Array Basics

For clarity in the paper we will restrict out attention to a linear array of isotropic antenna elements placed along the x-axis. An extension to two and three dimensions is given in [6]. The angle $\theta$ is measured with respect to the x-axis, with zero degrees lying perpendicular to the axis. The constant $c$ represents the velocity of radiation in free space.

Narrowband Array Pattern Response

The narrowband equation for the far-field pattern of a linear array of isotropic elements located at positions $\{x_{k}$} is usually given as

$$A(\theta )=\sum _{k}a_{k}e^{-j2\pi \frac{x_{k}}{\lambda }\sin (\theta )}$$ (1)

where $\lambda$ is the radiating wavelength. The complex coefficients $\{a_{k}\}$ are chosen to steer the beam in the desired direction and to control sidelobes. This equation has the form of a Fourier transform to the variable $\sin (\theta )$, often denoted as $u$. When $x_{k}=kd$, then (1) can be written

$$A(u)=\sum _{k}a_{k}e^{-j2\pi k\left( \frac{d}{\lambda }u\right) },$$ (2)

which is a discrete-time Fourier transform from index k to normalized frequency $\nu =ud/\lambda$. In the common case where $d=\lambda /2$, $\theta \in [-90^{\circ },90^{\circ }]$ corresponds to $\nu \in [-0.5,0.5]$, and the semicircle of physical angles maps exactly to one period of the Fourier transform response. If $d<\lambda /2$ the semicircle maps to less than a full period, and thus there exists a range of values of $\nu$ that do not correspond to any physical angle, and the transform response in that region (the nonvisible sidelobes) does not directly affect the array pattern (more on this later). If $d>\lambda /2$ the semicircle maps to greater than one period of the transform response. This is usually undesirable, since any part of the array pattern corresponding to a single period of the Fourier transform completely determines the rest of the pattern, leading to grating lobes at high angles.

In all three cases the design of the coefficients $\{a_{k}\}$ is directly analogous to the design of an FIR filter over an appropriate range of frequencies. FIR filter design is a mature subject, and the design technique of interest here is convex programming. Convex-optimization-based approaches to the design of narrowband array patterns and FIR responses can be found in [6,7,8,9,10,2,1,11,12,13].

Extending To Wideband

The above analysis is valid for a single wavelength $\lambda$, but in the wideband case $\lambda$ will cover a range of wavelengths. Thus we wish to consider the array pattern as a function of radiating frequency as well as angle. Substituting $\lambda =c/f$ in (1) results in

$$A(u,f)=\sum _{k}a_{k}e^{-j2\pi \frac{fx_{k}}{c}u},$$

and increasing or decreasing the radiating frequency has the effect of scaling the beam in angle. When the beam is not pointed to zero angle (boresight), this results in the beam changing both in width and direction with frequency, an effect known informally as beam squint. The solution to this problem is to replace complex weight $a_{k}$ with frequency response $H_{k}(f)$, so that the effective array weighting can vary over the bandwidth of interest. This specializes to the conventional time-delay array architecture in which

$$H_{k}(f)=b_{k}e^{j2\pi \frac{fx_{k}}{c}u_{0}},$$

with $\{b_{k}\}$ a set of real weights and $u_{0}=\sin (\theta _{0})$ the desired pointing direction. This choice for $H_{k}(f)$ corrects for the delay of a plane wave arriving from angle $\theta _{0}$ as seen by element $k$ (relative to the origin). The resulting array response is

$$A(u,f)=\sum _{k}b_{k}e^{-j2\pi \frac{fx_{k}}{c}(u-u_{0})},$$ (3)

which at any frequency $f$ is just the response of the narrowband boresight beam defined by the coefficients $\{b_{k}\}$ but scaled by $f$ and shifted in $u$ to the desired angle. Thus the beam direction problem is alleviated, but the scaling by $f$ results in an array pattern that compresses (becomes narrower) as the frequency increases. The ratio of the maximum to the minimum size (measured in $u)$ of a given feature is proportional to the ratio of high to low radiating frequency.

DSP Realization

We consider here primarily receive arrays (although analysis of transmit arrays is essentially identical) since the transmit power requirements of modern radar systems usually limits the transmit beamforming choice to straight time delay with a uniform weighting across the elements. There are several functionally equivalent ways to implement a wideband digital receive array, but for our purposes we consider a simplified model in which the bandpass signal at each antenna element is sampled and converted to an analytic (spectrally one-sided) bandpass digital signal prior to digital beam combining at the RF frequency. In practice the downconversion step may occur before or after beamforming, which primarily affects whether the beamforming filters have real coefficients and operate at the initial sampling rate (before downconversion) or have complex coefficients and operate at a lower sampling rate (after downconversion).

If we consider the array pattern formed using FIR filters at each element, the result is

$$\begin{array}{ll} \displaystyle A(u,f) & =\sum _{k}H_{k}(f)e^... ...n}h_{k,n}e^{-j2\pi nfT}e^{-j2\pi \frac{fx_{k}}{c}u} \end{array}$$ (4)

At all frequencies and angles $A(u,f)$ is linear in the filter coefficients $\{h_{k,n}\}$, which conveniently permits a great number of common constraints to be expressed in terms of convex functions of the coefficients as the optimization variables. This in turn allows design of the array pattern using convex optimization tools[1,2,7]. In the common symmetric-array case, for which $x_{-k}=-x_{k}$, the additional requirement that $H_{-k}(f)=H^{*}_{k}(f)$, or its equivalent $h_{-k,-n}=h^{*}_{k,n}$, ensures that $A(u,f)$ is real-valued (a linear-phase response). This not only reduces by one half the number of variables to optimize, but can be exploited to reduce real-time computation requirements as well. Unless a specific (nonlinear) phase response is desired, these significant benefits come with no ill effects.