Performance of Narrowband Signal Detection under Correlated Rayleigh Fading Based on Synthetic Array

Abstract

The performance of a single moving antenna receiver in detecting a narrowband signal under correlated Rayleigh fading is considered. The spatial motion of the antenna during signal capture provides a realization of a synthetic antenna array. As shown, there is a net processing gain obtained by using a synthetic antenna array compared to the equivalent static antenna in Rayleigh fading environments subject to constant processing time. The performance analysis is based on average Signal-to-Noise Ratio (SNR) metrics for design parameters of probability of detection (𝑃_𝑑) and probability of false alarm (𝑃_𝑓𝑎). An optimum detector based on Estimator-Correlator (EC) is developed, and its performance is compared with that of suboptimal Equal-Gain (EG) combiner in different channel correlation scenarios. It is shown that in moderate channel correlation scenarios the detection performance of EC and EG is identical. The sensitivity of the proposed method to knowledge of motion parameters is also investigated. An extensive set of measurements based on CDMA-2000 pilot signals using the static antenna and synthetic array are used to experimentally verify these theoretical findings.

1. Introduction

In a wireless mobile communication system, signals propagate from the transmitter to the receiver over multiple paths resulting in multipath fading. When there is no line of sight (LOS) path available from the transmitter and an antenna is located in a dense scattering environment (e.g., indoor and urban environments), the multipath fading appears to be spatially random conforming to Rayleigh statistics [1, 2]. A characteristic of multipath fading is fluctuations in received signal strength as a function of spatial dimensions, which results in signal reception problems for a static antenna. The use of multiple antennas can alleviate the fading problem to some degree by providing a means of diversity gain [3–5]. Diversity techniques are established based on receiving statistically independent signals on each diversity antenna denoted as diversity branch. In practice, this may be implemented by utilizing spatially separated antennas in dense multipath environments, which results in spatial diversity [6–8] or by utilizing antennas with different polarizations that map into polarization diversity [9]. A performance comparison of the spatial diversity and the polarization diversity in Rayleigh fading environment is investigated in [10]. The performance of a diversity system can be characterized by the correlation coefficient value among diversity branches. In the multipath fading environment, the correlation coefficient decreases spatially where the decorrelation rate depends on the geometry of the scatterers and array configuration. If multipath components arrive from a small sector in space, the antenna elements should have spacing on order of several wavelengths of the carrier frequency to yield spatial diversity, whereas in the ring of scatterers multipath model, less than half wavelength spacing is sufficient to ensure spatially uncorrelated samples. Hence, spatial correlation matrix is a function of channel model and antenna spacing. The performance of diversity systems reduces by increasing the correlation coefficient among diversity branches.

As stated before, an antenna array with multiple elements that samples the spatial field with statistically independent samples can be considered as a diversity system. Unfortunately, the physical size of antenna array necessary for achieving a reasonable spatial diversity gain is several carrier wavelengths, which is incompatible with the small form factors of typical handheld receiver devices. To overcome this size restriction, a single antenna can be physically moved while the receiver is capturing the signal, thereby realizing a synthetic antenna array. The mobile station (MS) operator may move the handheld receiver in an arbitrary trajectory or else the MS can be attached to a moving platform such as a vehicle. The concept of a synthetic array based on a single moving antenna has been utilized in wireless signal parameter estimation and radar signal processing [11, 12]. In previous work, antenna trajectories have either been mechanically fixed or precisely measured during the signal snapshot using accurate inertial navigation devices [12, 13] which restricts portability and applicability of implementation in handset receivers.

Authors in [14] have proposed a new method for detecting a narrowband signal in uncorrelated Rayleigh fading based on the synthetic array concept. In [14] the detection performance of a single channel receiver based on a moving antenna is compared to that of the equivalent receiver with a stationary antenna based on the diversity gain as a quantifiable metric. It is shown that if the antenna is held at a fixed position during the snapshot interval, then the signal is not subject to decorrelation. However, the signal will be subject to fading losses, which are statistically large in a Rayleigh fading environment. Conversely, if the antenna is translated along some arbitrary trajectory during the snapshot interval, then the coherency of the signal will be compromised as the channel gain will change randomly but the snapshot data will contain spatial diversity that can effectively counter the spatial fading effects. Based on this, it is shown that the tradeoff between increased diversity gain and loss of signal coherency will result in an optimum processing gain.

The primary assumption of previous work related to the spatial diversity gain was based on uncorrelated spatial signal samples that resulted in the Equal-Gain (EG) combining [14–16]. While this is optimum for uncorrelated Rayleigh fading, it is not optimum when the fading is spatially correlated [17]. However, in the synthetic array context uncorrelated spatial sample assumption is valid when the trajectory of the moving antenna and channel statistics are known. Authors in [18] have taken into account the performance degradation of the synthetic array in the correlated Rayleigh fading.

In this paper, the achievable processing gain of a synthetic array antenna where the single antenna is translated through an arbitrary trajectory of spatially correlated Rayleigh faded signal is analyzed. The net processing gain of the synthetic array antenna based on optimum estimator-correlator (EC) combining is contrasted with that of the static antenna in an equivalently faded propagation environment. The processing gain advantage of the EC is also compared with the EG combining for a range of spatial correlation. The sensitivity of the proposed method in terms of the trajectory estimation is also considered. The environment of interest is indoor where there is no well-defined LOS component and the multipath is diffuse. In addition, the signal bandwidth is narrow such that it is assumed to be less than the coherence bandwidth resulting in unresolved multipath components. The quantitative metric that is used for comparison is the SNR required at the receiver to achieve specific detection performance goals. That is target values of the probability of detection (𝑃_𝑑) and probability of false alarm (𝑃_𝑓𝑎) are fixed. The required SNR to achieve this detection performance is compared for the synthetic array with EG and EC combining and the static antenna. The required SNR is initially determined theoretically based on the assumption of Rayleigh fading. Subsequently, experimental measurements based on CDMA-2000 pilot signals propagated from an outdoor BS and received indoor are utilized to partially validate the theoretical conclusions.

The paper is organized as follows. In Section 2, the static antenna system model is given along with theoretical performances. In Section 3, the synthetic array system model with EC and EG combining is described. In Section 4, the processing gain advantage of the synthetic array based on the EC and EG combining in correlated Rayleigh fading is contrasted with the static antenna. The practical implementation issues of the synthetic array are also given in Section 4. Experimental measurement results are given in Section 5. Section 6 concludes the paper.

2. Static Antenna System Model and Detection Performance

Consider 𝑟(𝑡) as a complex baseband signal received by a single antenna, which is processed to decode between two states, 𝐻₀, where only noise is present, and 𝐻₁ where both signal and noise are present. The representation of 𝑟(𝑡) for a static antenna located at position 𝐩 is

𝑟(𝑡)=𝐴(𝐩)𝑠₀(𝑡)+𝑤(𝑡),(1) where 𝑠₀(𝑡) is the deterministic complex baseband component of the signal that is known to the receiver, and 𝐴(𝐩) is the channel gain as a function of spatial position. The channel gain is assumed to be temporally static during the time interval where the data is collected. However, the channel gain varies randomly with the spatial variable 𝐩 according to the assumed Rayleigh fading. Hence, 𝐴(𝐩) is assumed to be a zero mean circular normal random variable such that 𝐴(𝐩)∼CN(0,𝜎^2𝐴) where ∼ denotes the Probability Density Function (PDF) of the left-hand side variable, and CN(0,𝜎²) signifies a zero mean circular normal PDF with variance 𝜎².(Random variable 𝐵=𝑢+𝑗𝑣 distributed according to circularly normal PDF with zero mean and variance 𝜎², CN(0,𝜎²), where 𝑢 and 𝑣 are independent zero-mean normal random variables each with variance 𝜎²/2, 𝑢∼𝑁(0,𝜎²/2) and 𝑣∼𝑁(0,𝜎²/2).) The received signal is corrupted with additive noise denoted by 𝑤(𝑡), which is assumed to be circularly normal and spectrally white within the signal bandwidth of 𝑠₀(𝑡) with a double sided power spectral density of 𝑁₀/2. The static antenna receiver accumulates a temporal snapshot of 𝑟(𝑡) over the snapshot interval of 𝑡∈[0,𝑇]. Based on these assumptions, the optimal Neyman-Pearson (NP) detection processing is a matched filter based on correlation with 𝑠₀(𝑡)^∗ followed by a magnitude squared operation [17]. This processing results in the decision variable as 𝑧_𝑠 which is expressed as [19]

𝑧_𝑠=||||1𝑇^𝑇0𝑟(𝑡)𝑠₀(𝑡)^∗𝑑𝑡||||²=||𝑥_𝑠||²,(2) where 𝑥_𝑠=(1/𝑇)∫^𝑇0𝑟(𝑡)𝑠₀(𝑡)^∗𝑑𝑡 and subscript 𝑠 signifies static processing case. For convenience the signal energy of 𝑠₀(𝑡) is normalized as (1/𝑇)∫^𝑇0|𝑠₀(𝑡)|²𝑑𝑡=1. The noise at the output of the processor is circularly normal as ∫^𝑇0𝑤(𝑡)𝑠₀(𝑡)^∗𝑑𝑡∼CN(0,𝑇𝑁_𝑜). Based on this it is convenient to define 𝜌 as the average signal-to-noise ratio as

𝜌≡𝜎^2𝐴𝜎^2𝑤=𝑇𝜎^2𝐴𝑁_𝑜,(3) where 𝜎^2𝑤=𝑁_𝑜/𝑇 is the noise variance at the output of the correlation process. This definition will be used throughout the remainder of the paper. Without loss of generality, the normalization of 𝑇𝑁_𝑜=1 can be imposed such that 𝜌=𝑇²𝜎^2𝐴 which simplifies the expressions to follow [14]. Based on the above definitions and normalizations, the PDF of 𝑥_𝑠 conditioned on 𝐻₀ and 𝐻₁ is

𝑥_𝑠∼CN(0,1+𝜌),under𝐻¹CN(0,1),under𝐻₀,.(4)

Consequently, the PDF of 𝑧_𝑠 defined in (2) conditioned on H₀ and H₁ is Chi-Squared central with two degrees of freedom (DOF).(If 𝐵∼CN(0,𝜎²), |𝐵|² is distributed according to 𝑓(|𝐵|²)=1/𝜎²exp(−|𝐵|²/𝜎²).)

Hence [17],

𝑧_𝑠∼⎧⎪⎨⎪⎩𝑒^−𝑧01,under𝐻₀,𝜌+1𝑒^{−𝑧0/(𝜌+1)},under𝐻₁.(5) Assuming that 𝑧_𝑠 is compared with a threshold 𝛾, then the 𝑃_𝑓𝑎 and 𝑃_𝑑 can be determined by [17]

𝑃𝑃_𝑓𝑎=exp(−𝛾),_𝑑=exp−𝛾1+𝜌.(6) Let 𝜌_𝑠 denote the value of 𝜌 for the static antenna that can be expressed explicitly in terms of the given target values of 𝑃_𝑓𝑎 and 𝑃_𝑑 using (6) as

𝜌_𝑠=ln𝑃^𝑓𝑎ln𝑃_𝑑−1.(7)𝜌_𝑠 is the average SNR required to meet the target values of 𝑃_𝑓𝑎 and 𝑃_𝑑 for a static antenna assuming Rayleigh fading.

3. Synthetic Array System Model and Detection Performance

The synthetic array antenna is now considered where the single antenna is moving along an arbitrary trajectory while the snapshot data is being collected. The position vector to the antenna location at time t from the origin is now denoted as 𝐩(𝑡). The signal component of the complex baseband signal 𝑟(𝑡) is written as 𝑠(𝑡,𝐩(𝑡)), which is a function of time, 𝑡, and the antenna position, 𝐩(𝑡), which in turn is a function of 𝑡. The signal is assumed to be narrowband implying that the reciprocal of the maximum delay extent of the antenna trajectory is much larger than the bandwidth of 𝑠₀(𝑡). The signal bandwidth is also small relative to the coherence bandwidth of the propagation channel. Figure 1 illustrates a realization of the synthetic array antenna with an arbitrary trajectory. During the antenna movement, the receiver collects 𝑀 spatial samples, each with coherent integration intervals of Δ𝑇. Note that the 𝑀 spatial samples are taken sequentially.

Figure 1 

Synthetic array model illustrating signal collection over the 𝑚th and (𝑚+1)th subintervals.

The narrowband assumption justifies the decomposition of 𝑠(𝑡,𝐩(𝑡))=𝐴(𝐩(𝑡))𝑠₀(𝑡), which implies that the small delay changes due to 𝐩(𝑡) are insignificant in the context of 𝑠₀(𝑡). It is assumed that the signal snapshot of 𝑟(𝑡) is collected in 𝑀 subintervals, each of duration Δ𝑇. The constraint 𝑇=𝑀Δ𝑇 is imposed such that the static antenna and synthetic array antenna can be compared directly. Define 𝑡_𝑚 as the starting instance of the 𝑚th subinterval that extends over the interval of [𝑡_𝑚,𝑡_𝑚+Δ𝑇] for 𝑚∈[1,2,…,𝑀]. It is assumed that there can be arbitrary time gaps between the subintervals such that 𝑡_𝑚−𝑡_𝑚−1>Δ𝑇. The collection of signal over the 𝑚th and (𝑚+1)th subintervals is illustrated in Figure 1.

As stated earlier, Δ𝑇 is considered to be sufficiently small such that 𝐴(𝐩(𝑡)) can be approximated as constant over the subinterval of Δ𝑇. The signal captured in each subinterval is correlated with 𝑠₀(𝑡) resulting in a set of 𝑀 spatial array samples denoted by 𝑥_𝑚 and given as

𝑥_𝑚=1Δ𝑇^{𝑡𝑚𝑡+Δ𝑇}_𝑚𝑟(𝑡)𝑠₀(𝑡)^∗𝑑𝑡.(8)𝑥_𝑚 is expressed as

𝑥_𝑚=𝐴𝐩𝑡_𝑚𝑠_𝑚+𝑤_𝑚,(9) where

𝑠_𝑚=1Δ𝑇^{𝑡𝑚𝑡+Δ𝑇}_𝑚𝑤||𝑠₀(𝑡)||²𝑑𝑡,_𝑚=1Δ𝑇^{𝑡𝑚𝑡+Δ𝑇}_𝑚𝑤(𝑡)𝑠₀(𝑡)^∗𝑑𝑡.(10)

Based on these assumptions made, it follows that 𝑥_𝑚 forms a set of sufficient statistics of the accumulated snapshot signal in terms of optimal decoding between 𝐻₀ and 𝐻₁. The vector forms of the signals are defined as 𝐱=[𝑥₁,…,𝑥_𝑀]^𝑇, 𝐬=[𝑠₁,…,𝑠_𝑀]^𝑇, 𝐰=[𝑤₁,…,𝑤_𝑀]^𝑇,and 𝐀=[𝐴(𝐩(𝑡₁)),…,𝐴(𝐩(𝑡_𝑀))]^𝑇. With these definitions the detection problem is stated as [14]

𝐱𝐱||_𝐻1𝐻0=𝐀⊙𝐬+𝐰,=𝐰,(11) where ⊙ denotes the Hadamard vector product operator. The noise covariance matrix is denoted as 𝐂_𝐰 and given as

𝐂_𝐰=𝐸𝐰𝐰^𝐻=1𝑀𝐈^𝐌,(12) where 𝐈_𝐌 is an 𝑀×𝑀 identity matrix. The last step follows from the normalization 𝑇𝑁_𝑜=1 and that∫^{𝑡𝑚𝑡+Δ𝑇}_𝑚|𝑠₀(𝑡)|²𝑑𝑡≈𝑇/𝑀=Δ𝑇, which is based on the assumption that the bandwidth of the spreading signal 𝑠₀(𝑡) is much larger than 1/Δ𝑇. The signal covariance matrix is denoted as 𝐂_𝐬 and given as

𝐂_𝐬=𝐸𝐀⊙𝐬(𝐀⊙𝐬)^𝐻=Δ𝑇²𝐂_𝐀,(13) where 𝐂_𝐀 is the covariance matrix of 𝐀. As stated previously, the individual components of the vector 𝐀 are circularly normal such that [𝐀]_𝑚=𝐴(𝐩(𝑡_𝑚))∼CN(0,𝜎^2𝐴). The signal covariance matrix in a Rayleigh fading environment with ring of scatterer models can be defined by [20]

𝐂_𝐬≈𝜎^2𝐴𝑇²𝑀²Ψ=𝜌𝑀²[Ψ]Ψ,_𝑖,𝑗=𝐽₀2𝜋𝜅||𝐩^𝐢−𝐩_𝐣||,(14) where Ψ denotes normalized correlation coefficient matrix which is a function of antenna position, 𝐽₀ is zero-order Bessel function of the first kind, and 𝜅 is the carrier wavelength. Based on (12) and (14), the PDF of 𝐱 conditioned on 𝐻₁ and 𝐻₀ is

𝐱∼⎧⎪⎨⎪⎩CN0,𝐂^𝐬+𝜎^2𝑤𝐈=CN0,𝜌𝑀²Ψ+1𝑀𝐈under𝐻¹CN0,𝜎_2𝑤,𝐈=CN0,1𝑀𝐈under𝐻⁰.(15)

3.1. Optimum Detection Performance of a Narrowband Signal in Rayleigh Fading

The optimal detection processing based on the likelihood Ratio Test (LRT) chooses 𝐻₁ if [17]

𝐿(𝐱)=𝑝𝐱|^𝐻1𝑝𝐱|_𝐻0>𝛾,(16) where 𝑝(𝐱|_𝐻1) and 𝑝(𝐱|_𝐻0) are the conditional PDF’s of 𝐱 given 𝐻₁ and 𝐻₀, respectively. As both 𝐀 and 𝐰 are zero mean multivariate Circular Gaussian random vectors, so is 𝐱. Hence, 𝐿(𝐱) is a function of the covariance matrices of 𝐀⊙𝐬 and 𝐰.

After some manipulation and removing deterministic scaling and additive constants, the LRT reduces to the Estimator-Correlator (EC) formulation [19] resulting in a sufficient statistic given as

𝑧_EC(𝐱)=𝐱^𝐻𝐂_𝐬𝐂_𝐬+𝜎^2𝑤𝐈⁻¹𝐱,(17) where 𝜎^2𝑤 and 𝐂_𝐬 are defined in (15). Since 𝐂_𝐬 is a Hermitian matrix, the eigen-decomposition of 𝐂_𝐬 can be represented as

𝐕^𝐇𝐂_𝐬𝐕=𝚲_𝐬,(18) where 𝐕=[𝐯_𝟏𝐯_𝟐⋯𝐯_𝐌] is the orthogonal matrix of columnwise eigenvectors and Λ_𝐬 is the diagonal matrix of eigenvalues where the mth eigenvalue is denoted by 𝜆_𝑠𝑚. The test statistics in (17) can be shown as [17]

𝑧_EC(𝐲)=^𝑀_𝑚₌₁𝜆_𝑠𝑚𝜆_𝑠𝑚+𝜎^2𝑤||𝑦_𝑚||²,(19) where 𝐲=𝐕^𝐇𝐱. The vector 𝐲=[𝑦₁,𝑦₂,…,𝑦_𝑀]^𝑇 consists of 𝑀 independent circular Gaussian random variables such that [17]

𝐲∼⎧⎪⎨⎪⎩CN0,𝚲^𝐬+𝐈𝑀under𝐻¹CN0,𝐈,𝑀under𝐻⁰.(20) Therefore, the test statistics, 𝑧_EC becomes a scaled factor of Chi-Squared distribution. Figure 2 shows the synthetic array processing model and the canonical form of the EC process.

Figure 2 

Synthetic antenna channel model and subsequent EC processing.

Next consider the calculation of the 𝑃_𝑓𝑎 and 𝑃_𝑑. The characteristic function of 𝑧_EC conditioned on 𝐻₁ and 𝐻₀ is given as [19]

𝜙_𝑧|𝐻1(𝜔)=^𝑀_𝑚=11𝜙_𝑧|𝐻01−𝑗𝛼^𝐻1_𝑚𝜔,(𝜔)=^𝑀_𝑚=111−𝑗𝛼^𝐻0_𝑚𝜔,(21) where

𝛼_𝐻0𝑚𝛼^𝐻1_𝑚=𝜆_𝑠,𝑚,=𝜆^𝑠,𝑚𝑀𝜆_𝑠,𝑚+1,(22)

The values of 𝑃_𝑓𝑎 and 𝑃_𝑑 can be determined by taking the Fourier transform of the characteristic functions [17]. For a given threshold of 𝛾 applied to 𝑧_EC, the following is obtained:

𝑃_𝑑=^∞𝛾^∞−∞𝜙_𝑧|𝐻1(𝜔)𝑒^{−𝑗𝜔𝑧}𝑑𝜔𝑃2𝜋𝑑𝑧,_𝑓𝑎=^∞𝛾^∞−∞𝜙_𝑧|𝐻0(𝜔)𝑒^{−𝑗𝜔𝑧}𝑑𝜔2𝜋𝑑𝑧.(23) A closed form expression for 𝑃_𝑓𝑎 and 𝑃_𝑑 in case of distinctive eigenvalues of 𝐂_𝐬 is given in the appendix.

3.2. Detection Performance of the Equal Gain (EG) Combiner in Rayleigh Fading

As it is shown in (19) the EC emphasizes the stronger signal components corresponding to those with the larger eigenvalues. EC formulation requires approximate knowledge of signal and noise covariance matrices in order to compute the eigenvalues. This may not be applicable in many practical applications. When 𝐂_𝐰 and 𝐂_𝐬 are not known a priori, the EG combiner is a practical suboptimal alternative that may be applied. Note from (19) that the EG combiner becomes asymptotically optimal when the signal becomes more uncorrelated. On the other hand, EG combiner is an optimal approach for signal detection in uncorrelated Rayleigh fading channels. This is because 𝜆_𝑠𝑚/(𝜆_𝑠𝑚+𝜎^2𝑤) in (19) becomes identical for all spatial samples that leads to EG formulation. The test statistics of EG combiner can be represented by [6]

𝑧_EG=𝐱^𝐇𝐱=^𝑀_𝑚₌₁||𝑥_𝑚||².(24) As it is shown in (24) the test statistics of EG combiner is independent of 𝐂_𝐬. To determine the performance of EG in correlated Rayleigh fading, it is convenient to perform the following transformation, which decorrelates the signal covariance matrix [17]:

𝑧_EG==𝐱𝐲^𝐇_𝐇𝐱=𝐱^𝐇𝐕𝐕^𝐇𝐱𝐲=^𝑀_𝑚₌₁||𝑦_𝑚||²,(25) with 𝐲=𝐕^𝐇𝐱 where 𝐕 is the eigenvectors of signal covariance matrix defined earlier. The distribution of 𝐲 is given in (20). Consequently

𝑦_𝑚||∼𝑦CN0,𝜆^𝐻𝑖_𝑚,𝑖=0,1,_𝑚||²∼1𝜆^𝐻𝑖_𝑚𝜒²²𝜆𝑧^𝐻𝑖_𝑚,(26) where 𝜆^𝐻𝑖_𝑚 is the 𝑚th eigenvalue of 𝐂_𝐬 under 𝐻_𝑖 state and 𝜒²² denotes Chi-Squared distribution with two DOF. For a general signal covariance matrix 𝐂_𝐬, the characteristic function of 𝑧_EG conditioned on 𝐻₁ and 𝐻₀ is given as

𝜙_𝑧|𝐻1(𝜔)=^𝑀_𝑚=11𝜙_𝑧|𝐻01−𝑗𝜆^𝐻1_𝑚𝜔,(𝜔)=^𝑀_𝑚=111−𝑗𝜆^𝐻0_𝑚𝜔,(27) where

𝜆^𝐻1_𝑚=𝜆_𝑠𝑚+1𝜆_𝐻0𝑚𝑀,=1𝑀.(28) The values of 𝑃_𝑓𝑎 and 𝑃_𝑑 can be determined by inserting (27) into (23).

In summary the following points are made.

(i)In a correlated signal environment, the optimum detector for a signal in Gaussian noise is the EC, for which the test statistic is given in (19). This procedure requires a priori knowledge of the covariance matrix and signal power. The EC performances were determined by inserting (21) into (23). The closed form expressions of EC performance for distinctive eigenvalues of signal covariance matrix is given in the appendix.(ii)When the multipath fading is such that the channel gains associated with the 𝑀 samples of the synthetic array become uncorrelated, then the EC process reduces to the EG combiner. In this case, the test statistic becomes a random variable with a central Chi-squared PDF with 2𝑀 DOF.(iii)When the multipath fading is such that the channel gains associated with the 𝑀 samples of the synthetic array become fully correlated, then the EC combining reduces to that of a matched filter followed by a magnitude squaring operation. In this case, the test statistic is random with a central Chi-square PDF of two DOF. The test statistic and performance are therefore equivalent to that of the static antenna as represented in (2) and (6).(iv)When 𝐂_𝐬 and 𝐂_𝐰 are unknown, a suboptimal solution is provided by the EG combiner. The performances of the EG combiner in correlated Rayleigh fading were demonstrated by inserting (27) into (23).

4. Analysis of Processing Gains and Practical Implementation Issues

In this section, the detection performances of a receiver using a static antenna and a synthetic array are compared for a Rayleigh multipath fading. The EG and EC combiners are considered for the synthetic array. The covariance matrix of the signal samples 𝐂_𝐬 is assumed to be known. The performance comparison of the static antenna and synthetic array antenna receivers will be achieved by the following approach.

(i)Assume fixed target values for 𝑃_𝑓𝑎 and 𝑃_𝑑(ii)Determine the average SNR 𝜌 required to meet these target objectives. These will be denoted as 𝜌_s, 𝜌_EG, and 𝜌_EC for the static antenna, synthetic array with EG combining, and synthetic array with EC combining, respectively.(iii)The performance advantage of the synthetic array with EG or EC combining relative to the single static antenna is then given as 𝐺_EG=10log(𝜌_𝑠/𝜌_EG), 𝐺_EC=10log(𝜌_𝑠/𝜌_EC), respectively.

While the formulation presented thus far is for an arbitrary number of samples 𝑀 of the synthetic array, the special case of 𝑀=2 is considered in detail as this is compatible with the experimental results given in Section 5. For 𝑀=2, using (14), the signal covariance matrix is given as

𝐂_𝐬=𝜌41r1r,(29) where 𝑟 is the correlation coefficient which is a function of antenna spacing.

The left column of Figure 3 shows the average SNR required to achieve different target values of 𝑝_𝑓𝑎 and 𝑝_𝑑 given the correlation coefficient 𝑟 which varies between zero and one. The right column of Figure 3 also reveals the corresponding gain of the synthetic array processing schemes, 𝐺_EC and 𝐺_EG, from which several significant observations can be made. The required average SNR for a single antenna is significantly higher than that of the synthetic array schemes. For the uncorrelated case, 𝑟=0, the gain is about 4 dB for 𝑝_𝑓𝑎=0.05 and 𝑝_𝑑=0.95. This is essentially a result of the diversity gain possible. Note that there is no suppression of the channel noise for the dual antenna schemes possible when 𝑟=0 as the signal samples emerging from the antennas are uncorrelated. Hence, the observed gain corresponding to 𝑟=0 is strictly diversity gain. When 𝑟 approaches one, the signal components emanating from the pair of antennas are correlated and hence there is no diversity gain. In this case, there is identical gain on the EC scheme with a static antenna. This is a result of the averaging of the pair of uncorrelated noise samples due to the coherent combining of the EC. When high system performance is required, there is a negligible advantage of the EC processing over the EG processing for realistically encountered values of the correlation coefficient. This phenomenon is shown in Figures 3(a) and 3(b). Only when 𝑟 is close to one, there is an advantage in using EC over EG. This has a practical significance in that the parameters 𝜌 and 𝑟 do not have to be estimated by the receiver. By decreasing the probability of detection and increasing the probability of false alarm the advantage of using EC becomes more evident. As it is shown in Figures 3(c), 3(d), 3(e), and 3(f), the achievable gain from EC and EG is identical only when the correlation coefficient 𝑟 is less than 0.8 and 0.7, respectively, for (𝑃_𝑓𝑎=0.1𝑃_𝑑=0.9) and (𝑃_𝑓𝑎=0.15𝑃_𝑑=0.85).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

(a), (c), and (e) required SNR for the stationary and synthetic antenna with EC and EG for target values of 𝑃𝑓𝑎 and 𝑃𝑑; (b), (d), and (f) processing gain of synthetic array with EC and EG combiner.

Figure 3 demonstrated that the performance of the synthetic array with the EC process is better or identical to that of the static antenna for all range of 𝑟. Although the performances of the EG and EC combiners are almost indistinguishable for correlated cases with 𝑟 values of up to about 0.75, in highly correlated situations the performance of the EG combiner becomes worse than that of the static antenna which means that there is performance loss when moving the antenna. This happens when the processing gain of EG combiner shown in Figure 3 goes below zero dB. Thus, it is investigative to determine values of 𝑟 for which zero crossing occurs. Figure 4 shows values of correlation coefficient for a range of target detection performance metrics {𝑃_𝑑,𝑃_𝑓𝑎} for which the detection performance of the synthetic array based on the EG combiner becomes worse than that of the static antenna. By requiring higher detection performance, the zero crossing occurs in higher values of correlation coefficient. As an example when 𝑃_𝑑=0.99 and 𝑃_𝑓𝑎=0.01, the zero crossing happens at 𝑟=0.98, however for 𝑃_𝑑=0.8 and 𝑃_𝑓𝑎=0.1 zero crossing occurs at 𝑟=0.8. Figure 4 shows upper boundaries of correlation coefficient values for which the EG combiner provides diversity gain with respect to the static antenna. Resulting from Figure 4, if the correlation coefficient value is less than 0.8, the synthetic array based on the EG combiner has diversity gain over the static antenna. To satisfy this condition in a Rayleigh fading environment with the ring of scatterers model, pairwise antenna spacing must be greater than 0.15 of the carrier wavelength.

Figure 4 

Values of correlation coefficient where the performance of EG combiner becomes worse than static antenna for a range of target detection performance metrics.

The receiver operating characteristics (ROCs) curves, which are plots of 𝑃_𝑑 as a function of 𝑃_𝑓𝑎 for specific value of 𝜌, are commonly used to demonstrate the performance of a receiver. Figure 5 represents ROC curves versus 𝑟 for given values of SNR (16 dB) and 𝑀=2. By increasing the correlation coefficient 𝑟, the performance reduction due to correlation becomes more apparent. As expected in moderate correlated cases (𝑟<0.75) performance of EC and EG is identical. Figure 5 also shows ROC curves for almost coherent case where 𝑟=0.99. In this case the performance of EC is superior to that of the EG.

Figure 5 

ROC curves for given SNR and 𝑟.

An important implementation factor of the synthetic array is estimating the trajectory of the moving antenna to capture statically independent spatial samples. As it is shown in (14) the signal covariance matrix 𝐂_𝐬 depends only on pair wise distance between spatial sampling points, denoted here as 𝑑, which is determined by the approximate velocity and time interval between samples taken by a moving antenna. Thus, no matter the array shape and configuration, as long as any pairs of spatial samples have appropriate spacing, diversity gain are attainable. This resolves the problem of precise trajectory estimation and array calibration, which are practical implementation difficulties associated with beamforming techniques [12, 20]. However, in practice the receiver requires a rough estimate of the motion velocity such that it can reject highly correlated samples due to insufficient spatial separation. This can be accomplished by implementing a consumer grade accelerometer devices which only needs to estimate the spatial distance between samples [13].

To analyze the sensitivity of the proposed method to a trajectory estimation error, a scenario with two antenna positions is considered. The problem of interest is determining the degradation in processing gain of the synthetic array due to a trajectory estimation error. Since the signal correlation matrix is a function of antenna spacing, considering some errors in the trajectory estimation unit, the processed spatial samples may become correlated. Hence, the EG detector is no longer the optimum process. However, the EG combiner has numerous implementation benefits and it is preferred for implementation in practice. The performances of the EC and the EG combiner for different correlation coefficient values of the Rayleigh channel are compared in Figures 3 and 5, and it was shown for moderate correlation coefficient that the EG combiner is a practical option.

Normalized gain degradation (NGD) can be defined as a designing metric to quantify the performance degradation of a diversity system due to correlated fading. NGD is defined by the percentage of the diversity gain difference between uncorrelated and correlated processes normalized by the diversity gain in the uncorrelated case:

NGD=𝐺^𝑟=0EG−𝐺^𝑟=𝑞_EG𝐺^𝑟=0_EG×100,𝑞∈[0,1],(30) where 𝐺^𝑟=0_EG is the EG combiner processing gain for uncorrelated case.

Figure 6 shows NGD for desired range of 𝑃_𝑑 and correlation coefficient values. As expected, increasing 𝑟 results in performance degradation. This performance reduction is more severe for low values of 𝑃_𝑑. Figure 6 also gives a designing perspective regarding the performance degradation. As an example if a synthetic array diversity system must work within 10% of the maximum diversity gain, the amount of the correlation coefficient, which the diversity system can tolerate, is extracted directly from Figure 6. For nominal detection performances of 𝑃_𝑑=0.98, to have NGD value within 10 percent, the correlation coefficient should be less than 0.45 which means that based on the ring of scatters model the antenna spacing should be more than quarter of the carrier wavelength. Hence, by appropriate design of the unit trajectory estimation, the gain degradation due to a trajectory estimation error can be removed.

Figure 6 

NGD for different values of correlation coefficient and 𝑃𝑑 for 𝑃𝑓𝑎=0.05.

5. Experimental Results

In the previous sections, the theoretical gain of the synthetic array was determined based on the Rayleigh fading model with the assumption that the channel gain, 𝐴(𝐩), is a circular normal random process with respect to the antenna position 𝐩 but temporally static with respect to the snapshot interval. This led to usable expressions for the relative processing gain. It is shown that the processing gain due to synthetic array processing diminishes with increasing correlation coefficient, the latter being a function of antenna spacing and diffuse multipath characteristics. It was also shown that the performance of the optimal EC is almost the same as the performance of the suboptimal EG combiner in cases where the channel gain is moderately correlated. In this section the EG combiner is utilized to validate the theoretical achievements represented in the previous sections. The experimental measurements described in this section attempt to partially validate the application of these assumptions in the context of the synthetic array for indoor environments. The objective of the experimental measurements is to determine processing gain for a selection of typical indoor locations and compare these with corresponding theoretical values. Experimental measurements involving indoor multipath scenarios are generally plagued with the issue of attaining statistical significance.

The experimental measurements are based on the indoor reception of a terrestrial CDMA-2000 pilot signal emanating from an outdoor base station located about 1 km from the indoor location. The pilot channel has no data modulation and consists of only in-phase and quadrature phase pseudonoise (PN) codes. All BSs use the same PN code, distinguished by the different code offsets. The receiver is tuned to capture CDMA signals with a bandwidth of 1.25 MHz modulated by 1.2288 Mchip/s PN sequence with the period of 2¹⁵ chips at 1947.5 MHz [21]. Signals received at the antennas are amplified, filtered, down-converted, and sampled by 10 MHz digitizer board. The detection process in this case can be formulated as a binary multihypothesis test problem where the objective is to determine the correct code phase between received and replica signals and discarding all incorrect code phases under the constraint of a tolerable rate of false detections. The condition where the code phase of the locally generated despreading signal is different from that of the incoming signal corresponds to the 𝐻₀ state where equivalently there is no discernable signal. In this state the 𝑝_𝑓𝑎 can be evaluated. Likewise, if the code phase of the locally generated despreading signal is the same as that of the incoming signal, then it corresponds to an 𝐻₁ state where the signal is present from which the 𝑝_𝑑 can be evaluated. A vertically polarized omnidirectional antenna in horizontal plane was mounted on the linear motion table to capture the CDMA pilot signals. Figure 7 shows a photograph of the data collection environment and measurement equipment used and vertical antenna pattern. The theoretical findings represented in the previous sections were based on the ring of scatterers model which reduced to a signal correlation matrix represented in (14). Although, in general, indoor propagation model may be characterized by a sphere of scatterers model, due to vertical pattern of the antenna shown in Figure 7 the scatterers geometry can be approximately characterized by a ring model.

(a)

(b)

Figure 7 

(a) Data collection environment and measurement equipment. (b) Vertical pattern of the antenna.

As mentioned earlier, the diversity gain results as an effect of the independency of spatially separated samples. Therefore, it is essential to evaluate the channel spatial correlation coefficients. To this end, the spatial covariance matrix is estimated based on all spatial samples on the linear moving table. The measurement results agree fairly well with the theoretical model of ring of scatterers defined in (14). Thus, based on the measurement results, arrival signals onto two spatial samples separated by half of a wavelength are approximately uncorrelated.

The second set of experimental measurements was aimed at producing ROC curves. The results obtained for the static antenna and synthetic array for different antenna correlations, a process controlled by choosing different antenna spacings, were then compared with calculations based on the theoretical expressions for 𝑃_𝑑 and 𝑃_𝑓𝑎 derived earlier. The spatial samples were taken while the antenna was moving at constant speed with a velocity of 0.2 m/s. To evaluate the detection performance of a moving antenna, comprehensive data collections shown in Figure 7 at various locations in the hallway were collected and the conditional PDFs corresponding to the 𝐻₀ and 𝐻₁ states were numerically calculated based on the measured sample set. Figure 8 shows the measured PDFs of test statistics 𝑧_𝑠 under 𝐻₀ and 𝐻₁ for static antenna. For comparison, the PDFs of Chi-Squared central distributions with two DOFs (𝜒²²) are overlaid. Figure 9 shows measured and theoretical PDFs of test statistics for the moving antenna (𝑧_EG) under 𝐻₀ and 𝐻₁ when 𝑀=2 and 𝑑=𝜅/2 where 𝜅 is the carrier wavelength. The theoretical PDFs in Figure 9 are Chi-Squared central distributions with four DOFs (𝜒²⁴). The match with the theoretical Chi-Squared density functions, which results from the uncorrelated Rayleigh fading assumption, is reasonable and verifies the validity of the Rayleigh fading model.

Figure 8 

Numerically calculated PDF under 𝐻0 and 𝐻1 states for a static antenna.

Figure 9 

Numerically calculated PDF under H0 and H1 states for a moving antenna.

Based on the fitting to the Chi-Squared PDFs of Figures 8 and 9 results of (15), it can be shown that 𝜎^2𝐻₁/𝜎^2𝐻₀=((𝜌/𝑀)+1). Hence, the average SNR and 𝜌 can be extracted from the overlaid theoretical PDFs. Based on measurement results, for static and moving antennas, the average SNRs are approximately 𝜌_𝑠=10.2dB and 𝜌_EG=10dB, respectively. Figure 10 shows numerically calculated ROC curves based on measurements for the synthetic array and static antenna. The detection performance of the synthetic array with two statistically uncorrelated sensors is significantly better than that of the static antenna. To determine the synthetic array gain over the static antenna in term of required average SNR, the measured ROC curves of a synthetic array with average SNR of 𝜌_EG=8.6dB are also plotted in Figure 10, which fits to the ROC curves of the static antenna. The average SNR is decreased by adding noise to the process. From this, the synthetic array gain over static antenna can be obtained and is about 𝐺_EG=1.6dB. The target performance of 𝑝_𝑓𝑎=0.15 and 𝑝_𝑑=0.85 is nearly located on the ROC curve of the static antenna, as shown in Figure 10. The theoretical gain of the synthetic array with target values of 𝑝_𝑓𝑎=0.15 and 𝑝_𝑑=0.85 was shown in Figures 3(e) and 3(f). The theoretical gain for the uncorrelated case (𝑟=0) is about 1.6 dB, which matches fairly well with the measurements results.

Figure 10 

Numerically calculated ROC curves in an uncorrelated case.

Figure 11 shows measured PDFs for the synthetic antenna array under 𝐻₁ state for different antenna spacings 𝑑 in terms of carrier wavelength 𝜅. Figure 11 shows that, by decreasing antenna spacing or increasing the correlation coefficient, the mean value of the detection variable decreases. This phenomenon causes losing the diversity gain. Figure 11 also shows that, by increasing the correlation coefficient, the test statistics PDFs moves from a central Chi-Squared with four DOFs 𝜒²⁴ toward a central Chi-Squared with two DOFs 𝜒²², which again results in a lower diversity gain. Another observation is that the measured PDFs fits fairly well with theoretical PDFs for the different correlation coefficients and antenna spacings derived in (26).

Figure 11 

PDFs of the synthetic array under 𝐻1 state for different antenna spacings.

Figure 12 shows ROC curves obtained with different synthetic antenna spacings 𝑑. The corresponding ROC curves for the static antenna are generated in a similar manner based on coherent summations of stationary samples and are also shown in the figure. As expected the synthetic antenna with a synthetic element spacing of 𝜅/2 with approximately uncorrelated samples outperforms all other scenarios. Figure 12 shows that, by decreasing antenna spacing, the performance of the EG combiner degrades due to the correlated signal covariance matrix. It is also shown that the performance of the coherent integration in the static case is almost the same as that of the synthetic antenna array with a spacing of 𝜅/8 which can be directly compared with the theoretical results of Figure 4. By decreasing the antenna spacing to 𝜅/20, the performance worsens, which demonstrates that there is no advantage of using a synthetic array.

Figure 12 

Measured ROC curves for static antenna and synthetic array.

6. Conclusions

The performance enhancement of a synthetic antenna array as compared to a static antenna subjected to a correlated Rayleigh fading environment was demonstrated. A synthetic array of two antennas was selected for a direct comparison of experimental results with theoretical expectations. It was shown that, in such fading environments, a synthetic antenna with two elements for practical target performance values provides a net gain advantage of 4 dB, which is comprised essentially of diversity gain. More spatial samples would provide further gains due to further diversity gains. If the correlation of the signal samples is increased, then the diversity gain diminishes eventually to the point where the moving antenna has no advantage over the static antenna. Of interest was the determination of the performance degradation of the EG combiner due to the correlation with respect to an optimal EC approach. The results show that in moderate correlation environments, the performance of EC and EG combiners is almost identical. Experimental measurements were performed to verify the assumption of the Rayleigh fading and also to confirm the theoretical processing gain. A reasonable agreement between the experimental and theoretical results was observed.

The assumption utilized herein was based on the flat Rayleigh fading where there is not a well-defined LOS signal component and the signal bandwidth is much smaller than the coherence bandwidth of the channel. The results presented herein may be generalized by considering the presence of the LOS signal and resolvable multipath components. In addition, the synthetic array concept presented in this paper may be considered for other antenna diversity systems and combining methods.

Appendix

Closed Form Expression for 𝑃_𝑓𝑎 and 𝑃_𝑑 in Case of Distinctive Eigenvalues of the EC Process

To determine a closed form expression for 𝑃_𝑓𝑎 and 𝑃_𝑑 in case of distinctive eigenvalues of 𝐂_𝐬, using a partial fraction expansion, it can be shown that [17]

^𝑀_𝑚=111−𝑗𝛼^𝐻𝑖_𝑚𝜔=^𝑀_𝑚=1𝐴^𝐻𝑖_𝑚1−𝑗𝛼^𝐻𝑖_𝑚𝜔,𝑖=0,1,(A.1) where

𝐴^𝐻𝑖_𝑚=^𝑀_𝑘_{=1𝑘≠𝑚}11−𝛼^𝐻𝑖_𝑘/𝛼^𝐻𝑖_𝑚,𝑖=0,1,(A.2) where 𝛼^𝐻𝑖_𝑚 is defined in (22). For the general case of 𝑧=∑^𝑀𝑚=1𝛼^𝐻𝑖_𝑚𝑥^2𝑚 where 𝛼^𝐻𝑖_𝑚 are distinct with 𝛼^𝐻𝑖_𝑚>0, and 𝑥^2𝑚 are IID with PDF of Chi-Squared with two DOF, the PDF of 𝑧 is given by

𝑃_𝑧(𝑧)=^{∞𝑀−∞}_𝑚=111−𝑗𝛼^𝐻𝑖_𝑚𝜔𝑒^{−𝑗𝜔𝑧}𝑑𝜔=⎧⎪⎨⎪⎩2𝜋𝑑𝑧_𝑀𝑚₌₁𝐴^𝐻𝑖_𝑚𝛼^𝐻𝑖_𝑚0,𝑒^{−z/𝛼𝐻𝑖𝑚},𝑧𝑧><00,.(A.3)

Thus, the performance of EC can be represented in closed form as

𝑃_𝑓𝑎=^𝑀_𝑚₌₁𝐴^𝐻0_𝑚exp−𝛾𝑃_𝑑𝛼^𝐻0_𝑚,=^𝑀_𝑚₌₁𝐴^𝐻1_𝑚exp−𝛾𝛼^𝐻1_𝑚.(A.4)

List of variables:

𝐩(𝑡):	Position vector of a moving antenna
𝑟(𝑡):	Complex baseband signal received by an antenna
𝑠𝑜(𝑡):	Deterministic component of the received signal
𝐴(𝐩(𝑡)):	Channel response to the signal at the antenna position of 𝐩(𝑡)
𝑤(𝑡):	Additive noise
𝐻0:	State that the received signal and replica are not synchronized
𝐻1:	State that the received signal and replica are synchronized
𝑥𝑚∶	Correlator output of the 𝑚th subinterval
𝜎2𝐴:	Variance of the channel gain
𝜎2𝑤:	Variance of noise
𝐂𝐀:	Spatial covariance matrix of the channel gains
𝐂𝐬:	Signal covariance matrix
𝐂𝐰:	Noise covariance matrix
𝜌𝑠:	Required average SNR for the static antenna
𝜌EC:	Required average SNR for the synthetic array with the EC process
𝜌EG:	Required average SNR for the synthetic array with the EG process
Ψ:	Correlation coefficient matrix
𝑧𝑠:	Test statistics of the static antenna
𝑧EC:	Test statistics of the synthetic array for the EC process
𝑧EG:	Test statistics of the synthetic array for the EG process
𝑇:	Snapshot interval in seconds
Δ𝑇:	Snapshot subinterval in seconds
𝑀:	Number of subintervals
𝐽0:	zero-order Bessel function of the first kind
𝜅:	Carrier wavelength
𝐺EC:	Processing gain of the SA over the static antenna with the EC process
𝐺EG:	Processing gain of the SA over the static antenna with the EG process
𝜙:	Characteristic function
𝜆𝑠:	Eigenvalues of signal covariance matrix.