Tapering Theory#

Amplitude tapering (windowing) is a fundamental technique for sidelobe control in phased arrays. This section covers the theory behind common taper functions.

Why Taper?#

A uniform amplitude distribution has the narrowest main beam but relatively high sidelobes (~-13 dB for a rectangular aperture). The sidelobes arise from the sharp discontinuity at the aperture edges.

Tapering smooths the amplitude distribution, reducing sidelobes at the cost of:

  1. Increased beamwidth: Main lobe becomes wider

  2. Reduced directivity: Less aperture efficiency

  3. Reduced gain: Peak gain decreases

The Fourier Relationship#

The array factor is the Fourier transform of the aperture distribution:

\[AF(u) = \int_{-L/2}^{L/2} a(x) e^{jkux} \, dx\]

where \(a(x)\) is the amplitude distribution.

For discrete arrays:

\[AF(u) = \sum_{n=1}^{N} a_n e^{jkx_n u}\]

Sharp edges in \(a(x)\) produce high-frequency components (sidelobes). Smooth tapers reduce these high frequencies.

Aperture Efficiency#

The aperture efficiency (taper efficiency) is:

\[\eta = \frac{\left|\sum_n a_n\right|^2}{N \sum_n |a_n|^2}\]

For uniform weighting (\(a_n = 1\)): \(\eta = 1\)

For any taper: \(\eta < 1\)

The directivity loss in dB:

\[\text{Loss} = 10 \log_{10}(\eta)\]

Taylor Taper#

The Taylor distribution is designed for a specified sidelobe level with controlled sidelobe decay. It produces \(\bar{n}\) nearly-equal sidelobes before rolling off.

The continuous distribution:

\[a(x) = 1 + 2\sum_{m=1}^{\bar{n}-1} F(m, A, \bar{n}) \cos\left(\frac{2\pi m x}{L}\right)\]

where A is determined by the desired sidelobe level:

\[\text{SLL} = -20\log_{10}(\cosh(\pi A))\]

Properties:

  • Provides best efficiency for a given sidelobe level

  • Smooth rolloff (no discontinuities)

  • \(\bar{n}\) controls sidelobe behavior (higher = more equal-level sidelobes)

Typical values:

SLL (dB)

nbar

Efficiency

Beamwidth Factor

-25

3

94%

1.08

-30

4

90%

1.14

-35

5

87%

1.19

-40

6

84%

1.24

Chebyshev (Dolph-Chebyshev) Taper#

The Chebyshev distribution produces equi-ripple sidelobes (all at the same level). This provides the minimum beamwidth for a given sidelobe level.

The distribution is related to Chebyshev polynomials:

\[a_n = \frac{1}{N} \sum_{k=0}^{N-1} T_{N-1}(x_0 \cos(\pi k/N)) e^{j2\pi nk/N}\]

where \(x_0\) is determined by the sidelobe level:

\[x_0 = \cosh\left(\frac{1}{N-1}\cosh^{-1}(10^{\text{SLL}/20})\right)\]

Properties:

  • All sidelobes equal (equi-ripple)

  • Narrowest beamwidth for given SLL

  • Has discontinuities at aperture edges

  • Less efficient than Taylor for same SLL

Hamming and Hanning Windows#

Classic signal processing windows, simple to implement:

Hanning (Hann):

\[a(x) = 0.5 - 0.5\cos\left(\frac{2\pi x}{L}\right)\]

  • First sidelobe: -31 dB

  • Sidelobe rolloff: -18 dB/octave

Hamming:

\[a(x) = 0.54 - 0.46\cos\left(\frac{2\pi x}{L}\right)\]

  • First sidelobe: -42 dB

  • Sidelobe rolloff: -6 dB/octave (slower than Hanning)

Cosine Tapers#

Cosine (Sine) Taper:

\[a(x) = \sin\left(\frac{\pi x}{L}\right)\]

Cosine-on-Pedestal:

\[a(x) = p + (1-p)\sin\left(\frac{\pi x}{L}\right)\]

where p is the pedestal level (edge amplitude).

Gaussian Taper#

\[a(x) = \exp\left(-\frac{x^2}{2\sigma^2}\right)\]

Properties:

  • Very low sidelobes (theoretically none for infinite Gaussian)

  • Smooth decay in all regions

  • Practical truncation introduces sidelobes

2D Tapers#

For 2D arrays, tapers can be applied as:

Separable (product):

\[a(x, y) = a_x(x) \cdot a_y(y)\]

Circularly symmetric:

\[a(r) = f\left(\sqrt{x^2 + y^2}\right)\]

Separable tapers are simpler but produce slightly higher sidelobes in diagonal planes. The library uses separable tapers:

# 2D Taylor = product of 1D Taylors
taper_2d = pa.taylor_taper_2d(Nx, Ny, sidelobe_dB=-30)

Taper Selection Guidelines#

  1. Taylor: Best general-purpose choice for radar and communications

  2. Chebyshev: When minimum beamwidth is critical and equi-ripple sidelobes are acceptable

  3. Hamming: Good balance, simple implementation

  4. Gaussian: When very low far-out sidelobes are needed (low probability of intercept)

  5. Cosine-on-pedestal: When moderate sidelobe control with high efficiency is needed

Combining Tapers with Steering#

Amplitude taper and phase steering are multiplicative:

# Steering weights (complex, unit magnitude)
steer = pa.steering_vector(k, geom.x, geom.y, theta0, phi0)

# Amplitude taper (real, varying magnitude)
taper = pa.taylor_taper_2d(Nx, Ny, sidelobe_dB=-30)

# Combined weights
weights = steer * taper