For estimating and education only. Results come from the simplified models described below, are provided as is with no warranty, and are not a substitute for engineering analysis, simulation or measurement. Check them before relying on them in a design, purchase or safety decision. See the terms of use.
Equations
AF(\theta) = \sum_{n=0}^{N-1} w_n\, e^{\,j n \psi}, \qquad \psi = 2\pi d \sin\thetad in wavelengthsSteering
w_n = a_n\, e^{-j n \psi_0}, \qquad \psi_0 = 2\pi d \sin\theta_0Uniform
|AF| = \frac{\left|\sin(N\psi^{\prime}/2)\right|}{N\left|\sin(\psi^{\prime}/2)\right|}, \qquad \psi^{\prime} = \psi - \psi_0Dolph–Chebyshev
AF \propto T_{N-1}\!\left(x_0\cos\frac{\psi^{\prime}}{2}\right), \qquad x_0 = \cosh\!\left(\frac{\cosh^{-1} R}{N - 1}\right)R = main beam / sidelobeTaylor n̄
a(x) = 1 + 2\sum_{m=1}^{\bar n - 1} F_m \cos(2\pi m x), \qquad A = \frac{\cosh^{-1} R}{\pi}x = element position / apertureF_m = \frac{(-1)^{m+1}\prod_{n=1}^{\bar n - 1}\left(1 - \frac{m^2}{\sigma^2\,(A^2 + (n - \frac{1}{2})^2)}\right)}{2\prod_{n \ne m}\left(1 - \frac{m^2}{n^2}\right)}, \qquad \sigma^2 = \frac{\bar n^2}{A^2 + (\bar n - \frac{1}{2})^2}Nulls
\mathbf{w} = \mathbf{w}_q - \mathbf{C}^{H}\left(\mathbf{C}\mathbf{C}^{H}\right)^{-1}\mathbf{C}\,\mathbf{w}_q, \qquad C_{kn} = e^{\,j n \psi_k}zero at each θk, minimum changeDirectivity
D = \frac{|AF|^2_{\max}}{\sum_m \sum_n \mathrm{Re}\left(w_m w_n^{*}\right)\mathrm{sinc}[k d\,(m - n)]}isotropic elementsTaper efficiency
\eta_t = \frac{\left|\sum a_n\right|^2}{N \sum |a_n|^2}No grating lobes
\frac{d}{\lambda} < \frac{1}{1 + |\sin\theta_0|}True time delay
\Delta t = \frac{d\,\sin\theta_0}{c}s per element, d in metresSquint
\sin\theta(f) = \frac{f_0}{f}\,\sin\theta_0phase shifters onlyQuantization
\mathrm{LSB} = \frac{360^\circ}{2^{b}}, \qquad \text{peak lobe} \approx -6.02\,bdB\Delta G \approx -10\log_{10}\left(1 - \frac{\pi^2}{3\cdot 2^{2b}}\right)dBVariables
| Symbol | Meaning | Unit |
|---|---|---|
N | Number of elements | — |
d | Element spacing in wavelengths | λ |
f_0 | Center frequency, where the phases are set | Hz |
b | Phase shifter bits | — |
\theta,\ \theta_0 | Observation and steering angle from broadside | ° |
a_n | Amplitude taper | — |
w_n | Complex element weight after steering and nulling | — |
R | Main beam to sidelobe voltage ratio, 10SLL/20 | — |
\bar n | Number of nearly equal sidelobes next to the main beam (Taylor) | — |
\mathbf{w}_q | Quiescent weights: taper with steering, before nulling | — |
\psi_k | 2π d sin θk for each null direction θk | rad |
Worked example
| Input | Value |
|---|---|
| Elements N | 16 |
| Spacing d | 0.5 λ |
| Taper | Taylor, −30 dB, n̄ = 5 |
| Nulls | −40° and 25° |
| Result | Value |
|---|---|
| Half-power beamwidth | 8.15 ° |
| Peak sidelobe level | −28.3 dB |
| Directivity (isotropic elements) | 11.34 dBi |
| Main beam points to | 0 ° |
| Taper efficiency | −0.679 dB |
| Gain change from nulls | −0.0224 dB |
| Gain toward θ0 change | −0.0258 dB |
| Null depth (worst) | < −150 dB |
| Element spacing | 14.99 mm |
| Progressive phase per element | 0 ° |
| True time delay per element | 0 ps |
| Total delay across the array | 0 ps |
| Max scan without grating lobes | 90 ° |
| Squint across band, phase shifters | 0 ° |
| Squint / beamwidth | 0 |
| Phase LSB | 5.625 ° |
| Peak quantization lobe (≈) | −36.1 dB |
| Gain loss from quantization (≈) | 0.00349 dB |
The two nulls cost about 0.02 dB of directivity. The sidelobes next to them rise from −30 to about −28.3 dB.
Load this example in the calculatorValidity
- Isotropic elements, equal spacing, no mutual coupling. Multiply by an element pattern for a first-order estimate of a real array.
- The plotted cut is the plane containing the array axis; angles are from broadside.
- Taylor and Chebyshev weights are sampled from the continuous distributions, as in the usual window functions; small arrays land within a fraction of a dB of the requested sidelobe level.
- Nulls are exact for the ideal weights. Amplitude and phase errors, quantization and element mismatch fill them in; real null depth is set by calibration.
- Hann and Blackman are sampled at interior points so the end elements are never zero.
- Quantization figures assume a periodic phase error with no randomization. Randomizing spreads the lobe into the sidelobe floor.
Sources
- C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016. Chapter 6 (arrays) and §6.8 (Dolph–Chebyshev).
- T. T. Taylor, “Design of Line-Source Antennas for Narrow Beamwidth and Low Side Lobes,” IRE Trans. Antennas Propag., vol. AP-3, no. 1, pp. 16–28, 1955.
- C. L. Dolph, “A Current Distribution for Broadside Arrays Which Optimizes the Relationship Between Beam Width and Side-Lobe Level,” Proc. IRE, vol. 34, no. 6, pp. 335–348, 1946.
- H. Steyskal, “Synthesis of Antenna Patterns with Prescribed Nulls,” IEEE Trans. Antennas Propag., vol. AP-30, no. 2, pp. 273–279, 1982.
- R. J. Mailloux, Phased Array Antenna Handbook, 3rd ed., Artech House, 2018.
- F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proc. IEEE, vol. 66, no. 1, pp. 51–83, 1978.