Radar Waveforms and Ambiguity Theory

Introduction

Radar performance is determined not only by transmitter power, antenna gain, and receiver sensitivity, but also by the waveform used to illuminate a target. A waveform controls the radar’s ability to measure range, radial velocity, acceleration, target fluctuations, and interference environment. It also determines hardware complexity, spectral occupancy, probability of detection, range ambiguity, Doppler tolerance, and susceptibility to electronic attack.

A radar waveform is the time-varying electromagnetic signal transmitted by the radar. Its important attributes include duration, bandwidth, carrier frequency, modulation law, pulse-repetition interval, pulse-repetition frequency, phase structure, amplitude envelope, and coding. The received echo is a delayed, Doppler-shifted, attenuated, and possibly distorted version of this waveform. Waveform processing attempts to distinguish the echo from noise, clutter, interference, and other returns.

This chapter develops the principles of pulsed, continuous-wave, frequency-modulated continuous-wave, phase-coded, and frequency-coded waveforms. It then introduces matched filtering and the ambiguity function as central tools for evaluating range–Doppler resolution and measurement trade-offs.

Learning Objectives

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

1. Relate waveform duration and bandwidth to range and velocity performance.

2. Design basic pulsed, CW, FMCW, phase-coded, and frequency-coded waveforms.

3. Derive and interpret matched-filter output and pulse-compression gain.

4. Explain the ambiguity function and identify mainlobe, sidelobes, and Doppler tolerance.

5. Analyse range ambiguity, Doppler ambiguity, and range–Doppler coupling.

6. Select an appropriate waveform for surveillance, tracking, automotive, imaging, or instrumentation radar.

7. Evaluate practical limitations involving hardware, spectrum, clutter, interference, and regulation.

1. Signal Model and Fundamental Measures

Let the complex analytic transmitted waveform be

\[

s(t)=a(t)\exp[j(2\pi f_c t+\phi(t))],

\]

where \(a(t)\) is the amplitude envelope, \(f_c\) is carrier frequency, and \(\phi(t)\) is the modulation phase. The complex envelope is often written as \(u(t)=a(t)e^{j\phi(t)}\), with the carrier omitted for baseband analysis.

For a point target at delay \(\tau\), radial velocity \(v\), and complex reflectivity \(\alpha\), a simplified received signal is

\[

r(t)=\alpha s(t-\tau)e^{j2\pi f_D t}+n(t),

\]

where \(n(t)\) is receiver noise and, for a monostatic radar,

\[

f_D=\frac{2v}{\lambda}, \qquad \lambda=\frac{c}{f_c}.

\]

The factor of two arises because the wave travels to the target and back. The target range is

\[

R=\frac{c\tau}{2},

\]

where \(c\) is the speed of light. A delay change \(\Delta\tau\) corresponds to range change \(\Delta R=c\Delta\tau/2\).

1.1 Pulse duration and bandwidth

A rectangular pulse of duration \(T\) has approximate uncompressed range resolution

\[

\Delta R\approx \frac{cT}{2}.

\]

This expression assumes that resolution is defined by the ability to separate two equal-strength targets using a conventional matched filter and that pulse shape and detection criterion are not being used to specify a more exact resolution measure.

For a waveform with effective bandwidth \(B\), the fundamental range-resolution scale is approximately

\[

\Delta R\approx \frac{c}{2B}.

\]

The exact coefficient depends on the waveform spectrum and the adopted resolution criterion, such as null-to-null separation, 3-dB width, or loss-based separation. Thus, short pulses obtain fine range resolution directly, whereas long coded or chirped pulses obtain fine resolution through compression.

The time–bandwidth product is

\[

BT,

\]

where \(T\) is waveform duration and \(B\) is effective bandwidth. A large \(BT\) permits high transmitted energy, proportional to pulse duration for fixed power, while retaining resolution governed mainly by bandwidth.

1.2 Pulse repetition frequency and ambiguities

For a periodic pulsed radar,

\[

\mathrm{PRF}=f_r=\frac{1}{T_r},

\]

where \(T_r\) is the pulse-repetition interval. The unambiguous range is

\[

R_u=\frac{cT_r}{2}=\frac{c}{2f_r}.

\]

An echo from a target beyond \(R_u\) may arrive during a later listening interval and appear at an incorrect range. The unambiguous Doppler frequency interval for uniformly sampled pulse-to-pulse data is approximately \(-f_r/2\le f_D<f_r/2\). Consequently, the unambiguous radial-velocity interval is

\[

|v|<\frac{\lambda f_r}{4},

\]

for the stated principal Doppler interval. High PRF improves Doppler ambiguity performance but reduces unambiguous range; low PRF does the opposite. Medium- and staggered-PRF designs combine multiple measurements to resolve ambiguities.

2. Principal Radar Waveforms

2.1 Conventional pulsed waveforms

A conventional pulsed radar transmits a pulse and listens for its echo. For an approximately rectangular pulse,

\[

s(t)=A\,\mathrm{rect}\left(\frac{t}{T}\right)e^{j2\pi f_c t}.

\]

Its energy is approximately \(E_s=A^2T\) in normalized complex-envelope units. Shortening \(T\) improves range resolution but reduces energy and generally demands higher peak power for the same energy. Increasing peak power may be limited by transmitter breakdown, thermal stress, and electromagnetic compatibility.

A pulsed radar must provide a transmit–receive switch or duplexer, timing control, a receiver protected from transmitter leakage, and a signal processor that associates returns with the appropriate pulse. Pulsed waveforms are suitable for long-range surveillance because they naturally separate transmission and reception in time.

2.2 Continuous-wave radar

An unmodulated CW radar transmits continuously. A stationary target produces a return at the carrier frequency, while a moving target produces a Doppler shift. Coherent detection can therefore measure radial velocity accurately, but an unmodulated CW radar cannot determine range from delay because there is no time reference identifying when a particular transmitted feature was emitted.

CW radar is useful for speed measurement, traffic enforcement, motion sensing, and illuminators for bistatic systems. It requires isolation between transmitter and receiver and often uses separate transmit and receive antennas. Clutter suppression is possible through Doppler filtering, but a target with near-zero radial velocity may be difficult to distinguish from stationary clutter.

2.3 FMCW waveforms

In frequency-modulated continuous-wave radar, the carrier frequency changes continuously, commonly as a linear chirp:

\[

f(t)=f_0+Kt,

\]

where \(K=B/T_c\) is chirp rate, \(B\) is sweep bandwidth, and \(T_c\) is sweep duration. The instantaneous phase is obtained by integration:

\[

\phi(t)=2\pi\left(f_0t+\frac{K}{2}t^2\right).

\]

After mixing the received chirp with a delayed copy of the transmitted signal, the beat frequency for a stationary target is approximately

\[

f_b=K\tau.

\]

Thus,

\[

R=\frac{cf_b}{2K}.

\]

For a moving target, Doppler contributes to the beat frequency. In a triangular or up–down chirp scheme, the beat frequencies on the two slopes can be combined to estimate range and velocity separately. For an up-chirp and down-chirp, under a common sign convention,

\[

f_{b,u}=K\tau+f_D,

\qquad

f_{b,d}=-K\tau+f_D.

\]

Therefore,

\[

f_D=\frac{f_{b,u}+f_{b,d}}{2},

\qquad

K\tau=\frac{f_{b,u}-f_{b,d}}{2}.

\]

Sign conventions depend on the mixer and Doppler definition, but the separation principle is unchanged. FMCW radar provides fine range resolution through sweep bandwidth, operates with low peak power, and is compact. Its challenges include transmitter leakage, phase noise, beat-frequency coupling, limited maximum range due to sweep timing, and interference between nearby FMCW radars.

2.4 Phase-coded waveforms

A phase-coded pulse consists of a sequence of subpulses, each having a prescribed phase. For a binary code,

\[

s(t)=\sum_{n=0}^{N-1} a_n p(t-nT_c),

\qquad a_n\in\{+1,-1\},

\]

where \(p(t)\) is a subpulse of duration \(T_c\), and \(N\) is code length. The total pulse duration is approximately \(T=NT_c\), while the bandwidth is of order \(1/T_c\). Matched filtering compresses the pulse to a width governed by \(T_c\), yielding approximate resolution

\[

\Delta R\approx \frac{cT_c}{2}\approx\frac{c}{2B}.

\]

For a well-designed binary code, the idealized processing gain is approximately the code length \(N\) in power, or \(10\log_{10}N\) dB, subject to losses and sidelobe constraints. Barker codes have very low aperiodic sidelobes but exist only for certain short lengths. Longer codes, polyphase codes, and nonlinear codes provide larger time–bandwidth products but require careful sidelobe control.

Doppler shifts change the relative phase between code elements and can degrade compression. Consequently, a code must be evaluated using its two-dimensional delay–Doppler response, not only its zero-Doppler autocorrelation.

2.5 Frequency-coded and stepped-frequency waveforms

Frequency-coded waveforms divide a pulse into subpulses having different carrier frequencies. A stepped-frequency waveform transmits a sequence of narrowband pulses or bursts at frequencies

\[

f_k=f_0+k\Delta f,

\qquad k=0,1,\ldots,N-1.

\]

The phase of the return across frequency steps contains delay information. The synthesized bandwidth is approximately

\[

B_s\approx (N-1)\Delta f

\]

when the frequency span is measured from the first to last step. The nominal range-resolution scale is then approximately \(c/(2B_s)\), provided phase coherence is maintained and calibration is accurate.

Frequency coding can reduce instantaneous bandwidth requirements and support high-resolution imaging or synthetic wideband operation. However, it is sensitive to target motion during the frequency sequence, oscillator phase errors, frequency-dependent scattering, and ambiguities associated with frequency spacing. If the step spacing is \(\Delta f\), the delay periodicity associated with uniformly sampled frequency measurements is approximately \(1/\Delta f\), corresponding to an unambiguous range scale of

\[

R_{u,s}\approx \frac{c}{2\Delta f}.

\]

3. Matched Filtering and Pulse Compression

For additive white Gaussian noise and a known deterministic signal, the matched filter maximizes the output signal-to-noise ratio at a selected delay. Its impulse response is proportional to the time-reversed complex conjugate of the signal:

\[

h(t)=k s^*(t_0-t),

\]

where \(t_0\) is the chosen sampling time and \(k\) is a scale factor. The output is proportional to the correlation between the received waveform and a delayed replica of the transmitted waveform.

If the received signal is \(\alpha s(t-\tau)\), the matched-filter output versus trial delay is proportional to the autocorrelation

\[

R_s(\eta)=\int s(t)s^*(t-\eta)\,dt.

\]

The peak occurs when \(\eta=\tau\). The matched filter is optimal under its assumptions, but not universally optimal: colored noise, unknown signal parameters, clutter, mismatch, transmitter distortion, and nonlinear detection may require whitening, adaptive filtering, or robust processing.

For a rectangular pulse of duration \(T\), the peak output signal energy is proportional to \(T\), while the compressed response has width of order the inverse bandwidth. A long LFM or coded pulse can therefore transmit energy associated with \(T\) but resolve targets according to \(B\). The ideal pulse-compression ratio is approximately

\[

G_{pc}\approx BT,

\]

or in decibels,

\[

G_{pc,dB}\approx10\log_{10}(BT).

\]

This is an approximate processing-gain measure. Actual gain is reduced by weighting, quantization, Doppler mismatch, phase noise, timing error, finite filter implementation, and transmitter–receiver amplitude or phase distortion.

Weighting the matched-filter spectrum can reduce sidelobes, but it broadens the mainlobe and lowers peak signal-to-noise ratio. This is a fundamental resolution–sidelobe trade-off. Taylor, Kaiser, and other windows are commonly used when sidelobe suppression is more important than maximum resolution or peak gain.

Worked Example 1: Pulse compression

A radar transmits a linear-FM pulse of duration \(T=20\ \mu\text{s}\) and bandwidth \(B=10\ \text{MHz}\).

The time–bandwidth product is

\[

BT=(20\times10^{-6})(10\times10^6)=200.

\]

The ideal processing gain is

\[

10\log_{10}(200)=23.0\ \text{dB}\quad\text{(approximately)}.

\]

The range-resolution scale is

\[

\Delta R\approx\frac{3\times10^8}{2(10\times10^6)}=15\ \text{m}.

\]

The uncompressed 20-\(\mu\)s pulse would correspond to approximately

\[

\frac{cT}{2}=3000\ \text{m},

\]

so pulse compression improves the range-response width by roughly a factor of 200, subject to waveform and implementation losses.

4. The Ambiguity Function

The ambiguity function describes the response of a matched filter when the received signal differs from the reference by delay \(\tau\) and Doppler frequency \(f_D\). One common definition is

\[

\chi(\tau,f_D)=\int_{-\infty}^{\infty}s(t)s^*(t-\tau)e^{j2\pi f_D t}\,dt.

\]

Definitions may differ by a phase factor or sign convention; the magnitude \(|\chi(\tau,f_D)|\) is generally the principal object for resolution analysis. The origin is the matched response to a perfectly matched target. The mainlobe indicates the region over which delay and Doppler cannot be sharply distinguished. Sidelobes indicate responses from mismatched delays and Dopplers.

Important features include:

• Delay cut: \(|\chi(\tau,0)|\), describing range response at zero Doppler.
• Doppler cut: \(|\chi(0,f_D)|\), describing velocity tolerance at zero delay.
• Peak: normally at zero delay and zero Doppler for a normalized waveform.
• Mainlobe widths: measures of delay and Doppler resolution.
• Sidelobes: potential masking of weak targets by strong targets.
• Ridge or tilt: evidence of range–Doppler coupling.

The ambiguity function is not merely a mathematical plot. It is a design map showing which pairs of target delay and Doppler produce similar matched-filter outputs. A desirable waveform depends on the mission. A tracking radar may accept moderate sidelobes in exchange for Doppler tolerance, while a dense-surveillance radar may demand very low range sidelobes.

4.1 Rectangular pulse ambiguity behaviour

For a rectangular pulse, the zero-Doppler delay response has a triangular magnitude envelope, with width determined by pulse duration. The zero-delay Doppler response has a sinc-like form whose first null spacing is approximately inverse pulse duration. Hence a short pulse offers broad Doppler tolerance but poor energy, while a long pulse offers narrow Doppler response and greater Doppler sensitivity.

4.2 LFM ambiguity behaviour

An LFM pulse generally has a narrow delay mainlobe of scale \(1/B\), while its ambiguity function has a tilted ridge. Doppler can shift the apparent delay after matched filtering; this is range–Doppler coupling. For an up-chirp, a Doppler shift may resemble a time shift in the dechirped or matched-filter response. Up/down chirp pairs, alternative processing, and waveform design can reduce this coupling.

4.3 Phase-coded ambiguity behaviour

Phase-coded waveforms can provide a sharp delay response, but Doppler changes the phase progression across subpulses. A code with excellent zero-Doppler autocorrelation may have unacceptable sidelobes at operational target velocities. Therefore code selection should use ambiguity-surface simulations over the expected Doppler interval.

5. Range–Doppler Design Trade-offs

Range resolution is primarily controlled by bandwidth, whereas Doppler resolution in a coherent pulse train is controlled by coherent observation time. If \(M\) pulses are observed with pulse-repetition interval \(T_r\), the coherent processing interval is approximately

\[

T_{CPI}=MT_r.

\]

The Doppler mainlobe scale is of order \(1/T_{CPI}\), with exact width depending on weighting and definition. The corresponding velocity-resolution scale is approximately

\[

\Delta v\approx\frac{\lambda}{2T_{CPI}}.

\]

This formula assumes coherent processing, stable phase reference, approximately constant target velocity, and a Doppler spectrum whose width is characterized by inverse observation time.

Increasing CPI improves Doppler resolution but can cause target range migration, acceleration mismatch, scan loss, and reduced update rate. Increasing bandwidth improves range resolution but raises sampling rate, data volume, calibration difficulty, and spectral-compatibility concerns. Increasing PRF expands Doppler ambiguity limits but reduces unambiguous range. Waveform design is therefore a multidimensional optimization rather than a single-resolution problem.

A radar designer should distinguish:

1. Resolution: ability to separate nearby targets.

2. Accuracy: closeness of an estimate to the true value.

3. Precision: repeatability of the estimate.

4. Ambiguity: multiple parameter values producing equivalent or confusing observations.

5. Sidelobe level: response caused by a strong target at an incorrect parameter location.

6. Practical Design Procedure

A systematic waveform-design process is as follows:

1. Define mission requirements: maximum range, range resolution, velocity interval, velocity resolution, update rate, target fluctuation, clutter environment, and allowable false-alarm rate.

2. Select the operating band and available spectrum, considering propagation, atmospheric loss, antenna size, and regulation.

3. Choose a waveform family: pulsed, CW, FMCW, phase-coded, frequency-coded, or a hybrid.

4. Set bandwidth from the desired range resolution using \(B\approx c/(2\Delta R)\), with margin for practical response criteria.

5. Set pulse duration or code length from energy, peak-power, and duty-cycle requirements.

6. Select PRF or waveform timing to manage range and Doppler ambiguities.

7. Compute the ambiguity function, including expected Doppler, acceleration, clutter, and waveform mismatch.

8. Design matched-filter weighting and sidelobe control.

9. Assess transmitter linearity, receiver dynamic range, sampling rate, phase noise, calibration, and processing load.

10. Validate using hardware-in-the-loop and measured target or channel data rather than relying only on ideal simulations.

7. Illustrative Case: Automotive FMCW Radar

Background

Consider an illustrative 77-GHz automotive radar requiring approximately 0.5-m range resolution and measurement of vehicle speeds over a broad interval. The example is conceptual; actual products must meet applicable regulatory and automotive standards.

Problem

The radar must distinguish vehicles in adjacent lanes and estimate both range and radial velocity while operating with limited peak power and compact hardware. Multiple vehicles and nearby radars create interference and dynamic-range challenges.

Analysis

The required bandwidth is approximately

\[

B\approx\frac{c}{2\Delta R}

=\frac{3\times10^8}{2(0.5)}

=300\ \text{MHz}.

\]

A chirp with bandwidth near 300 MHz therefore provides the required range-resolution scale. Suppose the chirp duration is \(T_c=50\ \mu\text{s}\). The chirp slope is

\[

K=\frac{B}{T_c}=\frac{300\times10^6}{50\times10^{-6}}

=6\times10^{12}\ \text{Hz/s}.

\]

For a stationary target at 50 m, the delay is

\[

\tau=\frac{2R}{c}=\frac{100}{3\times10^8}=333.3\ \text{ns}.

\]

The range beat frequency is then

\[

f_b=K\tau=(6\times10^{12})(333.3\times10^{-9})\approx2.0\ \text{MHz}.

\]

For moving targets, Doppler shifts the beat frequency. Using up- and down-chirps or multiple chirps permits separation of range and velocity. A coherent processing interval across several chirps improves velocity estimation, but it must be short enough to limit motion-induced range migration and long enough to obtain useful Doppler resolution.

Approach and solution

The design uses a linear FMCW chirp sequence, calibrated transmit and receive chains, digital dechirping, fast-time FFT processing for range, and slow-time Doppler FFT processing for velocity. Windowing reduces sidelobes caused by finite observation intervals. Adaptive interference detection rejects corrupted chirps, while antenna arrays provide angular separation.

Outcome

The waveform provides fine range resolution without requiring a high-power pulsed transmitter. Its performance depends strongly on chirp linearity, phase noise, ADC bandwidth, leakage cancellation, synchronization, and interference management.

Learning points

The example demonstrates that bandwidth sets the range-resolution scale, chirp slope maps delay into beat frequency, and coherent processing across chirps supplies Doppler information. It also illustrates why nominal formulas must be followed by hardware, interference, and regulatory analysis.

8. Limitations, Governance, and Contemporary Developments

Real waveforms are affected by amplifier AM–PM distortion, oscillator instability, finite switching time, antenna mismatch, quantization, calibration error, clutter, multipath, and target micro-Doppler. A waveform with a theoretically excellent ambiguity function may perform poorly if its implementation generates spectral regrowth or phase noise.

Radar spectrum is regulated nationally and internationally. Designers must control occupied bandwidth, out-of-band emissions, interference to incumbent services, and coexistence with other radars. Automotive and aerospace systems also require electromagnetic compatibility, safety assurance, cybersecurity, and traceable configuration control. Waveform agility must not become an uncontrolled source of interference.

Cognitive and adaptive radar systems select waveforms in response to clutter, interference, target uncertainty, and mission priorities. Machine learning can assist waveform selection, interference classification, and ambiguity-surface prediction, but data-driven methods require representative training data, uncertainty assessment, explainability, and protection against adversarial manipulation. AI should support—not replace—validated physical models and safety-critical engineering judgement.

Modern systems increasingly use digital arbitrary-waveform generation, direct RF sampling, multistatic synchronization, MIMO waveform diversity, orthogonal or semi-orthogonal codes, and joint radar–communications designs. These developments increase flexibility but also intensify requirements for timing coherence, spectral control, computational efficiency, and secure waveform management.

Best-Practice Checklist

• Specify resolution criteria explicitly rather than quoting only \(c/(2B)\).
• Evaluate the full delay–Doppler ambiguity function.
• Include sidelobes and strong-target masking in performance budgets.
• Distinguish unambiguous range and velocity from actual operating coverage.
• Account for Doppler mismatch, acceleration, range migration, and calibration error.
• Use weighting only after quantifying its loss in resolution and signal-to-noise ratio.
• Verify spectral emissions and coexistence requirements early.
• Test the waveform with measured hardware impairments and representative interference.
• Preserve timing, frequency, and phase integrity throughout the processing chain.

Chapter Summary

Radar waveforms determine how transmitted energy is distributed in time, frequency, phase, and modulation structure. Pulse duration controls uncompressed range resolution, while effective bandwidth establishes the fundamental compressed range-resolution scale. PRF determines the principal range and Doppler ambiguity intervals, creating an unavoidable design trade-off.

CW radar measures Doppler but not range without modulation. FMCW radar maps delay into beat frequency and is attractive for compact, low-peak-power systems. Phase-coded and frequency-coded waveforms provide pulse compression, flexible energy management, and potentially large time–bandwidth products, but they require careful Doppler and sidelobe analysis. Matched filtering maximizes output SNR for a known signal in white noise, while practical weighting trades peak gain and resolution against sidelobe suppression.

The ambiguity function provides a unified delay–Doppler description of waveform behaviour. Its mainlobe indicates resolution and its sidelobes indicate masking risk; tilt or ridge structure reveals range–Doppler coupling. Professional waveform design therefore combines mathematical analysis, implementation constraints, spectral regulation, clutter and interference modelling, and experimental validation.

Key Terms

Waveform: Time-varying signal transmitted by a radar.

Pulse duration: Time interval over which a radar pulse is transmitted.

Bandwidth: Effective frequency span occupied by a waveform.

Time–bandwidth product: Product of waveform duration and effective bandwidth.

Pulse repetition frequency: Number of transmitted pulses per second.

Unambiguous range: Maximum range that can be associated with one pulse interval without range aliasing.

Doppler frequency: Frequency shift caused by radial target motion.

FMCW: Continuous-wave radar using frequency modulation, commonly a chirp.

Chirp: Signal whose instantaneous frequency varies with time.

Phase-coded waveform: Waveform whose subpulses carry prescribed phase states.

Frequency-coded waveform: Waveform containing prescribed frequency states or steps.

Matched filter: Filter matched to a known signal to maximize output SNR in white noise.

Pulse compression: Processing that combines long-pulse energy with short-pulse resolution.

Ambiguity function: Delay–Doppler response of a waveform and matched filter.

Mainlobe: Central high-response region of an ambiguity or correlation function.

Sidelobe: Secondary response away from the desired delay–Doppler point.

Range–Doppler coupling: Dependence of estimated range on target Doppler, or vice versa.

Coherent processing interval: Time over which phase-coherent samples are jointly processed.

Micro-Doppler: Modulation of Doppler caused by rotating, vibrating, or moving target parts.

Waveform agility: Adaptive selection or modification of radar waveforms.

Review Questions

Short Answer Questions

1. Why does effective bandwidth, rather than pulse duration alone, determine compressed range resolution?

2. State the relationship between PRF and unambiguous range.

3. What information can an unmodulated CW radar measure, and what information does it lack?

4. Define the ambiguity function and explain the meaning of its sidelobes.

5. Why does matched-filter weighting usually reduce sidelobes at the cost of resolution or peak gain?

Long Answer Questions

1. Explain the operating principles, advantages, and limitations of pulsed, CW, FMCW, phase-coded, and frequency-coded radar waveforms.

2. Derive the FMCW beat-frequency relationship for a stationary target and explain how up- and down-chirps separate range and Doppler.

3. Discuss pulse compression, matched filtering, time–bandwidth product, and practical processing losses.

4. Explain how PRF, coherent processing interval, bandwidth, and pulse duration create range–Doppler trade-offs.

5. Describe how the ambiguity function should be used to select and validate a waveform for a surveillance radar.

Analytical / Application Questions

1. A radar requires 3-m range resolution. Estimate the minimum effective bandwidth using the standard range-resolution scale, and discuss why the actual design may require additional bandwidth.

2. A radar operates at wavelength 0.03 m and PRF 20 kHz. Determine the principal unambiguous radial-velocity limit and explain the corresponding range trade-off.

3. Design a conceptual FMCW waveform for 1-m range resolution using a 40-\(\mu\)s chirp. Calculate the approximate bandwidth and chirp slope, then identify at least three implementation limitations.

Multiple Choice Questions with Answers

Question 1

For a monostatic radar, the range corresponding to a measured delay τ is

A. cτ

B. cτ/2

C. 2cτ

D. τ/(2c)

Correct Answer: B. cτ/2

Question 2

A waveform with effective bandwidth B has an approximate range-resolution scale of

A. cB/2

B. 2c/B

C. c/(2B)

D. B/(2c)

Correct Answer: C. c/(2B)

Question 3

If the pulse-repetition frequency is increased, the unambiguous range generally

A. increases

B. decreases

C. remains exactly unchanged

D. becomes equal to the wavelength

Correct Answer: B. decreases

Question 4

An unmodulated CW radar fundamentally measures target

A. range only

B. radial velocity but not delay-based range

C. acceleration only

D. pulse width

Correct Answer: B. radial velocity but not delay-based range

Question 5

For an FMCW chirp with slope K and stationary-target delay τ, the approximate beat frequency is

A. K/τ

B. τ/K

C. Kτ

D. K+τ

Correct Answer: C. Kτ

Question 6

The ideal pulse-compression processing gain in decibels is approximately

A. BT dB

B. 10 log10(BT) dB

C. 20 log10(B/T) dB

D. B+T dB

Correct Answer: B. 10 log10(BT) dB

Question 7

The matched filter for a known signal in white Gaussian noise is proportional to the signal’s

A. time-reversed complex conjugate

B. time derivative only

C. Fourier magnitude without phase

D. instantaneous frequency

Correct Answer: A. time-reversed complex conjugate

Question 8

In an ambiguity function, sidelobes primarily indicate

A. the speed of light

B. responses from mismatched delay–Doppler combinations

C. the transmitter carrier power only

D. antenna physical diameter

Correct Answer: B. responses from mismatched delay–Doppler combinations

Question 9

For a coherent processing interval T_CPI, Doppler resolution is approximately proportional to

A. T_CPI

B. 1/T_CPI

C. carrier frequency squared

D. pulse amplitude only

Correct Answer: B. 1/T_CPI

Question 10

A major consequence of increasing FMCW sweep bandwidth, all else equal, is

A. poorer range resolution

B. improved range resolution but greater sampling and spectral demands

C. elimination of Doppler effects

D. removal of transmitter leakage

Correct Answer: B. improved range resolution but greater sampling and spectral demands