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

Authors: Ali Broumandan; John Nielsen; Gérard Lachapelle
Journal: International Journal of Antennas and Propagation
 (2009)

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## 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(Pd)
and probability of false alarm (Pfa). 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.


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## Body

## 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(Pd) and probability of false alarm (Pfa) 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 Section2, 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

Considerr(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(1)r(t)=A(p)s0(t)+w(t),
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,σA2) 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 zs which is expressed as [19](2)zs=|1T∫0Tr(t)s0(t)*dt|2=|xs|2,
where xs=(1/T)∫0Tr(t)s0(t)*dt and subscript s signifies static processing case. For convenience the signal energy of s0(t) is normalized as (1/T)∫0T|s0(t)|2dt=1. The noise at the output of the processor is circularly normal as ∫0Tw(t)s0(t)*dt~CN(0,TNo). Based on this it is convenient to define ρ as the average signal-to-noise ratio as(3)ρ≡σA2σw2=TσA2No,
where σw2=No/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 TNo=1 can be imposed such that ρ=T2σA2 which simplifies the expressions to follow [14]. Based on the above definitions and normalizations, the PDF of xs conditioned on H0 and H1 is(4)xs~{CN(0,1+ρ),underH1,CN(0,1),underH0.Consequently, the PDF ofzs 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],(5)zs~{e-z0,underH0,1ρ+1e-(z0/(ρ+1)),underH1.
Assuming that zs is compared with a threshold γ, then the Pfa and Pd can be determined by [17](6)Pfa=exp(-γ),Pd=exp(-γ1+ρ).
Let ρs denote the value of ρ for the static antenna that can be expressed explicitly in terms of the given target values of Pfa and Pd using (6) as(7)ρs=ln(Pfa)ln(Pd)-1.ρs is the average SNR required to meet the target values of Pfa and Pd 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 timet 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 themth and (m+1)th  subintervals.The narrowband assumption justifies the decomposition ofs(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 tm as the starting instance of the mth subinterval that extends over the interval of [tm,tm+ΔT] for m∈[1,2,…,M]. It is assumed that there can be arbitrary time gaps between the subintervals such that tm-tm-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 xm and given as(8)xm=1ΔT∫tmtm+ΔTr(t)s0(t)*dt.xm is expressed as(9)xm=A(p(tm))sm+wm,
where(10)sm=1ΔT∫tmtm+ΔT|s0(t)|2dt,wm=1ΔT∫tmtm+ΔTw(t)s0(t)*dt.Based on these assumptions made, it follows thatxm 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,…,xM]T, s=[s1,…,sM]T, w=[w1,…,wM]T,and A=[A(p(t1)),…,A(p(tM))]T. With these definitions the detection problem is stated as [14](11)x|H1=A⊙s+w,x|H0=w,
where ⊙ denotes the Hadamard vector product operator. The noise covariance matrix is denoted as Cw and given as(12)Cw=E[wwH]=1MIM,
where IM is an M×M identity matrix. The last step follows from the normalization TNo=1 and that∫tmtm+ΔT|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 Cs and given as(13)Cs=E[A⊙s(A⊙s)H]=ΔT2CA,
where CA 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(tm))~CN(0,σA2). The signal covariance matrix in a Rayleigh fading environment with ring of scatterer models can be defined by [20](14)Cs≈σA2T2M2Ψ=ρM2Ψ,[Ψ]i,j=J0(2πκ|pi-pj|),
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(15)x~{CN(0,Cs+σw2I)=CN(0,ρM2Ψ+1MI)underH1,CN(0,σw2I)=CN(0,1MI)underH0.


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

The optimal detection processing based on the likelihood Ratio Test (LRT) choosesH1 if [17](16)L(x)=p(x|H1)p(x|H0)>γ,
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(17)zEC(x)=xHCs(Cs+σw2I)-1x,
where σw2 and Cs are defined in (15). Since Cs is a Hermitian matrix, the eigen-decomposition of Cs can be represented as(18)VHCsV=Λs,
where V=[v1v2⋯vM] is the orthogonal matrix of columnwise eigenvectors and Λs is the diagonal matrix of eigenvalues where the mth eigenvalue is denoted by λsm. The test statistics in (17) can be shown as [17](19)zEC(y)=∑m=1Mλsmλsm+σw2|ym|2,
where y=VHx. The vector y=[y1,y2,…,yM]T consists of M independent circular Gaussian random variables such that [17](20)y~{CN(0,Λs+IM)underH1,CN(0,IM)underH0.
Therefore, the test statistics, zEC 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 thePfa and Pd. The characteristic function of zEC conditioned on H1 and H0 is given as [19](21)ϕz|H1(ω)=∏m=1M11-jαmH1ω,ϕz|H0(ω)=∏m=1M11-jαmH0ω,
where(22)αmH1=λs,m,αmH0=λs,mMλs,m+1,The values ofPfa and Pd can be determined by taking the Fourier transform of the characteristic functions [17]. For a given threshold of γ applied to zEC, the following is obtained:(23)Pd=∫γ∞∫-∞∞ϕz|H1(ω)e-jωzdω2πdz,Pfa=∫γ∞∫-∞∞ϕz|H0(ω)e-jωzdω2πdz.
A closed form expression for Pfa and Pd in case of distinctive eigenvalues of Cs 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 Cw and Cs 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 λsm/(λsm+σw2) in (19) becomes identical for all spatial samples that leads to EG formulation. The test statistics of EG combiner can be represented by [6](24)zEG=xHx=∑m=1M|xm|2.
As it is shown in (24) the test statistics of EG combiner is independent of Cs. 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]:(25)zEG=xHx=xHVVHx=yHy=∑m=1M|ym|2,
with y=VHx where V is the eigenvectors of signal covariance matrix defined earlier. The distribution of y is given in (20). Consequently(26)ym~CN(0,λmHi),i=0,1,|ym|2~1λmHiχ22(zλmHi),
where λmHi is the mth eigenvalue of Cs under Hi state and χ22 denotes Chi-Squared distribution with two DOF. For a general signal covariance matrix Cs, the characteristic function of zEG conditioned on H1 and H0 is given as(27)ϕz|H1(ω)=∏m=1M11-jλmH1ω,ϕz|H0(ω)=∏m=1M11-jλmH0ω,
where(28)λmH1=λsm+1M,λmH0=1M.
The values of Pfa and Pd 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 theM 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 theM 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)
WhenCs and Cw 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).


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

The optimal detection processing based on the likelihood Ratio Test (LRT) choosesH1 if [17](16)L(x)=p(x|H1)p(x|H0)>γ,
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(17)zEC(x)=xHCs(Cs+σw2I)-1x,
where σw2 and Cs are defined in (15). Since Cs is a Hermitian matrix, the eigen-decomposition of Cs can be represented as(18)VHCsV=Λs,
where V=[v1v2⋯vM] is the orthogonal matrix of columnwise eigenvectors and Λs is the diagonal matrix of eigenvalues where the mth eigenvalue is denoted by λsm. The test statistics in (17) can be shown as [17](19)zEC(y)=∑m=1Mλsmλsm+σw2|ym|2,
where y=VHx. The vector y=[y1,y2,…,yM]T consists of M independent circular Gaussian random variables such that [17](20)y~{CN(0,Λs+IM)underH1,CN(0,IM)underH0.
Therefore, the test statistics, zEC 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 thePfa and Pd. The characteristic function of zEC conditioned on H1 and H0 is given as [19](21)ϕz|H1(ω)=∏m=1M11-jαmH1ω,ϕz|H0(ω)=∏m=1M11-jαmH0ω,
where(22)αmH1=λs,m,αmH0=λs,mMλs,m+1,The values ofPfa and Pd can be determined by taking the Fourier transform of the characteristic functions [17]. For a given threshold of γ applied to zEC, the following is obtained:(23)Pd=∫γ∞∫-∞∞ϕz|H1(ω)e-jωzdω2πdz,Pfa=∫γ∞∫-∞∞ϕz|H0(ω)e-jωzdω2πdz.
A closed form expression for Pfa and Pd in case of distinctive eigenvalues of Cs 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 Cw and Cs 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 λsm/(λsm+σw2) in (19) becomes identical for all spatial samples that leads to EG formulation. The test statistics of EG combiner can be represented by [6](24)zEG=xHx=∑m=1M|xm|2.
As it is shown in (24) the test statistics of EG combiner is independent of Cs. 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]:(25)zEG=xHx=xHVVHx=yHy=∑m=1M|ym|2,
with y=VHx where V is the eigenvectors of signal covariance matrix defined earlier. The distribution of y is given in (20). Consequently(26)ym~CN(0,λmHi),i=0,1,|ym|2~1λmHiχ22(zλmHi),
where λmHi is the mth eigenvalue of Cs under Hi state and χ22 denotes Chi-Squared distribution with two DOF. For a general signal covariance matrix Cs, the characteristic function of zEG conditioned on H1 and H0 is given as(27)ϕz|H1(ω)=∏m=1M11-jλmH1ω,ϕz|H0(ω)=∏m=1M11-jλmH0ω,
where(28)λmH1=λsm+1M,λmH0=1M.
The values of Pfa and Pd 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 theM 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 theM 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)
WhenCs and Cw 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 samplesCs 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 forPfa and Pd(ii)
Determine the average SNRρ required to meet these target objectives. These will be denoted as ρs, ρEG, and ρEC 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 asGEG=10log(ρs/ρEG), GEC=10log(ρs/ρEC), respectively.While the formulation presented thus far is for an arbitrary number of samplesM 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(29)Cs=ρ4[1rr1],
where r is the correlation coefficient which is a function of antenna spacing.The left column of Figure3 shows the average SNR required to achieve different target values of pfa and pd 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, GEC and GEG, 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 pfa=0.05 and pd=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 (Pfa=0.1Pd=0.9) and (Pfa=0.15Pd=0.85).(a), (c), and (e) required SNR for the stationary and synthetic antenna with EC and EG for target values ofPfa and Pd; (b), (d), and (f) processing gain of synthetic array with EC and EG combiner.
(a)(b)(c)(d)(e)(f)Figure3 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 {Pd,Pfa} 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 Pd=0.99 and Pfa=0.01, the zero crossing happens at r=0.98, however for Pd=0.8 and Pfa=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 ofPd as a function of Pfa 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 andr.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 Cs 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 Figures3 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:(30)NGD=(GEGr=0-GEGr=qGEGr=0)×100,q∈[0,1],
where GEGr=0 is the EG combiner processing gain for uncorrelated case.Figure6 shows NGD for desired range of Pd and correlation coefficient values. As expected, increasing r results in performance degradation. This performance reduction is more severe for low values of Pd. 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 Pd=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 andPd  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 of215 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 pfa 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 pd 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) Data collection environment and measurement equipment. (b) Vertical pattern of the antenna.
(a)(b)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 forPd and Pfa 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 zs 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 (zEG) 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 (χ42). 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 underH0 and H1 states for a static antenna.Figure 9
Numerically calculated PDF underH0 and H1 states for a moving antenna.Based on the fitting to the Chi-Squared PDFs of Figures8 and 9 results of (15), it can be shown that σ2H1/σ2H0=((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 ρs=10.2dB and ρEG=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 ρEG=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 GEG=1.6dB. The target performance of pfa=0.15 and pd=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 pfa=0.15 and pd=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.Figure11 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 χ42 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 underH1 state for different antenna spacings.Figure12 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.



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*Source: 610109-2009-10-15.xml*