Compressive Joint Angular-Frequency Power. Spectrum Estimation for Correlated Sources. Dyonisius Dony Ariananda and Geert Leus. Delft University of ...
Prosiding Seminar SISTI 2014
Compressive Joint Angular-Frequency Power Spectrum Estimation for Correlated Sources Dyonisius Dony Ariananda and Geert Leus Delft University of Technology, The Netherlands {d.a.dyonisius, g.j.t.leus}@tudelft.nl Abstract — In this paper, the power spectrum of the received wide-sense stationary (WSS) signals produced by possibly correlated sources is compressively estimated as a function of direction of arrival (DOA) and frequency. Here, we compress the received WSS signals in both the time and the spatialdomain. The spatial-domain compression is performed by using the so-called dynamic linear array of active antennas and the time-domain compression is implemented by sampling the signal received by each active antenna at sub-Nyquist rate using multi-coset sampling. In each time slot, we then calculate the correlations between the resulting sub-Nyquist rate samples both in the spatial-domain and the time-domain and use them to recover the two-dimensional (2D) power spectrum matrix that gives the power spectrum information in both the frequency and the angular dimensions. Under the full rank condition of the system matrices, the 2D power spectrum reconstruction can generally be done using simple least squares approach without applying any sparsity constraint on the true power spectrum. In addition, the resulting angular power spectrum at each frequency can also be used for DOA estimation of more correlated sources than active sensors by locating the peaks in the power spectrum. Keywords — Multi-coset sampling, dynamic linear array, power spectrum, wide-sense stationary.
I. INTRODUCTION Perfect reconstruction of the analog signal from its subNyquist rate samples is important especially when there is a demand to alleviate the requirements on the analog-to-digital converters (ADC). This has been shown to be possible through compressive sampling as long as the signal is inherently sparse [1]. Some possible implementations for compressive sampling have been proposed, such as random demodulator [2] and multi-coset sampling [3], [4]. Further progress, however, has been found in [5] for wide-sense stationary (WSS) signals where perfect reconstruction of the power spectrum of the signal sampled below the Nyquist rate is generally possible even without any sparsity constraint on the original signal. This new result is beneficial for some applications such as cognitive radio (CR) networks [6]. In a CR network, a secondary user (SU) who does not have any official ownership on frequency spectrum is allowed to borrow a portion of a frequency spectrum when the owner (called primary user (PU)) is not active. Here, the SU has to perform spectrum sensing to search for unoccupied bands in the licensed spectrum and once the free band is found, this spectrum sensing process has to be continuously performed as the SU has to monitor in case the PU is suddenly active (in which case the SU has to release the rented spectrum). As the sensed frequency band is generally wide, sampling the signal at sub-Nyquist rate is of interest. Note however, that the SU is only
interested in the spectrum occupancy and not in the PU signal, which implies that perfect original signal reconstruction is overkill since the information about power spectrum plot showing which bands are occupied is sufficient. The concept of power spectrum or temporal autocorrelation reconstruction from sub-Nyquist rate samples has also been introduced in the spatial-domain. It has been shown in [7], [8] that if we arrange the position of the antennas in the linear array based on the so-called nested array and coprime array, respectively, it is possible to use the spatial correlation values between the antennas outputs in the array to compute the spatial correlation values between the antennas outputs in a virtual array (which is uniform), called difference co-array, having more antennas and larger aperture than the physical array. This feature increases the degrees of freedom allowing [7] and [8] to estimate the direction of arrival (DOA) of more uncorrelated sources than physical sensors. The minimum redundancy array (MRA) introduced by [9] can also be employed to produce such feature although it does not allow us to formulate a closed-form expression for the array geometry and the achievable number of lags in the resulting difference co-array, as the nested and coprime arrays do. While [7] and [8] have exploited the reconstruction of the second-order statistics to estimate the DOA of more uncorrelated sources than sensors, their approaches cannot handle more correlated sources than physical antennas. In [10], a new DOA estimation approach for more correlated sources than active sensors has been proposed by introducing the so-called dynamic linear array (DLA). In this paper, we compressively reconstruct both the frequency-domain and angular-domain power spectrum by using a uniform linear array (ULA) as the underlying array. We adopt the DLA introduced in [10] and employ a periodic scanning where one scanning period contains several time slots and in different time slots, we activate different sets of antennas in the ULA resulting in a dynamic array with possibly less active sensors than correlated sources. This leads to a spatial-domain compression in every time slot. Next, in each active antenna, the received signal is sampled at sub-Nyquist rate by using the minimal sparse ruler sampling [5]. For every time slot, the correlations between the collected sub-Nyquist rate samples at all active antennas are calculated both in the spatial-domain and the timedomain. The computed correlation values at all time slots are then used to reconstruct the two-dimensional (2D) power spectrum matrix where each row contains the power spectrum in the frequency-domain for a given angle and
each column gives the power spectrum in the angulardomain for a given frequency. The resulting angular power spectrum at every frequency point can also be exploited for DOA estimation by locating the peaks in this angular power spectrum. Note that this 2D power spectrum reconstruction can be performed without applying any sparsity constraint on the true power spectrum. While this approach is applicable even when we have more correlated sources than active sensors, the upper bound on the number of detectable correlated sources is given in Section V. II. UNDERLYING MODEL As our underlying array, we consider a ULA having Ns antennas receiving signals transmitted by K possibly correlated WSS sources and assume that the distance between the sources and the ULA is large enough compared to the ULA aperture allowing us to assume the sources as point sources and the incident waves on the ULA as plannar waves. Moreover, the inverse of the bandwidth of the aggregated incident signals is also assumed to be larger than the signal propagation time across the ULA and thus we can define the delay introduced between the antennas as a phase shift. Using these assumptions, the output of the ULA at time index t can be written as: +
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where the Ns x 1 output vector x(t) contains the received signal at the Ns antennas of the ULA, s(t)=[s1(t), s2(t), ..., sQ(t)]T is the Q x 1 extended source vector with sq(t) the incoming signal from the investigated angle θq, n(t) is the Ns x 1 vector containing additive white Gaussian noise, and A= [a(θ1), a(θ2), ..., a(θQ)] is the Ns x Q extended array response matrix with a(θq) the Ns x 1 array response vector containing the phase shifts experienced by sq(t) at each ULA element. Here,
is generally known by the receiver and it
might only approximately contain the actual DOAs. It is assumed that the impact of the wireless channel has been taken into account in s(t), that the noises at different antennas are uncorrelated with variance σn2, i.e., E[n(t) n(t)H]= σn2 where is the Ns x Ns identity matrix, and that n(t) is independent from s(t). Further, the first element of the ULA is used as a reference point and the array response vector a(θq) is expressed as = 1,
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where a(θq) = exp(j2π sin(θq)) and d is the distance between two consecutive antennas in wavelengths, which is set to d ≤ 0.5 to prevent spatial aliasing. III. DYNAMIC LINEAR ARRAY AND TIME DOMAIN COMPRESSION In this section, we perform the spatial-domain compression by adopting the DLA in [10], which is
Prosiding Seminar SISTI 2014 implemented by performing a periodic scanning where a single scanning period consists of L time slots and in different time slots, we activate different sets of Ms out of Ns available antennas in the ULA leading to a possibly nonULA of less active antennas than sources in each time slot. For a given time slot, the receiver branches corresponding to the active antennas sample the received signal at subNyquist rate leading to a time-domain compression. Multicoset sampling discussed in [3,4] is one possible option to conduct the sub-Nyquist rate sampling and this can be implemented using the practical sampling device proposed in [5]. However, we here simply demonstrate the multi-coset sampling by applying a selection matrix on the Nyquist-rate samples. We perform a periodic scanning on the ULA model given by (1). While we have different sets of Ms active antennas in different time slots of a scanning period, the same antennas set is activated in the l-th time slot of different scanning periods. Let us introduce x[m] = x(mT), n[m] = n(mT), and s[m] = s(mT) as a digital representation of x(t), n(t), and s(t), respectively, with 1/T the Nyquist sampling rate at each ADC corresponding to every antenna. In addition, denote the number of Nyquist-rate samples received by each active antenna in one time slot by Nt and the total number of scanning periods by Np. Our next step is to express the outputs of the Ms active antennas in the array in the l-th time slot as the following Ms x 1 vector &' ()* +, + ), - = ./,' ( )* 0 + 1 +, + ), - 2 for l = 0, 1, ..., L-1, where the Ms x Ns spatial-domain selection matrix for the l-th time slot Cs,l is constructed by selecting Ms rows of , np = 0, 1, ..., Np-1, and nt = 0, 1, ..., Nt -1. Note that the indices of the selected rows of , used to form Cs,l correspond to the indices of the Ms active antennas in the l-th time slot that are selected from the Ns available antennas in the underlying ULA. If we collect all Nt output vectors x[(npL + l)Nt + nt] into the Ns x Nt matrix X[npL + l] = [x[(npL + l)Nt], x[(npL + l)Nt +1], ..., x[(npL + l + 1)Nt -1], we can then rewrite (2) in a matrix form as 3' ()* - = ./,' 4()* 0 + 1= ./,' 5()* 0 + 1- + 6()* 0 + 1- 3 where the Ms x Nt matrix Yl[np] is given by Yl[np]=[ yl[npNt], yl[npNt + 1], ..., yl[(np + 1)Nt -1]], S[npL + l] = [s[(npL + l) Nt], s[(npL + l)Nt + 1], ..., s[(npL + l +1)Nt - 1]] is the Q x Nt matrix, s[(npL + l)Nt + nt] = [s1[(npL + l)Nt + nt], s2[(npL + l)Nt + nt], …, sQ[(npL + l)Nt + nt]]T is the Q x 1 vector with sq[m] = sq(mT) a digital representation of sq(t), and N[npL + l] = [n[(npL + l)Nt], n[(npL + l)Nt + 1], …, n[(npL + l + 1)Nt 1]] is the Ns x Nt noise matrix. Our next step is to construct the Mt x Nt time-domain selection matrix Ct by selecting Mt rows of the Nt x Nt identity matrix 8 , and apply a timedomain compression on Yl[np] in (3) leading to an Ms x Mt matrix Zl[np]
9' ()* - = 3' ()* -.,% = ./,' 4()* 0 + 1-.,% . 4 Note that Mt can be interpreted as the number of subNyquist rate samples per time slot produced by each receiver branch corresponding to each active antenna. IV. POWER SPECTRUM RECONSTRUCTION Based on (3), let us denote the j-th row of Yl[np] and % Zl[np] in (4) as the 1 x Nt vector &',< ()* - = [>',< [)* +, ], >',< [)* +, + 1], . . ., >',< [ )* + 1 +, − 1]] and the 1 x Mt %
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