Design Approach

In this section, we demonstrate the flexibility of a convex-optimization approach to array pattern design through a series of examples. The goal here is not the realization of an actual design so much as it is to develop insight into the design issues associated with directly optimizing the wideband array pattern. We first discuss an example design in which the beam is steered with ideal time delays and then a second in which those delays are approximated with individually optimized FIR filters. Then we present two designs in which the FIR coefficients are jointly optimized to directly tailor the array function. In the first of these, the main-beam characteristics of the ideal time-delay design are essentially duplicated but with a performance gain in the sidelobes resulting from the joint optimization. The final example, which represents our recommended approach, illustrates the idea of really tailoring the constraints to the application to take advantage of the flexible control over the pattern afforded by the optimization approach.

The system parameters are the same for all cases except the first, which uses ideal time delays instead of FIR filters. The RF center frequency is at $1.25$ GHz, the system bandwidth is $400$ MHz, and the data is sampled at a rate of $1$ GHz. The array is composed of 15 identical isotropic elements, each feeding a 21-tap, real-coefficient, nonlinear-phase FIR filter. The filters obey the symmetry discussed above, so that $H_{0}(f)$ is itself linear-phase and $H_{-k}(f)=H_{k}^{*}(f)$ for $k=1,\ldots ,7$. The spacing between the elements is set to one half wavelength at the highest in-band frequency of $1.45$ GHz in order to suppress grating lobes at lower frequencies. In each case a pattern is designed with a center at $45^{\circ }$ and a maximum sidelobe height of $-25$ dB.

Ideal Time Delay

As seen previously, when ideal time delays are used, wideband pattern synthesis reduces to the narrowband case, with each element's delayed waveform receiving a single real weight. The result is a wideband pattern which at each frequency has an angular response of the form of the narrowband pattern, scaled by the frequency. In classic narrowband pattern synthesis, two classic approaches stand out. The first is Chebychev beam design, which seeks the narrowest beamwidth for a given sidelobe level, or alternatively the lowest sidelobe level for a given beamwidth. In practice a Chebychev design is usually easily improved upon, since a small increase in the close-in sidelobes can often lead to large reductions in the outer sidelobes [7]. Taylor [14] proposed a now-classic alternative: require (nearly) equiripple sidelobes close to the mainlobe, with monotonic falloff of sidelobe peaks outside of some interval. The resulting continuous Taylor distribution and its sampled version are often used in arrays.

In the spirit of Taylor's approach, but to facilitate comparison with later examples, we can design a narrowband beam according to:

minimize	$$\int _{\theta \in \Theta _{\mathrm{sb}}}\left\vert A(\theta )\right\vert ^{2}d\theta$$
subject to	$$\left\vert A(\theta )\right\vert \leq 10^{-25/20}$$,    $\theta \in \Theta _{\mathrm{sb}}$
	$$A(0)=1$$

where $\Theta _{\mathrm{sb}}$ denotes the stopband or sidelobe region. Since the coefficients are real, $A(\theta )$ and $\Theta _{\mathrm{sb}}$ are symmetric about zero degrees. The boresight pattern thus optimized has the lowest sidelobe energy for the given maximum sidelobe level and beamwidth, however this is no longer true once the pattern has been shifted by $u_{0}$ as in (3). These weights can then be used along with ideal time delays to form a wideband array pattern, as illustrated in the top half of fig. 1, which shows the resulting pattern as a two-dimensional function of $\theta$ and $f$ in addition to one-dimensional slices along both dimensions.

Two features of this response are especially notable. The first is that the beam does in fact narrow with increasing frequency, a consequence of the scaling by $f$ in the wideband equation. Moreover, the compression effect and the main beam itself are not symmetric about the center of the beam--the higher angles compress more rapidly. This is because the quantity scaled by $f$ is $u=\sin (\theta )$, and not $\theta$ itself, and at higher angles $\theta$ is a more sensitive function of $u$. The second notable feature is that the frequency response at off-center angles rolls off at higher frequencies, a direct consequence of the beam-narrowing effect. This rolloff becomes more pronounced at angles further from the center of the beam. In some applications this rolloff may be unacceptable, and so in a later example (the last one) we will consider ways to remove it.

The primary advantage of time-delay beamforming is that it only requires the design of a single narrowband weighting function, which is used for all steering directions. The disadvantages are the high-frequency response rolloff and the suboptimality of the antenna pattern, as well as the difficulty in obtaining high-precision, high-resolution delay elements.

FIR Time Delay Approximation

Digital beamsteering architectures replace traditional switched analog delay lines with digital filters to approximate arbitrary time delays. One can either use a library of filter weights corresponding to the needed set of closely spaced delays, or one can use a single filter tunable in delay [15]. The optimization is more easily discussed assuming the fixed delays of a library, and so that is the approach taken here, but similar strategies could also be used to improve a tunable delay filter.

A straightforward design approach for a minimum mean square error FIR filter approximating delay $\tau$ is

minimize

$$\int _{f\in \mathcal{F}_{\mathrm{pb}}}\left\vert H(f)-e^{-j2\pi f\tau }\right\vert ^{2}df$$,

where $\mathcal{F}_{\mathrm{pb}}$ is the passband region. The bottom half of fig. 1 shows the array pattern that results from using a set of delay filters so designed. Amplitude ripple has been introduced into the frequency response in the main beam, and some fine sidelobe structure differs, but otherwise the pattern resembles that of the ideal case. The approximation errors in the delay filters determine both the size of the amplitude ripples in the main beam and the fidelity with which the pattern's sidelobe structure duplicates that of the ideal frequency-scaled narrowband pattern. With this architecture then, the main-beam and sidelobe error structures cannot be independently controlled or traded off against each other. The overall pattern is also suboptimal since the delays are designed independently and not jointly.

Direct Beam Optimization

While it is true that having each filter approximate a delay results in a useful pattern in angle-frequency space, there are many other filter combinations that would presumably do so as well, and since there is no reason to presume that the time-delay approach results in the best one, it amounts to an inherent structural restriction in the design that holds that design away from optimality. Here we consider the much more flexible approach of directly constraining the wideband array pattern. By only enforcing constraints that are meaningful for a given application and avoiding inherent structural restrictions, we obtain better designs.

The following optimization setup was used for the last two examples:

min.	$$\int _{f\in \mathcal{F}_{\mathrm{pb}}}\int _{\theta \in \Theta _{\mathrm{sb}}(f)}\left\vert A(\theta ,f)\right\vert ^{2}d\theta df$$
s.t.	$$\left\vert A(\theta ,f)\right\vert \leq 10^{-25/20}$$,   $\theta \in \Theta _{\mathrm{sb}}(f), f\in \mathcal{F}_{\mathrm{pb}}$
	$$\int _{f\in \mathcal{F}_{\mathrm{pb}}}\left\vert A(\theta _{m},f)-\beta _{m}\right\vert ^{2}df\leq \delta , m=1,\ldots ,M$$
	$$\frac{1}{M}\sum _{m}\beta _{m}=1$$
	$$\sum _{k}\left\Vert h_{k}\right\Vert ^{2}\leq \epsilon$$

Here we seek to minimize the total energy in the in-band sidelobe region defined by $\mathcal{F}_{\mathrm{pb}}$ and $\Theta _{\mathrm{sb}}$ (which may be a function of frequency), subject to several constraints. The first sets a maximum sidelobe level of $-25$ dB for all in-band frequencies and for angles in $\Theta _{\mathrm{sb}}$. The second and third constraints control the mainlobe frequency response in the passband. The second constraint introduces auxiliary variables $\{\beta _{m}\}$, which represent desired gain levels of the array pattern at angles $\{\theta _{m}\}$ in the main beam. The square passband error at each angle $\theta _{m}$ is limited to some constant $\delta .$ By using auxiliary variables, the desired gain at each angle is allowed to float rather than being fixed. The third constraint then fixes the average gain of the main beam, which is needed to prevent the optimization from returning all zeros.

The last constraint limits the white-noise gain of the set of filters to a constant $\epsilon$. Why might this constraint be needed? Recall from before that when $d<\lambda /2$ for a narrowband array, part of the DTFT of the array coefficients does not map to any physical angles. Since $d$ here is half of the smallest wavelength of interest, this condition will hold for all frequencies in the band of interest except for the upper band edge. None of the other constraints directly limits the response in this nonvisible region, and left unconstrained it can grow quite large. While it has no direct effect on the antenna pattern (which measures the response to incident plane waves), the entire DTFT does respond to independent white receiver noise, and the result is an excessive noise gain and filter coefficients with a higher dynamic range. Choosing a modest limit on the total noise gain solves the problem, while still allowing the optimization to adjust the nonvisible portion of the response as needed to optimize the visible portion. This is an advantage over time-delay designs, which suppress the nonvisible sidelobes as much as the visible ones.

As a bridge from the FIR time-delay design example, we first consider a design in which the sidelobe region is chosen to match that of the time-delay case so that $\Theta _{\mathrm{sb}}$ consists of the interval $[-90^{\circ },90^{\circ }]$ minus an asymmetric interval about $45^{\circ }$ that narrows with increasing frequency. Optimizing as shown above with a single constraint limiting the rms frequency response error at $45^{\circ }$ to $1\%$ results in the response in the top plot of Fig. 2. The main beam is essentially identical to that of the time-delay design, as is the frequency response at the beam's center. The overall sidelobe level is lower, especially at lower frequencies, and the sidelobe structure is no longer forced to be unchanging across frequency. This is a result of designing the filters jointly and not separately as in the time-delay design and of removing the implicit structural restriction on the mainbeam shape. The design cost is now significantly higher, as a complete set of filter coefficients must be computed for each look direction, but the implementation cost is as before.

While better than the time-delay design, the design just described has obvious drawbacks. The frequency response at off-center angles in the main beam could be improved, and the beam still changes width with frequency. For the final design example, then, the sidelobe region is chosen to be constant (not a function of frequency) with a beamwidth slightly narrower than the maximum beamwidth of the previous designs, and the rms frequency response error is constrained to $0.1\%$ for $\theta \in [42^{\circ },48^{\circ }]$. The noise gain was limited to the same level as in the previous design. To demonstrate the ability to custom tailor the sidelobe distribution in both frequency and angle an additional constraint was used, limiting the response on the region between $-10^{\circ }$ and $10^{\circ }$ in angle and $1.1$ GHz and $1.2$ GHz in frequency to less than $-45$ dB. The optimized response is shown in the bottom plot of Fig. 2. Note that the beamwidth is in fact nearly constant and that the frequency response in the main beam is much better than that of previous designs. Even though the improved passband and custom sidelobe constraints necessarily increase the sidelobe energy, it is roughly the same as for the previous design and better than for the time-delay examples. Removing the frequency-dependent constraint on the sidelobe region and directly optimizing the wideband array pattern has resulted in a substantial performance gain over the conventional time-delay method.

Figure 1: Ideal time-delay (top) and optimized FIR time-delay (bottom) steered array patterns. The upper-left corner of each response grid show cuts across angle, and the lower-right corner show cuts across frequency.

Figure 2: Optimized wideband array patterns: time-delay-like narrowing main beam (top) and constant-beamwidth (bottom). The small white rectangle in the bottom plot outlines the additional sidelobe constraint region.