Antenna Arrays and Classical Synthesis

Introduction

An antenna array combines spatially distributed radiating elements so that their fields interfere constructively in desired directions and destructively elsewhere. Arrays provide capabilities unavailable from a single element: higher directivity, beam steering, shaped coverage, interference rejection, null placement and adaptive spatial processing. These capabilities underpin radar, satellite links, radio astronomy, direction finding, massive-MIMO systems and millimetre-wave communications.

Array analysis separates geometry, excitation and element radiation whenever the assumptions of pattern multiplication are valid. Practical designs must additionally account for grating lobes, scan loss, mutual coupling, finite phase resolution, manufacturing tolerance and frequency dependence. Classical synthesis methods—binomial, Dolph–Chebyshev and Fourier synthesis—establish the foundations for modern numerical and adaptive array design.

Learning Objectives

After studying this chapter, the reader should be able to:

• formulate array factors for linear, planar and conformal arrays;
• apply pattern multiplication and identify its limitations;
• determine beam directions, nulls, beamwidths and grating-lobe conditions;
• evaluate electronic scanning and beam squint;
• explain how mutual coupling modifies array currents and patterns;
• compare uniform, binomial and tapered amplitude distributions;
• calculate Dolph–Chebyshev excitation coefficients for a specified sidelobe level;
• apply Fourier-transform concepts to continuous and discrete array synthesis; and
• develop a technically defensible array design under practical constraints.

1. General Array Formulation

Consider N elements at position vectors rₙ. Adopt the time convention exp(jωt), wavenumber k = 2π/λ and observation unit vector r̂. If element n has complex excitation aₙ and isolated far-field vector pattern Eₙ(r̂), the total far field, apart from the common radial factor exp(−jkr)/r, is

E(r̂) = Σₙ₌₀ᴺ⁻¹ aₙ Eₙ(r̂) exp(jk r̂·rₙ).

The coefficient aₙ includes the excitation magnitude and phase. This expression is also useful for conformal arrays because Eₙ may depend on element orientation and location.

1.1 Pattern multiplication

If all elements are identical, identically oriented and sufficiently unaffected by their neighbours, then Eₙ(r̂) = Eₑ(r̂). The total field becomes

E(r̂) = Eₑ(r̂) AF(r̂),

where

AF(r̂) = Σₙ₌₀ᴺ⁻¹ aₙ exp(jk r̂·rₙ)

is the array factor. Pattern multiplication is therefore an exact algebraic separation only under the common-element-pattern assumption. In a coupled array, embedded element patterns generally differ, so a single isolated element factor multiplied by an ideal array factor may be inaccurate.

The array factor controls the spatial interference pattern, whereas the element factor limits usable coverage. A mathematically steerable array-factor maximum may consequently lie near an element-pattern null and produce little realized gain.

2. Uniform Linear Arrays

Let N elements lie along the z-axis with spacing d and positions zₙ = nd. Let θ be measured from the positive z-axis. For progressive excitation

aₙ = Aₙ exp(jnα),

the array factor is

AF(θ) = Σₙ₌₀ᴺ⁻¹ Aₙ exp[jn(kd cosθ + α)].

Define

ψ = kd cosθ + α.

For a uniform-amplitude array, Aₙ = 1, and the finite geometric series gives

AF(ψ) = exp[j(N−1)ψ/2] sin(Nψ/2)/sin(ψ/2).

The normalized magnitude is

|AFₙ(ψ)| = |sin(Nψ/2)/(N sin(ψ/2))|.

The exponential factor affects phase but not magnitude.

2.1 Beam steering

A principal maximum occurs when ψ = 2πm. To steer the m = 0 beam toward θ₀,

α = −kd cosθ₀.

Thus,

ψ = kd(cosθ − cosθ₀).

Special cases include:

• Broadside: θ₀ = 90°, so α = 0.
• Endfire toward +z: θ₀ = 0°, so α = −kd.
• Endfire toward −z: θ₀ = 180°, so α = +kd.

These phase conditions identify array-factor maxima; the total-pattern maximum can shift when the element factor is strongly directional.

2.2 Nulls and broadside beamwidth

For a uniform array, nulls occur when

Nψ/2 = ±π, ±2π, …,

provided the denominator is nonzero. The first nulls surrounding the principal maximum satisfy ψ = ±2π/N.

For a broadside array,

kd cosθ = ±2π/N,

so

cosθ = ±λ/(Nd).

When λ/(Nd) ≤ 1, the exact first-null beamwidth is

FNBW = 2 sin⁻¹[λ/(Nd)].

For a sufficiently narrow broadside beam,

FNBW ≈ 2λ/(Nd) radians.

The one-sided angular displacement from broadside is therefore approximately λ/(Nd), not 2λ/(Nd). For a large uniform broadside array, the approximate half-power beamwidth is

HPBW ≈ 0.886λ/(Nd) radians.

These approximations apply near broadside and are most accurate for electrically long arrays. They should not be generalized to strongly scanned beams without accounting for the angular transformation from direction cosine to physical angle.

Worked Example 1: First-null beamwidth

An eight-element uniform broadside array has d = λ/2.

The first-null direction-cosine displacement is

λ/(Nd) = λ/[8(λ/2)] = 0.25.

Hence the exact full first-null beamwidth is

FNBW = 2 sin⁻¹(0.25) = 28.96°.

The narrow-beam approximation gives

FNBW ≈ 2λ/(Nd) = 0.5 rad = 28.65°.

Thus, each first null is approximately 14.5° from broadside. The approximate uniform-array HPBW is

HPBW ≈ 0.886λ/[8(λ/2)] = 0.2215 rad = 12.69°.

3. Grating Lobes and Scanning

Grating lobes are additional maxima having approximately the same array-factor magnitude as the intended principal maximum. They arise when spatial sampling is too coarse.

Introduce the direction cosine u = cosθ for a linear array along z. For steering toward u₀,

ψ = kd(u − u₀).

All maxima satisfy

u = u₀ + mλ/d, m ∈ ℤ.

A grating lobe exists when any solution with m ≠ 0 lies in the visible interval −1 ≤ u ≤ 1.

For a specified scan magnitude |u₀|, strict exclusion of additional visible solutions requires

d < λ/(1 + |u₀|).

At equality, an unwanted maximum lies at a visible-region boundary. For broadside, u₀ = 0, strict avoidance therefore requires d < λ. If d = λ, maxima occur at u = ±1, corresponding to endfire boundary directions. The frequently used d = λ/2 spacing avoids interior grating lobes over the full scan range; strict exclusion even at the limiting endfire boundary requires d < λ/2.

Worked Example 2: Scan-limited spacing

A linear array must scan to θ₀ = 45° from its axis. Then |u₀| = cos45° = 0.7071. The strict spacing condition is

d < λ/(1 + 0.7071) = 0.5858λ.

A spacing of 0.5λ is acceptable, whereas 0.7λ produces a visible grating lobe. For d = 0.7λ and u₀ = 0.7071, the m = −1 maximum is

u = 0.7071 − 1/0.7 = −0.7215,

which lies in the visible interval.

3.1 Scan loss and beam broadening

When a broadside beam is scanned away from broadside, its angular width generally increases because equal increments in direction cosine correspond to larger angular increments near endfire. Projected aperture also decreases. For a planar aperture scanned by angle θₛ from broadside, an idealized projected-area argument gives a gain reduction approximately proportional to cosθₛ, before element-pattern and coupling losses are included.

3.2 Beam squint

A phase-shifter array establishes the required phase at a design frequency f₀. If its progressive phase remains fixed with frequency, then

u(f) = (f₀/f)u₀.

The beam therefore changes direction across a wide bandwidth. True-time-delay networks impose actual delays rather than frequency-independent phases and can substantially suppress beam squint.

4. Planar Arrays

For a rectangular array in the xy-plane, define

u = sinθ cosφ, v = sinθ sinφ.

Let elements be located at xₘ = mdₓ and yₙ = ndᵧ. The array factor is

AF(u,v) = Σₘ Σₙ aₘₙ exp[jk(mdₓu + ndᵧv)].

If the excitations are separable, aₘₙ = bₘcₙ, then

AF(u,v) = AFₓ(u)AFᵧ(v).

This property permits independent synthesis in the two principal dimensions. Steering toward (u₀,v₀) is obtained using

aₘₙ = Aₘₙ exp[−jk(mdₓu₀ + ndᵧv₀)].

The visible region is not the full u-v square but the unit disk

u² + v² ≤ 1.

Consequently, a replicated spectral maximum is a physical grating lobe only if it falls inside this disk. For nonrectangular lattices, reciprocal-lattice analysis provides the most general grating-lobe test.

Planar arrays enable pencil beams and two-dimensional scanning. Their practical feed networks, however, introduce conductor loss, phase error, routing complexity and polarization constraints.

5. Conformal Arrays

A conformal array follows a curved platform such as an aircraft fuselage, vehicle body or satellite surface. Element positions and orientations both vary. A useful field model is

E(r̂) = Σₙ aₙ Eₙ,emb(r̂) exp(jk r̂·rₙ),

where Eₙ,emb is the vector embedded element pattern with all other ports terminated in specified reference impedances.

Unlike an ideal regular array, a conformal array generally cannot be described completely by a scalar array factor multiplied by one common element pattern. Local surface normals change polarization and element coverage, while platform scattering alters amplitude and phase. Accurate analysis often requires full-wave simulation, measured embedded patterns or hybrid electromagnetic techniques.

Beam synthesis can be expressed in matrix form. If G is a steering or field-response matrix and a is the excitation vector, sampled fields satisfy

e = Ga.

Constrained least squares, convex optimization or regularized inverse methods may then determine a while controlling sidelobes, nulls, power and polarization.

6. Mutual Coupling and Active Array Behaviour

Each element interacts electromagnetically with the others. For an N-port array,

v = Zi,

where v and i are port-voltage and port-current vectors and Z is the mutual-impedance matrix. The diagonal terms are self-impedances; off-diagonal terms describe coupling. Therefore, specifying feed voltages does not generally produce the intended uncoupled currents.

For element n, active input impedance may be written

Zactive,n = Vₙ/Iₙ = Σₘ Zₙₘ(Iₘ/Iₙ),

provided Iₙ ≠ 0. It depends on the complete excitation state and thus changes with scan angle.

Mutual coupling can cause:

• excitation-amplitude and phase errors;
• active impedance mismatch and reduced efficiency;
• scan-dependent polarization;
• sidelobe and null degradation;
• scan blindness, where a surface-wave or array mode prevents efficient radiation; and
• changes in noise correlation in receiving arrays.

Mitigation techniques include increased isolation, electromagnetic band-gap structures, decoupling networks, coupling-aware synthesis, embedded-pattern calibration and digital predistortion. Calibration must be repeated when temperature, packaging or the platform environment changes appreciably.

7. Amplitude Distributions and Design Trade-offs

Excitation taper controls the trade-off among directivity, beamwidth, sidelobes and dynamic range.

7.1 Uniform distribution

Uniform excitation provides a narrow main beam and high directivity for a given aperture. For a long uniformly excited linear aperture, the first sidelobe is approximately −13.26 dB relative to the main-beam peak. This sidelobe level may be unacceptable in radar, radio astronomy or interference-sensitive links.

7.2 Binomial distribution

For N elements, binomial amplitudes are proportional to the coefficients of

(1 + x)ᴺ⁻¹.

Thus,

Aₙ ∝ C(N−1,n).

The array factor is

AF(ψ) = (1 + exp(jψ))ᴺ⁻¹

= 2ᴺ⁻¹ exp[j(N−1)ψ/2] cosᴺ⁻¹(ψ/2).

Within a grating-lobe-free visible region, the binomial pattern has no sidelobes. The price is a broad main beam and a rapidly increasing excitation dynamic range. For example, a six-element binomial array has amplitudes 1:5:10:10:5:1. Such a ratio may be inefficient or difficult to implement accurately.

7.3 Dolph–Chebyshev distribution

Dolph–Chebyshev synthesis produces the narrowest main beam for a specified equal-ripple sidelobe level among broadside, equally spaced, symmetric linear arrays of a given size. Equivalently, it minimizes the maximum sidelobe for a specified main-beam width under its classical assumptions.

Let R be the voltage ratio between the main-beam peak and equal sidelobes:

R = 10^(S/20),

where S is the positive sidelobe attenuation in decibels. Define

x₀ = cosh{cosh⁻¹(R)/(N−1)}.

The desired polynomial form is

AF(ψ) ∝ Tₙ₋₁[x₀ cos(ψ/2)],

where Tₙ₋₁ is a Chebyshev polynomial of the first kind. Since

Tₙ(cos q) = cos(nq),

its magnitude oscillates with equal ripple over the sidelobe region. The excitation coefficients are obtained by expanding the polynomial into harmonics corresponding to the physical element locations.

Worked Example 3: Calculating Chebyshev excitations

Design a six-element symmetric broadside array with a −20 dB equal-ripple sidelobe level.

The required voltage ratio is

R = 10^(20/20) = 10.

For N = 6,

x₀ = cosh[cosh⁻¹(10)/5] ≈ 1.1846.

Because the array has an even number of elements, place the elements symmetrically at ±d/2, ±3d/2 and ±5d/2. Let the outer, intermediate and inner pair amplitudes be a, b and c. Then

AF(ψ) = 2a cos(5ψ/2) + 2b cos(3ψ/2) + 2c cos(ψ/2).

Use

T₅(x) = 16x⁵ − 20x³ + 5x

and set x = x₀ cos(ψ/2). The identities

cos⁵t = [10cos t + 5cos3t + cos5t]/16,

cos³t = [3cos t + cos3t]/4

give

T₅(x₀ cos t) = x₀⁵ cos5t

+ (5x₀⁵ − 5x₀³)cos3t

+ (10x₀⁵ − 15x₀³ + 5x₀)cos t.

Equating coefficients with t = ψ/2 gives values proportional to

a = x₀⁵/2 ≈ 1.1664,

b = (5x₀⁵ − 5x₀³)/2 ≈ 1.6760,

c = (10x₀⁵ − 15x₀³ + 5x₀)/2 ≈ 2.1576.

Normalizing the inner-element amplitude to unity produces

[a, b, c, c, b, a]

= [0.5406, 0.7768, 1, 1, 0.7768, 0.5406].

These are actual feed-current amplitudes for the ideal uncoupled array. The resulting sidelobes have equal peaks at approximately −20 dB. In a physical array, coupling-aware optimization or calibration is required to realize these currents accurately.

8. Fourier Synthesis

Classical array synthesis can be understood as spatial spectral design. For a continuous line aperture of length L with distribution A(x),

F(u) = ∫₋ᴸ⁄₂ᴸ⁄₂ A(x) exp(jkxu) dx.

With spatial-frequency variable q = ku, this is a Fourier transform. If a complete desired spectrum F(q) is specified, the formal inverse is

A(x) = (1/2π)∫₋∞∞ F(q) exp(−jqx) dq.

Only |u| ≤ 1 is visible in free space. Desired-pattern specifications outside this interval are not directly observable, yet they affect inverse reconstruction. A finite aperture also truncates the ideal distribution, producing ripple and limiting transition sharpness.

For a discrete array,

AF(u) = Σₙ aₙ exp(jkxₙu).

When positions and spatial-frequency samples are uniformly spaced, discrete Fourier transform methods can calculate aₙ. If sampled desired values are Fₘ and the forward relation is

Fₘ = Σₙ₌₀ᴹ⁻¹ aₙ exp(j2πmn/M),

then

aₙ = (1/M)Σₘ₌₀ᴹ⁻¹ Fₘ exp(−j2πmn/M).

Fourier synthesis is intuitive and computationally efficient, but arbitrary desired patterns may demand complex, negative or high-dynamic-range excitations. Truncation, sampling and element-pattern effects mean that the synthesized total pattern must always be verified rather than inferred solely from the ideal transform.

9. Illustrative Case: Millimetre-Wave Access-Point Array

Background

An illustrative engineering team is developing a 28 GHz access point using an 8 × 8 planar patch array. The array must scan to 45° from broadside while suppressing interference toward adjacent sectors.

Problem

The design must avoid grating lobes, maintain approximately −20 dB sidelobes in principal cuts and limit beam-squint degradation over the operating bandwidth. Mutual coupling and regulatory EIRP limits must also be considered.

Analysis

At 28 GHz,

λ = c/f ≈ 10.71 mm.

For a 45° scan from broadside, the relevant principal direction cosine has magnitude sin45° = 0.7071. The strict grating-lobe condition is

d < λ/(1 + 0.7071) = 0.5858λ.

Selecting d = 0.5λ gives d ≈ 5.36 mm and satisfies the scan requirement.

For an eight-element uniform principal cut at broadside,

FNBW = 2sin⁻¹[λ/(8d)] = 2sin⁻¹(0.25) = 28.96°,

while the approximate HPBW is 12.69°. Chebyshev tapering to −20 dB will broaden the main beam relative to the uniform case.

Approach and solution

A separable two-dimensional taper is adopted so that the excitation matrix is the outer product of two one-dimensional Chebyshev distributions. Full-wave embedded patterns are then used to revise the ideal coefficients. Factory calibration estimates the complex response of every channel, while temperature-dependent correction tables compensate phase drift.

For moderate bandwidth, digital subband beamforming reduces beam squint. A wider-band version would require true-time-delay hardware. The controller enforces EIRP and scan-angle limits because element gain, array gain and active impedance all vary with steering direction.

Outcome

Simulation predicts no visible grating lobes within the required scan region. Coupling-aware recalibration restores the principal sidelobes close to their −20 dB target, although corner-plane sidelobes and cross-polarization require separate verification. The tapered design sacrifices some peak gain but improves sector isolation.

Learning points

• Element spacing must be selected from the maximum scan requirement, not broadside operation alone.
• A one-dimensional sidelobe specification does not guarantee compliant two-dimensional coverage.
• Ideal Chebyshev currents must be translated into realized currents through calibration.
• Beam squint, EIRP and thermal variation are system-level rather than purely array-factor issues.

10. Practical Limitations, Governance and Best Practice

Array performance is limited by phase-shifter quantization, amplifier variation, feed loss, position error, radome scattering and finite calibration accuracy. Deep nulls are especially sensitive because they depend on precise cancellation. Wideband arrays also require impedance and pattern stability over frequency.

Machine-learning methods can assist fault detection, calibration interpolation and environment-aware beam selection. They do not replace electromagnetic validation: training data may omit rare scan states, hardware ageing or changing platform conditions. Safety-critical beam controls should retain deterministic limits and auditable fallback modes.

Professional design must address spectrum licences, out-of-band emissions, national exposure limits, EIRP restrictions and interference coordination. Adaptive nulling should not be used to conceal noncompliant emissions. Secure beam-control interfaces are essential because malicious steering can disrupt service or increase exposure in unintended directions.

Recommended practice is to:

1. define scan volume, bandwidth, polarization and sidelobe metrics before selecting geometry;

2. test grating lobes over frequency and the complete visible region;

3. distinguish intended excitations from coupling-modified realized currents;

4. use embedded element patterns for final prediction;

5. include quantization, tolerance and thermal analyses;

6. validate simulations with calibrated near-field or far-field measurements; and

7. verify regulatory compliance for every operational beam state.

Chapter Summary

An array pattern results from coherent combination of fields radiated by spatially distributed elements. For identical, identically oriented and effectively uncoupled elements, pattern multiplication separates the element factor from the array factor. A uniform linear array has a closed-form array factor, enabling calculation of steering phases, nulls and beamwidths. For a broadside array, the first-null displacement is approximately λ/(Nd) on each side, giving a full FNBW of approximately 2λ/(Nd).

Grating lobes are visible replicas caused by excessive element spacing. For steering direction cosine u₀, strict avoidance requires d < λ/(1 + |u₀|). Planar arrays extend synthesis to two dimensions, while conformal arrays require orientation-dependent embedded element patterns and generally cannot be represented by a single scalar array factor alone.

Mutual coupling modifies currents, active impedances and radiation patterns. Uniform excitation offers a narrow beam but appreciable sidelobes; binomial excitation removes sidelobes at the cost of beamwidth and dynamic range. Dolph–Chebyshev synthesis provides equal-ripple sidelobes and an optimal classical beamwidth trade-off. Fourier synthesis interprets aperture design as spatial spectral transformation and supports both continuous and discrete synthesis. Practical arrays require coupling-aware calibration, wideband analysis, tolerance assessment and regulatory control.

Key Terms

Array factor: The field-interference function determined by element positions and complex excitations.

Pattern multiplication: Representation of the total pattern as a product of a common element pattern and an array factor.

Progressive phase: Constant excitation-phase difference between adjacent elements.

Broadside array: An array whose principal beam is perpendicular to its axis or aperture plane.

Endfire array: An array whose principal beam is directed along its axis.

First-null beamwidth: Angular separation between the first nulls surrounding the main beam.

Half-power beamwidth: Angular width between points 3 dB below the main-beam peak.

Direction cosine: Projection of an observation unit vector onto a specified coordinate axis.

Grating lobe: An unwanted visible-region maximum caused by spatial undersampling.

Beam squint: Frequency-dependent beam-direction change in a phase-steered array.

Scan loss: Reduction in realized gain as a beam is steered away from broadside.

Mutual coupling: Electromagnetic interaction among array elements.

Embedded element pattern: Pattern of one driven element with all other array ports terminated in specified loads.

Active impedance: Scan- and excitation-dependent input impedance of an operating array element.

Binomial array: An array excited according to binomial coefficients to obtain a sidelobe-free ideal pattern.

Dolph–Chebyshev array: A symmetric array with equal-ripple sidelobes and minimum classical beamwidth for a specified sidelobe level.

Amplitude taper: Deliberate variation of element amplitudes across an aperture.

Fourier synthesis: Determination of aperture excitation from a desired spatial spectrum.

Conformal array: An array whose elements follow a curved supporting surface.

Scan blindness: Severe radiation or matching degradation associated with coupled array or surface-wave modes.

Review Questions

Short Answer Questions

1. State the assumptions required for pattern multiplication.

2. What progressive phase steers a linear array toward θ₀?

3. Why can d = λ produce boundary grating lobes for broadside operation?

4. Define an embedded element pattern.

5. What is the principal advantage of Dolph–Chebyshev synthesis?

Long Answer Questions

1. Derive the closed-form array factor of a uniformly excited N-element linear array.

2. Explain grating-lobe formation and derive the spacing condition for a specified scan direction.

3. Compare uniform, binomial and Dolph–Chebyshev amplitude distributions in terms of beamwidth, sidelobes and implementation.

4. Develop the array-factor formulation for a separable rectangular planar array and explain its visible region.

5. Discuss the effects of mutual coupling on impedance, scan performance, calibration and pattern synthesis.

Analytical / Application Questions

1. A 12-element broadside array has d = 0.5λ. Calculate its exact and approximate FNBW and its approximate HPBW.

2. Determine whether an array with d = 0.65λ can scan to 40° from broadside without a visible grating lobe. Clearly define the direction cosine used.

3. Using the Chebyshev-polynomial method, derive normalized excitation coefficients for a five-element symmetric array with a specified −25 dB sidelobe level.

Multiple Choice Questions with Answers

Question 1

A ten-element uniform broadside array has spacing d = λ/2. What is its approximate full first-null beamwidth?

A. 0.2 rad

B. 0.4 rad

C. 0.8 rad

D. 1.0 rad

Correct Answer: B. 0.4 rad

Question 2

For a broadside linear array, which statement about d = λ is correct?

A. It eliminates all grating-lobe solutions.

B. It produces only one principal maximum at broadside.

C. It permits grating-lobe maxima at visible-region boundaries.

D. It is equivalent to d = λ/2 for every scan angle.

Correct Answer: C. It permits grating-lobe maxima at visible-region boundaries.

Question 3

What strict spacing condition avoids visible grating lobes when a linear array is steered to direction cosine u₀?

A. d < λ/(1 + |u₀|)

B. d < λ/(1 − |u₀|)

C. d > λ(1 + |u₀|)

D. d = λ/|u₀|

Correct Answer: A. d < λ/(1 + |u₀|)

Question 4

Which six-element amplitude ratio represents a binomial distribution?

A. 1:1:1:1:1:1

B. 1:2:3:3:2:1

C. 1:4:6:6:4:1

D. 1:5:10:10:5:1

Correct Answer: D. 1:5:10:10:5:1

Question 5

A desired sidelobe attenuation is 30 dB. What main-beam-to-sidelobe voltage ratio R is used in Chebyshev synthesis?

A. 10

B. 31.62

C. 1000

D. 3.162

Correct Answer: B. 31.62

Question 6

Which quantity most directly describes the excitation-dependent input behaviour of an element in a coupled array?

A. Free-space impedance

B. Isolated-element directivity

C. Active impedance

D. First-null beamwidth

Correct Answer: C. Active impedance

Question 7

When a rectangular planar array has separable excitations aₘₙ = bₘcₙ, its array factor can be written as:

A. AFₓ + AFᵧ

B. AFₓAFᵧ

C. AFₓ/AFᵧ

D. AFₓ − AFᵧ

Correct Answer: B. AFₓAFᵧ

Question 8

Which expression is appropriate for the total field of a general conformal array?

A. A common scalar element pattern multiplied by spacing alone

B. Only the scalar array factor, independent of element orientation

C. A vector sum using each element's orientation-dependent embedded pattern and spatial phase

D. The Fourier transform of element positions without excitation coefficients

Correct Answer: C. A vector sum using each element's orientation-dependent embedded pattern and spatial phase

Question 9

A phase-shifter array is steered to direction cosine u₀ at frequency f₀. Neglecting other effects, its beam direction cosine at frequency f is:

A. u₀f/f₀

B. u₀f₀/f

C. u₀(f − f₀)

D. u₀ for every frequency

Correct Answer: B. u₀f₀/f

Question 10

Which normalized amplitude sequence corresponds approximately to the six-element, −20 dB Chebyshev example in this chapter?

A. 1, 1, 1, 1, 1, 1

B. 0.1, 0.5, 1, 1, 0.5, 0.1

C. 0.5406, 0.7768, 1, 1, 0.7768, 0.5406

D. 0.7768, 0.5406, 1, 1, 0.5406, 0.7768

Correct Answer: C. 0.5406, 0.7768, 1, 1, 0.7768, 0.5406