ch7: shaping the beam

date: May 17 2026

the question driving this chapter: how do I shape the beam I want (narrower main lobe, lower sidelobes)?

1. The pattern is already shaped, just not always the shape you want

An array of $N$ uniformly excited isotropic elements at $\lambda/2$ spacing produces a fixed angular pattern: one main lobe of width $\sim\lambda/(N \cdot d)$ , side lobes 13 dB down from the peak, and grating lobes if $d$ goes past the limits from Chapter 6. If that pattern is what is needed, the design is done.

The pattern is rarely exactly what is needed. Common reshaping goals:

Lower side lobes so a strong off-axis target does not appear as a ghost in the main beam. The 13-dB-down side lobes of a uniform array are too high for many sensing applications.
Narrower main lobe when the array geometry is fixed and adding elements is not possible.
A null placed at a specific angle, for example to suppress a strong interferer or to avoid radiating into a regulated direction.
Asymmetric beam shape, such as a wide beam in azimuth and a narrow one in elevation, with fewer elements than a fully filled 2D array would need.

Three knobs are available without changing the physical geometry: amplitude weighting on the elements (sections 2 and 3), independent pointing of TX and RX sub-apertures (section 4), and physical shaping of the radiated field with reflectors and absorbers (section 5). Each one has its own tradeoffs, and combining them is the standard practice in real systems.

2. Amplitude tapering is spatial windowing

The array factor from Chapter 6 is

\text{AF}(\theta) \;=\; \sum_{n=0}^{N-1} w_n \, e^{\,j n \psi}, \qquad \psi = k d \sin\theta - \beta,

which is a discrete Fourier transform of the element weights $\{w_n\}$ evaluated at spatial frequency $\sin\theta$ . Setting $w_n = 1$ for all $n$ produces the sinc-like pattern with $-13$ dB sidelobes. Any other choice of $\{w_n\}$ produces a different pattern, and the relationship is exactly the same as time-domain DSP windowing.

This direct correspondence between array tapering and DSP windowing means every result from spectral analysis transfers: Hann widens the main lobe but cuts the first sidelobe to $-32$ dB; Hamming gets $-43$ dB; Blackman drops further but widens the main lobe more. Dolph-Chebyshev is the window that minimizes beamwidth for a specified sidelobe ceiling; it is the theoretically optimal taper. Taylor is a smoothed Dolph-Chebyshev that is less sensitive to amplitude errors and more common in practice. Kaiser parameterizes the tradeoff with a single knob $\beta$ .

The cost of tapering

Three things are lost when tapering is applied.

Main lobe widens. Concentrating the energy at the array center spreads the spatial-frequency response. For Hamming, the broadening factor is about 1.4 over a uniformly excited array of the same length.
Directivity drops. Total radiated power is lower because the edge elements contribute less. For Hamming, the loss is around 1.3 dB compared to the uniform case.
Power amplifiers run below capacity. A common design constraint is that each element’s amplifier must run at the same peak power for cost reasons. Tapered weights leave the edge amplifiers idle.

The standard mitigation is to apply the taper on receive only. RX-side tapering reduces sidelobes in the received pattern (which is what matters for clutter rejection) without forcing the TX amplifiers to run unevenly. For radar, the round-trip pattern is $\text{AF}_\text{TX} \cdot \text{AF}_\text{RX}$ , so even one-sided tapering still suppresses sidelobes effectively, just by half as many dB.

3. The standard tapers and what they buy

taper	weight formula	1st sidelobe	HPBW factor	directivity loss
Uniform	$w_n = 1$	−13 dB	1.00	0 dB
Triangular	$1 - \lvert 2n/(N-1) - 1 \rvert$	−26 dB	~1.45	~1.25 dB
Hann	$0.5 - 0.5\cos(2\pi n/(N-1))$	−32 dB	~1.50	~1.5 dB
Hamming	$0.54 - 0.46\cos(2\pi n/(N-1))$	−43 dB	~1.45	~1.3 dB
Blackman	$0.42 - 0.5\cos(\cdot) + 0.08\cos(2\cdot)$	−58 dB	~1.7	~1.8 dB
Dolph-Chebyshev	Chebyshev polynomial	set by user	min for chosen SLL	varies
Taylor	smoothed Dolph-Chebyshev	set by user	~Dolph + 5%	varies
Kaiser	$I_0(\beta\sqrt{1-(2n/(N-1)-1)^2}) / I_0(\beta)$	tunable via $\beta$	tunable	tunable

Numbers are for large $N$ . At small $N$ the sidelobe floor rises slightly because there are too few elements to approximate the continuous window shape.

Tapering comparison: pattern, sidelobes, beamwidth

A 16-element uniform array at $\lambda/2$ spacing with selectable taper. The polar plot shows the resulting pattern in dB; the bar plot shows the element weights. First sidelobe level and half-power beamwidth update with the choice.

Taper N elements 16 Kaiser β 6.0

Switching from uniform to Hamming knocks the first sidelobe from −13 dB to about −43 dB but widens the main lobe by roughly 45%. The Kaiser slider continuously trades sidelobe level against beamwidth.

4. The TX/RX pattern product

A radar’s effective angular response is the product of the transmit and receive patterns. This is a fact about how the two-way SNR is computed, not a design trick by itself. The product structure shows up in every radar, monostatic or bistatic.

For colocated co-pointed TX and RX (the standard case), both apertures point at the same target. The round-trip power pattern is $|F(\theta)|^2 \cdot |F(\theta)|^2 = |F(\theta)|^4$ . The squaring suppresses sidelobes: a uniform array’s $-13$ dB sidelobes become $-26$ dB in the round-trip pattern; combining a Hamming RX taper with a uniform TX brings round-trip sidelobes to roughly $-56$ dB. This is just the existence of the product, not an extra design knob.

Offsetting TX and RX is real but not free

If TX and RX are steered to different angles $\theta_T \neq \theta_R$ , the product pattern peaks somewhere between them and the overlap region of the two main lobes is narrower than either factor. The overlap-narrowing intuition can be approximated, for small offsets and roughly Gaussian-shaped main lobes, as

\theta_\text{HPBW}^\text{prod} \;\approx\; \sqrt{\theta_\text{HPBW}^2 - \Delta^2},

where $\Delta$ is the angular separation between the two steering directions. The formula is decent intuition for the Gaussian-lobe case; it is not a general law for real array-factor sinc-like main lobes, where the exact result depends on aperture taper, element pattern, scan angle, and whether the discussion is about field amplitude or power. The demo below uses the actual sinc pattern, not the Gaussian approximation.

The offset trick has real costs. Three of them.

Peak two-way gain drops. Each factor at the product’s peak direction is off-boresight from its own main lobe, so neither contributes its full $N$ -element gain. The SNR loss at the target compared to co-pointed operation can be several dB even for modest offsets.
Coverage changes shape. The narrowed product lobe sees only a sliver of the angular range that either single beam covers. For surveillance applications this means more time spent scanning to cover the same field of view.
Calibration tightens. The product peak sits between two main lobes that have to remain mutually aligned. Phase or amplitude drift on one side moves the product peak in ways that drift on a co-pointed system would not.

In practice, deliberate TX/RX offset is uncommon in monostatic radar. It shows up more often in passive bistatic configurations (where the geometry forces $\theta_T \neq \theta_R$ ) and in some pencil-beam imaging systems where the SNR margin is large enough to spend on beamwidth.

TX/RX offset: beam narrowing vs SNR loss

Two $N$ -element ULAs at $\lambda/2$ spacing. TX is steered to $-\Delta/2$ , RX to $+\Delta/2$ . The three polar plots show TX pattern, RX pattern, and product. The metrics panel reports the product’s HPBW and the SNR penalty at $\theta = 0$ compared to the co-pointed ( $\Delta = 0$ ) baseline.

TX/RX separation Δ (deg) 6.0° N (each array) 8

At $\Delta = 0$ the SNR loss is zero by definition and the product is the standard $|F(\theta)|^4$ pattern. Increasing $\Delta$ narrows the product main lobe (HPBW drops) but the SNR penalty grows. For an 8-element array the SNR loss crosses 3 dB around $\Delta \approx 7°$ and 10 dB around $\Delta \approx 13°$ .

5. Physical beam shaping: reflectors, ground planes, blockers

Tapering and product tricks only re-weight the array factor and element pattern. The other approach is to physically alter the field with conductors and absorbers placed near the antenna.

Ground planes

A conductor behind the array acts as a mirror. By image theory, every element above the conductor pairs with a virtual image below, doubling the effective aperture in the direction perpendicular to the conductor. This concentrates radiation into the hemisphere facing the conductor (typically up or forward) and suppresses back-hemisphere radiation. The element pattern in front goes from omnidirectional to roughly $\cos\theta$ -shaped.

Spacing the array a quarter-wavelength above the ground plane gets the right reflection phase for constructive front-hemisphere combining. Closer or further reduces effectiveness, and the reflection phase drift with frequency limits the bandwidth of ground-plane-backed arrays.

Parabolic reflectors and lenses

A parabolic dish converts a small feed antenna into a large effective aperture by directing all forward-radiated rays into a parallel beam. The dish does not change the feed’s element pattern; it provides geometric optics that take whatever the feed emits in its forward cone and reshapes it into a narrow beam. Gain scales with the dish’s aperture area in square wavelengths, $G \approx 4\pi A_\text{eff}/\lambda^2$ , regardless of the underlying element.

Dielectric lenses do the same job through refraction rather than reflection. They are used at mmWave to keep antennas planar.

Radar absorbing material

Foam or rubber loaded with carbon, ferrite, or other lossy material can be placed near the antenna to absorb energy from directions where it would otherwise reflect into unwanted sidelobes. Typical performance is 10 to 30 dB of absorption per inch of thickness at GHz frequencies. RAM is heavy and bulky for low-frequency operation: a 20 dB absorber at 1 GHz is several inches thick.

Blockers and edge diffraction

A flat conducting sheet placed between the antenna and an unwanted direction blocks radiation in that direction. But the sheet has edges, and edges diffract. Geometric theory of diffraction (GTD) predicts that the field behind a blocker is dominated by diffraction from the edges, not by transmission through the sheet. The diffracted field is roughly proportional to $1/\sqrt{r}$ at the diffraction edge, falling off slower than the geometric-optics shadow would suggest.

For a knife-edge blocker positioned to suppress a specific sidelobe, the blocking ratio is set by the geometry: deep shadow behind the centre of the blocker (40 dB or more), much less suppression near the edges (10 to 15 dB). The pattern in the shadow region is the original pattern minus the diffracted contribution from each blocker edge, which produces a fringe pattern rather than a clean null. This is the technical reason a piece of metal cannot kill a single sidelobe cleanly: the very edges that define the blocked region scatter into the region you are trying to keep clean.

6. Why metal cannot fix a single sidelobe cleanly

The intuitive picture is appealing: a strong sidelobe points in one direction, so block that direction with a metal sheet. In practice this rarely works as cleanly as expected, for three reasons.

Edge diffraction lights up the shadow. Any finite-extent blocker scatters from its edges. Behind the blocker, the field is the sum of the geometric-optics shadow (close to zero) and the diffracted edge waves (not zero). The diffracted contribution typically caps suppression at 30 to 40 dB even for ideal blockers.
The shadow is not isolated to one direction. A blocker at distance $r$ from the antenna casts a shadow over an angular range $\sim w/r$ where $w$ is the blocker width. For a sidelobe at $\theta = 30°$ from broadside, the blocker also shadows other nearby angles, including some you might want to keep illuminated.
Reflection back into the antenna. A metal sheet reflects whatever does not pass through. Standing waves form between the blocker and the antenna, retuning the antenna’s impedance and shifting the rest of the pattern in unpredictable ways. A thin layer of RAM in front of the blocker mitigates this at the cost of even more bulk.

The cleaner ways to kill a specific sidelobe are either (a) amplitude or phase tapering, which can place a null arbitrarily close to a chosen angle, or (b) adaptive beamforming (next section) which solves for weights that put a null exactly where measurement says interference is coming from. Physical blockers are reserved for situations where the offending source has a much wider angular extent than the array pattern can null, or where the blocker is mechanically convenient (a wall, an enclosure, a radome panel).

Every shaping mechanism in this chapter (amplitude taper, offset pointing, ground plane, parabolic reflector, blocker) costs gain in some direction to buy suppression in another. There is no free reshaping. The conservation rule is that the total integrated radiated power, set by what the source delivers, has to end up somewhere; reshaping just chooses where.

7. When intuition runs out

Everything up to this point has been hand-tuned. Pick a taper, set a steering angle, place a reflector. For a uniform 1D array these intuitive choices are close to optimal; for irregular arrays, multi-objective constraints, or scenarios with measured interference patterns, intuition stops being enough.

Two questions drive the rest of this chapter. Can a solver find better array configurations than the hand-tuned ones? And what can you actually transmit at the power levels these designs require if you live somewhere like New York City? The two constrain each other. A solver that finds an array geometry hitting 5° beamwidth at 2.4 GHz is useless if FCC Part 15 caps the transmit power so low that SNR at the target collapses. The reverse is also true: a perfectly legal low-power transmission with a hand-tuned uniform array can be improved by ~5 dB by adaptive nulling, which can be the difference between a useful sensor and a useless one.

Solvers for array synthesis

Three classes of problem appear when designing arrays beyond the hand-tuned regime.

Sparse array synthesis. Given a target pattern (peak gain at boresight, sidelobes below $-30$ dB, nulls at $\pm 45°$ ), find the minimum number of elements and their positions that achieve it. This is a non-convex combinatorial problem in general. Convex relaxations using $\ell_1$ minimization (Lebret-Boyd 1997) work well when the target pattern is feasible; metaheuristic methods (genetic algorithms, simulated annealing, particle swarm) handle harder cases at the cost of compute time.
Beam pattern synthesis with weights. Geometry is fixed, weights are free. The forward problem (weights to pattern) is linear, so the inverse problem can often be cast as a convex optimization: minimize sidelobe peak subject to main-lobe constraints, or minimize beamwidth subject to a sidelobe ceiling. Dolph-Chebyshev solves a specific instance of this in closed form; convex solvers handle the general case.
Adaptive beamforming. The interference environment is measured in real time, and weights are recomputed on the fly to null whatever directions are active. Capon’s MVDR (minimum-variance distortionless response) is the standard formulation: minimize total received power subject to a unit-gain constraint in the look direction. Solves in closed form per snapshot; standard in modern phased-array radar and 5G base stations.

The pattern from a solver is rarely qualitatively different from a hand-tuned Dolph-Chebyshev or Taylor. It is quantitatively better: 3 to 6 dB more sidelobe suppression, or 10 to 20% narrower main lobe, or specific nulls placed at observed interferer directions. Solvers shine when constraints are hard (a specific FCC mask, a known interferer angle) or when the array is geometrically irregular (sparse, deformed, conformal to a curved surface).

next: ch8: planning →