Chapter Two Sensor Array Systems An array of sensors, distributed over a horizontal plane surface, is used to receive a propagating wavefield with the following objectives: 1) To localize a source. 2) To receive a message from a distant source. 3) To image a medium through which the wavefield is propagating. In this chapter we shall study the basic structure of a sensor array system and in the sequel learn how the above objectives are achieved. The most commonly used array geometries are uniform linear array (ULA) and uniform circular array (UCA). A uniform planar array (UPA) where sensors are placed on an equispaced rectangular grid is more common in large military phased array systems. A wavefront which propagates across the array of sensors is picked up by all sensors. Thus, we have not one but many outputs which constitute an array signal. In the simplest case, all components of the array signals are simply delayed replicas of a basic signal waveform. In the worst case, individual sensor outputs are strongly corrupted with noise and other interference, leaving a very little resemblance among them. Array processing now involves combining all sensor outputs in some optimal manner so that the coherent signal emitted by the source is received and all other inputs are maximally discarded. The aperture of an array, that is, the spatial extent of the sensor distribution, is a limiting factor on resolution. However, the aperture can be synthetically increased by moving a source or sensor. The synthetic aperture concepts are extensively used in mapping radars and sonars. In this chapter we concentrate on sensor array systems which will form the basic material for the subsequent chapters. §2.1 Uniform Linear Array (ULA): 2.1.1 Array Response: Consider a plane wavefront, having a temporal waveform f(t) incident on a uniform linear array (ULA) of sensors (see fig. 2.1) at an angle θ. In signal processing literature the angle of incidence is also known as direction of arrival (DOA). Note that the DOA is always measured with respect to the normal to array aperture, while another related quantity azimuth, which was introduced in chapter 1, is measured with respect to the x- axis, independent of array orientation. In this work θ stands for DOA and ϕ stands for azimuth. We shall assume that a source emits a stationary stochastic signal f(t). Let f m (t), m=0, 1, 2,..., M-1 be the outputs of the sensors. The signal arrives at successive sensors with an incremental delay. The output of the Broadside f(t) θ wave vector ULA 7 2 1 0 Reference sensor Figure 2.1: Uniform linear array of sensors. Note the convention of sensor indexing. The left most sensor is the reference sensor with respect to which all time delays are measured. = = −∆ first sensor is f 0 (t) f (t), the output of the second sensor is f 1 (t) f (t t) th = − ∆ and so on. Thus, the output of the m sensor is f m (t) f (t m t). Some times it is convenient to represent the sensor output in the frequency domain ∞ md 1 jω(t − sin θ) f (t) = dF(ω)e c (2.1) m π ∫ 2 −∞ where we have used the spectral representation of a stationary stochastic process [1]. The simplest form of array signal processing is to sum all sensor outputs without any delay. ∞ M −1 M −1 − ω md θ 1 1 ω 1 j sin g(t) = ∑ f (t) = dF(ω)e j t ∑e c m π ∫ M m=0 2 −∞ M m=0 ∞ 1 ω = dF(ω)H(ωτ)e j t (2.2) π ∫ 2 −∞ d where H(ωτ) is the array response function, τ= sin θ, and d is sensor c spacing. The array response function for a ULA is given by M M −1 md sin( ωτ) M −1 1 jω sin θ j ωτ ωτ = c = 2 2 H( ) ∑e ωτ e (2.3a) M m=0 M sin 2 When the sensor output is weighted with complex coefficients, am , m=0,1,...,M-1, the array response becomes − md 1 M 1 jω sin θ ωτ = c H( ) ∑ame (2.3b) M m=0 A few samples of the frequency response function (magnitude only) are shown in fig. 2.2 for different values of M, that is, array size. The response function is periodic with a period 2π . The maximum occurs at ωτ = 2nπ. The peak at n=0 is known as the main lobe and other peaks at n =±1, ±2,... are known as grating lobes. Since the magnitude of the array response is plotted, the period becomes π as seen in fig. 2.2. The grating lobes can be avoided if we restrict the range of ωτ to ±π, that is, at a fixed frequency the direction of arrival d 1 π must satisfy the relation sin θ≤ . For θ in the interval ± this λ 2 2 d 1 requirement is satisfied if ≤ . If the range of θ is reduced it is possible to λ 2 π π increase the sensor spacing, for example, for − ≤θ+ the sensor spacing 4 4 d 1 need satisfies the constraint ≤ . The phase of the frequency response is a λ 2 linear function of ωτ. This useful property of a ULA is lost when the sensors are nonuniformly spaced (see p. 94). The array response is a function of the product of frequency ω and d delay τ or, more explicitly, ω sin θ . The implication of this dependence is λ that two wavefronts whose waveform is a simple sinusoid but with different ω ω θ θ frequencies ( 1, 2 )arriving at different angles ( 1, 2 ) will produce identical ω θ =ω θ array response if 1 sin 1 2 sin 2 . We shall discuss later such ambiguity issues when we look into the broadband beamformation. The response function has a main lobe which is surrounded by many sidelobes of decreasing magnitude just as we find in spectral windows. The first zero is at 1 M=8 0.8 0.6 0.4 0.2 0 1 0.8 M=64 0.6 0.4 0.2 Response (magnitude) 0 1 0.8 M=128 0.6 0.4 0.2 0 -6 -4 -2 0 2 4 6 Radians Figure 2.2: Array response function (magnitude) for different values of M. Notice that the main lobe becomes sharper as the array size is increased. − λ θ = sin 1 (2.4) zero Md which, for large M, becomes inversely proportional to the array length expressed in terms of wavelength. The first sidelobe is 13.5 dB below the main lobe. It is well known that both width of the main lobe and magnitude of the sidelobes can be controlled by using a suitable weight function as in spectrum analysis [1]. 2.1.2 Array Steering: We have seen that the array response is maximum when the direction of arrival (DOA) is on broad side ( θ=0). The maximum, however, can be changed to any direction through a simple act of introducing a time delay to each sensor output before summation. This is known as array steering. Let an incremental delay of τ per channel be introduced. The sum output of the array is now given by M −1 = 1 + τ g(t) ∑ f m (t m ) M m=0 ∞ M −1 τ−d θ ω 1 ω 1 j( sin 0 ) m = dF(ω)e j t ∑e c (2.5) π ∫ 2 −∞ M m=0 ∞ 1 d ω = dF(ω)H((τ− sinθ )ω)e j t π ∫ 0 2 −∞ c d where we have assumed that the DOA is θ . Let τ= sin θ. Then the array 0 c θ=θ response is maximum whenever 0 . We say that the array is steered in the θ direction 0 , that is, in the direction of arrival of the incident wavefront. The array response is now a function of DOA. This is demonstrated in fig. 2.3. It is interesting to note that the width of the main lobe increases with increasing DOA. To further understand this broadening effect we shall study the array d response function around its maximum, that is, at τ= sin θ . The first zero c 0 will occur at M d d ω sinθ −ω sin(θ −∆θ) =π (2.6a) 2 c 0 c 0 Upon simplifying (2.6a) we get an equation λ sinθ − sin(θ −∆θ) = (2.6b) 0 0 Md whose solution is given by λ ∆θ = θ − −1 θ − 0 sin sin 0 (2.6c) Md The dependence of ∆θ on the DOA for different array sizes is illustrated in fig. 2.4. The broadening of the main lobe is due to reduction in the array aperture for a wavefront which is incident away from the broadside. The response is maximum whenever d ω (sinθ − sinθ) = 2πn c 0 or d (sinθ − sinθ) = n λ 0 d 1 d 1 For ≤ , the acceptable solution is θ=θ for which n=0. For > there λ 2 0 λ 2 d is more than one solution, one for each grating lobe. For example, let = 1; a λ θ= −1 θ − ± solution of sin (sin 0 n) exists only for n=0 and 1. d k Now, let τ= . The array response function can be written as a c M 2πd discrete Fourier transform of a complex sinusoid, exp(− j sin θ m), λ 0 π π M −1 − 2 d θ 2 km d 1 j sin 0 m j H(k) = ∑e λ e M λ (2.7) M m=0 Now H(k) is the kth discrete Fourier transform coefficient which should − k correspond to the array response at a steering angle, sin 1 ( ).