eigenvalue corresponding to is (12) On the other hand, the generalized eigenvalue problem can be reduced to the standard eigenvalue problem [19] by (13) where the generalized eigenvalues are expressed as functions of matrix pencil and matrix product, provided that the pseudoinverse matrices of and exist. Thus the generalized eigenvalue in (11) can be written as (14) From (12) and (14), we have (15)

We have seen from above that both and are full rank, so there are totally generalized eigenvalues of the pencil [19, page 375], corresponding to frequencies.

Since the SVD of a matrix exhibits the spectral distribution of the comprised signal in harmonic retrieval problems [11], the principal singular values and vectors reflect the information of the frequencies with largest power. This intuitively explains why the generalized eigenvalues are associated with the frequencies with largest energy.

So far, we have established the links between the angles of the generalized eigenvalues and the frequencies. However, an extra condition has to be emphasized in the above proof: whether those generalized eigenvalues of the pencil exist or not? There may not exist a clear answer since in our experiments, it varies from time to time.

If the generalized eigenvalues of do not exist, the obtained eigenvalues become good approximations to the actual ones when is not very small compared to . Because in this case, the pencil can be viewed as an approximation of the original one, or can be regarded as the frequency estimates of the harmonics with larger energy under the interference of the remaining harmonics with lower energy. To characterize the errors of this approximation, the general perturbation analysis [19] could be used. However, we note that it is not very suitable here because the elements in the perturbation matrix are not small enough.

3.3. Energy/amplitude Estimation of the Harmonics

In the case when only out of frequencies are known, the amplitude estimates obtained by solving the under-determined linear equations of (1) will comprise large errors. Alternatively, when and are formed as the correlation matrices of , for example, (16) the energy of the harmonics can be estimated in a subspace method according to the following proposition.

Proposition 3. When and are constructed in the way similar to (16), the energy of th harmonic, , can be well approximated as (17) where is the generalized eigenvector corresponding to the generalized eigenvalue (and then frequency ), and is defined in (4).

Proof. See the appendix.

From the proof, we can see that a necessary condition for the above proposition is that the product needs to resemble an identity matrix. Actually, the th element of is given by (18) Figure 1 demonstrates the magnitude of these elements. From the figure, it is obvious that, only when is large enough and there is no frequency close to zero or , can be approximated as an identity matrix and the above method works. In practical applications, when this condition is not satisfied, we need to consider alternative approaches.

The two key factors in the derivation of (17) are that (1) is symmetric and (2) is fully contained in a diagonal matrix, and each of them can be mapped to one of the diagonal elements uniquely. These observations motivate us to construct the following data matrices (19) where and

These two matrices have the shift-invariant property, and the diagonal elements of can be determined by the generalized eigenvalues of the matrix pencil . The reduced rank algorithms described in Proposition 2 are also applicable to this pencil. Now, if we let , and assume is a real matrix ( are real), will be a Hermitian matrix. For a Hermitian but not necessarily positive-definite matrix, the eigenvalues are real but not necessarily positive. Therefore, to maintain its singular values positive, the left and right singular vectors of the matrix are equal up to a constant diagonal matrix . This matrix has diagonal entries or corresponding to the polarity of the eigenvalues. For example, for the principal singular vectors.

Then, similar to the proof of Proposition 3, the following proposition can be proven. Note that the matrices in (A.1) in the proof of Proposition 3 will be replaced by . This change leads to the estimates of amplitudes rather than squared amplitudes.

Proposition 4. When and are constructed in the way similar to (19) with , and is a real diagonal matrix with diagonal entries equal to the amplitudes of harmonics, the amplitude of th harmonic, , can be determined by (20) where is the generalized eigenvector corresponding to the generalized eigenvalue (and then frequency ).

It is obvious that this result is superior to the one in Proposition 3 in the estimation of . However, there is another problem associated with it. Since is a Hermitian matrix directly constructed from the samples, the performance of the frequency estimation might be inferior to the one in Proposition 3 when the dimensions of these two matrices are equal. This happens when the added noise matrix is also Hermitian, because in this case, the number of effective samples in Proposition 4 equivalently reduces to half. Even so, it might still be worthy of constructing a double size matrix and using our RIPC algorithms when fast algorithms can largely reduce the cost of computation, compared to the general -rank algorithms. This is confirmed by some experimental results to be given in Section 5.

3.4. Fast Algorithms to Find the Principal Signal Space

Since only out of principal singular values and vectors are required, the computation can be simplified by applying fast algorithms with lower complexity, such as the power method [19]. For each dominant singular value and vector, the power method has a computational order of for an Hermitian matrix. To be stated, in the power method, the speed of convergence depends on the ratio between the two largest singular values of the matrix. The larger the ratio is, the faster it converges.

For an Hermitian matrix , the power method generates principal singular values and vectors as shown in Algorithm 1.

When is not a Hermitian matrix, a similar algorithm is applicable in which the left and right singular vectors should be generated by constructing and , respectively.

On the detailed implementation of the power method, we have some interesting findings in our experiments.

(i) After the th eigenvector is generated, if we let it be the initial iterative vector in solving the next eigenvalue and vector rather than randomly chosen , the iteration usually converges very fast. For positive Hermitian matrices, or iterations are enough. (ii) Even when the first several estimated eigenvalues contain larger errors, the remaining eigenvalues can still be estimated with higher accuracy due to the stability of eigenvalues to the perturbation errors. (iii) If not all eigenvalues are positive, the power method might output eigenvalues in a nonordered manner. This usually implies relatively larger errors in these eigenvalues. However, the estimated frequencies can still have good accuracy. It should be noted that although the generalized eigenvalues of the pencil are equal to the eigenvalues of , the power method is ineffective in directly solving the first eigenvalues of because there are not large enough gaps between adjacent eigenvalues (the magnitudes of all eigenvalues equal ).

4. Joint Synchronization and Channel Identification

We consider a general transmitted UWB signal in a single-user system. The signal could be a spread spectrum (SS) signal (e.g., time-hopping or direct sequence spread) or non-SS signal (e.g., single pulse), but it should be unmodulated or modulated with known constant data. For randomly modulated signals, the sampled channel impulse response can be estimated using the least squares criterion first as discussed in [4]. We assume that the spread spectrum codes are known in an SS system.

Here, the used UWB multipath channel model is a simplified version of the IEEE802.15.3a channel model [15], which is a modified Saleh-Valenzuela model where multipath components arrive in clusters. For synchronization and channel estimation, the IEEE model can be simplified to a TDL model, represented by (21) where is the th multipath delay, is the th multipath gain with phase randomly set to with equal probability, is the number of multipaths, and is the Dirac delta function. The multipath delay and gain are regarded as deterministic parameters to be estimated.

When a symbol sequence is transmitted over this channel, the received signal is (22) where is the additive white Gaussian noise (AWGN), is the synchronization delay between the receiver and the transmitter, and is the symbol period.

To set up the connection between (22) and (1), we can transform (22) from time domain to frequency domain by applying the Discrete Fourier Transform (DFT) upon the samples of

4.1. Sampling of Signals

Since the system is not synchronized yet, whatever the signal is, the width of the sampling window should be chosen to equal the integral multiple of the symbol period and be larger than the maximal multipath spread . Assume that the sampling period is , the number of samples is , and the samples from (22) are . Two scenarios regarding to the sampling need to be considered.

(1) Sampling of Widely Separated Pulses

When the intervals between the continuously transmitted pulses are larger than , there is no ISI in the samples. Let the sampling length equal the symbol period be the samples of , and be the samples of the noise , then the DFT coefficients of (22) can be represented as (23) where is the basic frequency, and are the DFT coefficients of and , respectively.

(2) Sampling of Closely Spaced Pulses

When the intervals between the transmitted pulses are smaller than , ISI is generated. Assume that the multipath can be fully covered by at most symbols, that is, . Represent the symbols as (24) where is the index of any symbol, and let be the samples of . In this case, the samples of , contain ISI terms. However, when symbols are transmitted continuously without interruption, it can be proven that , the DFT coefficients of , are ISI-free due to the Circular Shift Property [20, page 536] of DFT, and (23) also holds.

This finding enables continuous transmission of the training sequence to speed the synchronization process. This is also another advantage of the proposed algorithms compared to conventional algorithms which generally require the interval between two impulses to be larger than the multipath delay spread.

4.2. Summary of Joint Synchronization and Channel Identification Schemes Using RIPC Algorithms

Deconvolution is defined as the operation of dividing by in (23), the reverse of convolution viewed in the frequency domain. After the deconvolution operation, we get some equations identical to (1) in the harmonic retrieval problem. Then the synchronization and channel identification algorithm can be summarized as follows:

(1) in a window with width , sample the received signal with period . Make sure equals an integral multiple of the symbol period and larger than the multipath spread (2) apply the FFT to the samples and select DFT coefficients carefully; (3) after deconvolution, form the Hankel data matrix , and use principal components tracking algorithms to estimate the delays with largest energy (sum of and ). (If the amplitudes are required, correlation matrices or Hermitian data matrices should be used.) (4) resolve and from the estimated delays.

The last step is necessary as each estimated delay in step (3) is the sum of the synchronization delay and one of the multipath delays . There is a phase-ambiguity problem with these sums as the delays may become circularly shifted. This could happen when sampling starts in the middle of multipath delays. Our solution is first to choose much larger than the maximal multipath delay , then separate and according to the following criteria.

(i) Sort the estimates in ascending order and get . If the gap between any two adjoining estimates is larger than a threshold , for example, , then equals the sum of the synchronization delay and the first desired multipath delay. And all the estimates need to be updated to(25)that is, the original are updated by adding to themselves. Now, the receiver can synchronize to the multipath with delay which implicitly assumes the delay of the first multipath of interest is zero, and the differences between the updated estimates and the first desired multipath are the relative multipath delays. (ii) Otherwise, the smallest estimate is the first multipath of interest and no update is needed. This judgement is based on the assumption that the gap between any two multipath signals is smaller than the threshold , which is generally close to the difference between the sampling window width and the maximal multipath delay . In practice, the multipath components with larger energy usually have smaller delays, so the threshold needs not be very large.

4.3. Complexity of Our Schemes

The complexity of our algorithms depends on the required resolution ability and performance of estimation. The resolution ability is roughly determined by the sampling period. The smaller the sampling period is, the higher the resolution ability is. The performance of estimation is mainly influenced by the SNR, and the dimension of the matrices and . Then the sampling period is the key parameter in both the complexity and performance since the main computation cost of our algorithm is associated with FFT, SVD, and GED. For a -point FFT, the computational workload is when a Cooley-Tukey radix- algorithm [21] is used. equals a power of . The complexity of GED for a matrix is . Plus the complexity of the power method (suppose that DFT coefficients are used), the total complexity is in the order of . Accordingly, the complexity of the general -rank algorithms is in the order of . When , the saving is considerable.

5. Simulations

First, we show some experimental results of the RIPC algorithms using the harmonic retrieval model. The performance can act as a basis for evaluating the performance loss in many applications of RIPC algorithms. Simulation results of the joint synchronization and channel identification for UWB signals are given in Section 5.2.

5.1. Simulations of RIPC Algorithms

The simulations in this subsection are based on the harmonic retrieval model in (1). The algorithms evaluated are classified as follows.

(A1) Algorithms A1 are full -rank algorithms with a general matrix (not Hermitian nor positive-definite), where amplitudes are obtained by solving a Vandermonde problem using the least squares criterion. (A2) Algorithms A2 are full -rank algorithms with a Hermitian matrix (not positive-definite), where amplitudes can be estimated via (20) or by solving a Vandermonde matrix. (A3) Algorithms A3 are RIPC algorithms with a general matrix , where amplitudes cannot be estimated. (A4) Algorithms A4 are RIPC algorithms with a Hermitian matrix , where amplitudes are estimated by (20). (A5) Algorithms A5 are RIPC algorithms with a Hermitian matrix , where the power method is applied and amplitudes are estimated by (20).

We first generate amplitudes randomly using a Gaussian distribution with mean zero and variance . These amplitudes are normalized such that their squared sum is unity. The frequencies are generated randomly using a uniform distribution on the interval . In most cases, harmonics are generated, and matrices and are constructed.

Figures 2 and 3 demonstrate some detailed implementations of algorithms A1–A5. Each figure consists of subfigures. Figures 2(a) and 3(a) shows the hit rate of the frequency estimates. When an estimate has an estimation error within a predefined threshold (named as “hitting threshold” hereafter, set to ), we say it “hits” the true value. The hit rate is then defined as the ratio between the number of the hit estimates and the total estimates. The hit rate is thus conceptually similar to the outage probability that is commonly used in the literature. Figures 2(b) and 3(b) shows the mean squared error (MSE) of the hit estimates (nonhit estimates are excluded) averaged over realizations. The results obtained by algorithms A1–A5 are denoted by diagonals, triangles (down), circles, stars, and squares, respectively. Figures 2(c) and 3(c) shows the energy ratio of the principal harmonics out of the total ones. Figures 2(d) and 3(d) shows the averaged number of iterations in the power method. In the power method, the maximal number of iterations in computing every eigenvalue and vector ( in Algorithm 1) is set to , and the threshold is set to to control the number of iterations.

In Figure 2, simulation results in the noise-free case are illustrated with and . It is clear that full -rank algorithms A1 and A2 can achieve perfect estimation with high hit accuracy and near zero MSEs (not plotted in Figure 2(b)). Comparatively, our reduced-rank RIPC algorithms can not achieve perfect estimation in the noise-free case, while they are relatively stable with respect to the change of SNR.

Even when the samples are corrupted by noise, the algorithms A1 and A2 can normally achieve good frequency estimates for some harmonics, as can be observed in Figures 2 and 3. However, their amplitude estimates usually contain relatively larger error due to the following two reasons. On the one hand, the frequency estimates with higher accuracy normally correspond to the harmonics with larger energy according to the Cramer-Rao bound. For frequencies with smaller energy, the estimates inevitably contain larger errors. On the other hand, the accuracy of frequency estimates is due to the inherent stability of eigenvalues and singular values. The amplitude estimates, however, are susceptible to the noise. Thus, in the sense of determining principal frequencies with largest energy, is less effective than RIPC algorithms.

The ratio of collected energy shown in Figures 2 and 3(b) indicates that the hit rate and MSE are actually weakly dependent of the collected harmonics energy. This implies that an analytical analysis using an approximation theory (or the perturbation theory) for Proposition 2 might not work. Simultaneously, it implies that the stability of the RIPC algorithms is high with respect to the number of the desired principal signals.

Figure 4 demonstrates how the hit rate and MSE vary with SNR where is a state space based algorithm within the framework of A1 used in [3]. From the figure, we see that when the SNR is larger (than dB), the performance of the RIPC algorithms are satisfactory and stable.

In experiments, we find that the amplitude estimates in the RIPC algorithms are not so accurate as the frequency estimates because the errors in the frequency estimates are actually transferred into the amplitude estimates. In most cases, the polarity of the amplitude can be determined accurately, while the magnitude can suffer an error as large as of the true value in the SNR range dB. This is a general problem in the subspace-based harmonic retrieval algorithms, which could be mitigated by averaging over multiple realizations. However, this problem does not influence the determination of principal harmonics in RIPC algorithms as they have been automatically tracked and picked out during the frequency estimation.

5.2. Simulations of Joint Synchronization and Channel Identification

The second-order Gaussian monocycle is used as the basic pulse (26) where parameterizes the effective pulse width. The dB bandwidth of this pulse is about Hz, dB bandwidth is about Hz, and center frequency is about Hz.

When sampling this pulse with period , we get roughly six samples per pulse, and this sampling rate is already above the Nyquist rate in terms of the dB bandwidth. To reduce the sampling rate without introducing aliasing, similar to [3], a low-pass filter with bandwidth much smaller than the signal bandwidth can be applied at the cost of reduced energy collection. When choosing “clean” DFT coefficients to minimize interference due to residual alias, coefficients near the normalized frequency should be excluded. On the other hand, DFT coefficients with larger energy should be chosen to avoid blowing up the noise in the deconvolution operation. When strong narrowband interference is present and the interference spectrum is known, the interference can be readily removed by selecting those coefficients in the unaffected spectrum.

To test the performance of our algorithms in practical implementations, we use the channel model CM1 proposed in [15] by IEEE802.15.3a. The channel impulse response (CIR) is reproduced using . The first multipath signals in each CIR are used to simulate the channel.

Before the actual implementations of synchronization and channel estimation, we first feed these multipath parameters into the harmonic retrieval model, that is, substitute with the multipath gains and with in (1), where is chosen to be slightly larger than the maximal multipath delay . The achievable performance can serve as upper bounds in practical implementations.

Figure 5 shows the hit rate and MSE of the frequency estimates in this case, and the actual collected energy by these algorithms is shown in Figure 6. To make the estimates independent of , estimates are kept in the form of frequencies rather than delays. It can be seen that there is not much difference between the performance here and that shown in Figure 4. This indicates the stability of our RIPC algorithms. From Figure 6, we can also see that about 80% energy of the largest channel taps can be collected (and exploited then), which is consistent with the hit rate.

To check the performance loss in practical implementations, let us examine the noise-free case first. In the noise-free experiments, DFT coefficients from to are chosen, the hitting threshold is set to , and the estimates are compared with principal multipath signals to determine the hits. Figure 7 shows the hit rate, root MSE (RMSE) of the delay estimates and mean error of the gain estimates obtained by A2, A4, and A5 with , and . The RMSEs of the delay estimates are normalized with respect to . The sampling rate is . In Figure 8, we show the RMSEs of the delay estimates versus the SNR for A2, A4, and A5. For comparison, the Cramer-Rao low bound (CRLB) in an AWGN channel [16] is also plotted. From the figures, we can see that an accuracy of about of can be obtained at average hit rate above . This means that the timing accuracy is mostly within one sample distance, which is only slightly inferior to the full-rank approach in [3]. When the SNR is as large as dB, the RMSEs are already very close to those in the noise-free case. Overlapping of the CRLB curve with other performance curves is due to the lower hit rate at smaller SNRs, where quite a few estimates with larger errors are excluded from the computation of the MSE. With the hit rate increasing, the CRLB curve becomes a good reference for evaluating the performance of the proposed schemes.

6. Conclusions

To reduce the complexity of general subspace-based delay estimation algorithms, we proposed reduced-rank shift-invariant techniques which can track the principal components automatically. Amplitude estimation schemes are also proposed based on subspace methods. Application of the proposed techniques in synchronization and channel estimation for UWB signals is investigated. Experiments show that our proposed algorithms can achieve performance comparable to full-rank algorithms, but with significantly reduced complexity.

Appendix

Proof of Proposition 3. Substitute (3) into (16), we get (A.1) where is the diagonal matrix defined in (4), approximates an identity matrix up to a multiplicative scalar since over intervals of infinite support, cisoids of different frequencies are orthogonal [8]. Thus, the product can be replaced by a diagonal matrix with diagonal entries equal to the energy of harmonics, that is, . Temporarily, we denote by for brevity.

Since is a Hermitian matrix and positive-definite, its left and right singular vectors are identical. Let be the generalized eigenvector corresponding to the generalized eigenvalue . According to the definition of the generalized eigen-problem, for the pencil , we have (A.2) where the expressions of and in (A.1) are used. Left multiplied by , (A.2) becomes(A.3) Since is an diagonal matrix with only the th diagonal element equal to zero, the vector has the form (A.4) that is, except for the th element, all others equal zero.

Notice that and is a diagonal matrix with the th diagonal element equal to one. Hence, (A.3) can be rewritten as (A.5) which establishes (17).

Acknowledgments

NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

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