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Radar Sensing Fundamentals

The fundamental task of radar sensing is to extract information about targets — their range (distance), radial velocity, and angle — from the echoes of a transmitted waveform. The key tool is the matched filter, which maximises the output signal-to-noise ratio (SNR) for a known waveform in additive white Gaussian noise.

Consider a transmitted baseband signal $s(t)$ with bandwidth $B$ and duration $T$ . A point target at range $R$ with radial velocity $v$ produces an echo:

$y(t) = \alpha \, s(t - \tau) \, e^{j2\pi f_d t} + n(t)$

where $\alpha$ is the complex target reflectivity (incorporating path loss and radar cross section), $\tau = 2R/c$ is the round-trip delay, $f_d = 2v/\lambda$ is the Doppler frequency shift, and $n(t) \sim \mathcal{CN}(0, \sigma^2)$ is additive white Gaussian noise. The factor of 2 in both $\tau$ and $f_d$ arises from the two-way (transmit and receive) propagation in monostatic radar.

The matched filter for delay estimation correlates the received signal with a time-shifted copy of the transmitted waveform:

$z(\hat{\tau}) = \int_{-\infty}^{\infty} y(t) \, s^*(t - \hat{\tau}) \, dt$

This correlation peaks at $\hat{\tau} = \tau$ , yielding the target range $R = c\tau/2$ . The width of the correlation peak determines the range resolution.

Definition:
Ambiguity Function

The ambiguity function of a waveform $s(t)$ is defined as:

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

The squared magnitude $|\chi(\tau, f_d)|^2$ is called the ambiguity surface and characterises the joint range-Doppler resolution of the waveform. It satisfies the following properties:

Peak normalisation: $|\chi(0, 0)|^2 = E_s^{2}$ , where $E_s = \int |s(t)|^2 \, dt$ is the waveform energy.
Symmetry: $|\chi(\tau, f_d)| = |\chi(-\tau, -f_d)|$ .
Volume conservation (Moyal's identity): $\iint |\chi(\tau, f_d)|^2 \, d\tau \, df_d = E_s^{2}$ This fundamental constraint means that the ambiguity volume is fixed: narrowing the main lobe in one dimension necessarily raises sidelobes elsewhere. The ambiguity surface cannot be made arbitrarily narrow in both delay and Doppler simultaneously.
Marginals: The zero-Doppler cut $\chi(\tau, 0)$ is the autocorrelation of $s(t)$ ; the zero-delay cut $\chi(0, f_d)$ is related to the spectral autocorrelation.

The ambiguity function is the central tool for waveform design in radar signal processing. An ideal "thumbtack" ambiguity function — a sharp peak at the origin with uniformly low sidelobes — cannot exist due to Moyal's identity. In practice, waveform design aims to shape the ambiguity function to concentrate energy near the origin with acceptable sidelobe levels.

Definition:
Range-Doppler Map

A range-Doppler map is a two-dimensional representation of the radar scene obtained by evaluating the matched filter output across a grid of delay-Doppler hypotheses. For a sequence of $M$ received pulses (or OFDM symbols), the range-Doppler map is:

$\mathbf{Z}(p, q) = \sum_{m=0}^{M-1} \sum_{k=0}^{K-1} Y_{m,k} \, e^{-j2\pi k p / K} \, e^{j2\pi m q / M}$

where $Y_{m,k}$ is the received signal on the $k$ -th subcarrier (or range bin) of the $m$ -th symbol (or pulse), $p$ indexes the range bins, and $q$ indexes the Doppler bins. The map displays peaks at the range-Doppler coordinates of each target. In matrix form, this is a 2D-FFT applied first across the subcarrier (fast-time) dimension for range processing, then across the symbol (slow-time) dimension for Doppler processing.

The range-Doppler map is the radar analogue of the time-frequency representation used in communication signal processing. Each bin in the map corresponds to a hypothesis about a specific delay-Doppler pair, and targets appear as peaks above the noise floor. Target detection is performed by applying a threshold (e.g., CFAR — constant false alarm rate) to the map.

Theorem: Radar Range and Velocity Resolution

For a waveform with bandwidth $B$ and coherent observation time $T$ :

Range resolution: $\Delta r = \frac{c}{2B}$ Two targets separated by less than $\Delta r$ in range cannot be resolved. This follows from the width of the autocorrelation (zero-Doppler cut of the ambiguity function), which is approximately $1/B$ in delay.
Velocity resolution: $\Delta v = \frac{\lambda}{2T}$ equivalently, the Doppler resolution is $\Delta f_d = 1/T$ . Two targets separated by less than $\Delta v$ in radial velocity cannot be resolved.
Maximum unambiguous range (for pulsed radar with PRI $T_p$ ): $R_{\max} = \frac{c T_p}{2}$
Maximum unambiguous velocity (for pulsed radar with PRI $T_p$ ): $v_{\max} = \frac{\lambda}{4 T_p}$

The time-bandwidth product $BT$ characterises the waveform's degrees of freedom: a large $BT$ enables simultaneously fine range and velocity resolution. The resolution cell volume in the range-velocity plane is $\Delta r \cdot \Delta v = c\lambda/(4BT)$ , which shrinks as $BT$ increases.

Proof

Derivation

Range resolution. The matched filter output for a signal with bandwidth $B$ has a main lobe width of approximately $1/B$ in the delay domain. Two point targets at delays $\tau_1$ and $\tau_2$ can be resolved when $|\tau_1 - \tau_2| > 1/B$ , i.e., when $|R_1 - R_2| > c/(2B) = \Delta r$ .

Velocity resolution. The coherent integration of $M$ pulses over a total observation time $T$ produces a sinc-like peak in the Doppler domain with a main lobe width of $1/T$ in frequency. Two targets with Doppler shifts $f_{d,1}$ and $f_{d,2}$ can be resolved when $|f_{d,1} - f_{d,2}| > 1/T$ . Since $f_d = 2v/\lambda$ , the velocity resolution is $\Delta v = \lambda/(2T)$ .

Maximum unambiguous range. For a pulse repetition interval (PRI) $T_p$ , echoes from targets beyond $R_{\max} = cT_p/2$ arrive after the next pulse is transmitted, causing range ambiguity.

Maximum unambiguous velocity. The Doppler phase shift between consecutive pulses is $\phi = 2\pi f_d T_p$ . Unambiguous estimation requires $|\phi| < \pi$ , i.e., $|f_d| < 1/(2T_p)$ , yielding $|v| < \lambda/(4T_p)$ .

Example: Resolution of a 77 GHz Automotive Radar

A 77 GHz automotive radar has bandwidth $B = 150$ MHz and a coherent processing interval of $T = 10$ ms (corresponding to $M = 128$ chirps at a PRI of $T_p = 78.125\;\mu\text{s}$ ). Compute the range resolution, velocity resolution, maximum unambiguous range and velocity, and the time-bandwidth product.

Solution

Wavelength and range resolution

$\lambda = c/f_0 = 3 \times 10^8 / 77 \times 10^9 = 3.896$ mm.

$\Delta r = \frac{c}{2B} = \frac{3 \times 10^8}{2 \times 150 \times 10^6} = 1.0\;\text{m}$

Velocity resolution

$\Delta v = \frac{\lambda}{2T} = \frac{3.896 \times 10^{-3}}{2 \times 10 \times 10^{-3}} = 0.195\;\text{m/s} \approx 0.7\;\text{km/h}$

Maximum unambiguous range and velocity

$R_{\max} = \frac{c T_p}{2} = \frac{3 \times 10^8 \times 78.125 \times 10^{-6}}{2} \approx 11.7\;\text{km}$

$v_{\max} = \frac{\lambda}{4 T_p} = \frac{3.896 \times 10^{-3}}{4 \times 78.125 \times 10^{-6}} = 12.47\;\text{m/s} \approx 44.9\;\text{km/h}$

Time-bandwidth product

$BT = 150 \times 10^6 \times 10 \times 10^{-3} = 1.5 \times 10^6$ , providing approximately $1.5$ million degrees of freedom in the delay-Doppler plane. In practice, automotive radars use up to 4 GHz bandwidth to achieve sub-5 cm range resolution. $\blacksquare$

Ambiguity Function Geometry — The ambiguity function $|\chi(\tau, f_d)|^2$ for three representative waveforms: (a) a linear frequency modulated (LFM) chirp showing the characteristic ridge tilted at $45°$ in the delay-Doppler plane, (b) an unweighted rectangular pulse with a "bed of nails" ambiguity along the Doppler axis, and (c) a random phase-coded waveform with a thumbtack-like shape (sharp main lobe, uniformly distributed sidelobes). Moyal's identity guarantees that the total volume under each surface is identical; only the distribution of energy differs.

Ambiguity Function: Bandwidth vs Resolution

Watch the ambiguity function main lobe narrow as the signal bandwidth sweeps from 10 MHz to 500 MHz. The narrower the main lobe, the finer the range resolution (

\Delta r = c/(2B)

). This animation illustrates the fundamental bandwidth-resolution relationship.

As bandwidth increases, the zero-Doppler cut of the ambiguity function narrows, improving range resolution.

Ambiguity Function Surface

Explore the ambiguity function $|\chi(\tau, f_d)|^2$ for different waveform types. Select among three waveforms: (1) a linear frequency modulated (LFM) chirp, which produces a ridge-shaped ambiguity tilted in the delay-Doppler plane; (2) an OFDM waveform with random data, yielding a near-thumbtack shape with data-dependent sidelobes; and (3) a random phase-coded waveform with low autocorrelation sidelobes. Adjust the bandwidth and duration to observe how these parameters control the width of the main lobe in the delay and Doppler dimensions, respectively. Note that Moyal's identity ensures the total volume is conserved: narrowing the main lobe in one dimension spreads energy to sidelobes.

Parameters

Waveform (1=chirp, 2=OFDM, 3=random)1

Bandwidth (MHz)100

Duration (us)10

Range-Doppler Processing via 2D-FFT

For a coherent processing interval (CPI) consisting of $M$ pulses (or OFDM symbols), each sampled at $K$ range bins, the received data can be arranged into a $K \times M$ matrix $\mathbf{Y}$ where the $(k, m)$ entry is the signal at the $k$ -th range sample of the $m$ -th pulse. A single point target at range bin $p_0$ with Doppler bin $q_0$ contributes:

$Y_{k,m} = \alpha \, e^{-j2\pi k p_0 / K} \, e^{j2\pi m q_0 / M}$

The range-Doppler map is obtained by applying:

An IFFT across rows (subcarrier/fast-time dimension) for range compression;
An FFT across columns (symbol/slow-time dimension) for Doppler processing.

This 2D-FFT has computational complexity $O(KM \log(KM))$ and produces a $K \times M$ range-Doppler map where each target appears as a peak at coordinates $(p_0, q_0)$ . The peak height is proportional to $|\alpha| KM$ (coherent processing gain), while the noise floor is proportional to $\sigma\sqrt{KM}$ , yielding an output SNR gain of $KM$ over a single sample.

Range-Doppler Map with Targets

Visualise the range-Doppler map produced by 2D-FFT processing of a simulated OFDM radar signal. Place multiple targets at random range-velocity positions and observe how they appear as peaks in the map. Adjust the number of targets to see how closely spaced targets interact. Vary the SNR to observe the transition from clear peaks to noise-limited detection. Increase the bandwidth to sharpen the range resolution (narrower peaks along the range axis).

Parameters

Targets3

SNR (dB)20

BW (MHz)100

Ambiguity Function

The function $\chi(\tau, f_d) = \int s(t) s^*(t-\tau) e^{j2\pi f_d t} dt$ that characterises the joint range-Doppler resolution of a radar waveform. Its squared magnitude $|\chi(\tau,f_d)|^2$ is the ambiguity surface; Moyal's identity constrains its total volume to be constant.

Related: ISAC, Range-Doppler Map

Range-Doppler Map

A two-dimensional representation of the radar scene produced by applying a 2D-FFT to the received signal matrix. Each pixel corresponds to a range-velocity hypothesis; targets appear as peaks above the noise floor.

Related: Ambiguity Function

Common Mistake: Confusing Resolution with Maximum Unambiguous Range/Velocity

Mistake:

Assuming that finer range resolution (larger bandwidth) also increases the maximum unambiguous range.

Correction:

Range resolution $\Delta r = c/(2B)$ and maximum unambiguous range $R_{\max} = cT_p/2$ depend on different parameters. Increasing bandwidth improves resolution but does not affect $R_{\max}$ , which depends on the pulse repetition interval $T_p$ . Similarly, velocity resolution $\Delta v = \lambda/(2T)$ depends on the coherent processing interval, while $v_{\max} = \lambda/(4T_p)$ depends on the PRI.

In OFDM radar: $\Delta r$ depends on total bandwidth $B = K\Delta f$ , while $R_{\max} = c/(2\Delta f)$ depends on subcarrier spacing. Increasing $K$ (more subcarriers) improves resolution without affecting the unambiguous range.

Quick Check

A radar system operates at a carrier frequency of 28 GHz with a signal bandwidth of $B = 200$ MHz and a coherent processing interval of $T = 5$ ms. What are the range resolution $\Delta r$ and the velocity resolution $\Delta v$ ?

$\Delta r = 0.75$ m, $\Delta v = 1.07$ m/s

$\Delta r = 1.5$ m, $\Delta v = 0.54$ m/s

$\Delta r = 0.75$ m, $\Delta v = 2.14$ m/s

$\Delta r = 1.5$ m, $\Delta v = 1.07$ m/s

Correction:

\Delta r = 0.75

m,

\Delta v = 1.07

m/s

Range resolution: $\Delta r = c/(2B) = 3 \times 10^8 / (2 \times 200 \times 10^6) = 0.75$ m. Wavelength: $\lambda = c/f_0 = 3 \times 10^8 / 28 \times 10^9 = 10.71$ mm. Velocity resolution: $\Delta v = \lambda/(2T) = 10.71 \times 10^{-3} / (2 \times 5 \times 10^{-3}) = 1.071$ m/s.

Signal Processing Foundations for Sensing

Radar Sensing Fundamentals

Definition: Ambiguity Function

Definition: Range-Doppler Map

Theorem: Radar Range and Velocity Resolution

Derivation

Example: Resolution of a 77 GHz Automotive Radar

Wavelength and range resolution

Velocity resolution

Maximum unambiguous range and velocity

Time-bandwidth product

Ambiguity Function Geometry

Ambiguity Function: Bandwidth vs Resolution

Ambiguity Function Surface

Parameters

Range-Doppler Processing via 2D-FFT

Range-Doppler Map with Targets

Parameters

Ambiguity Function

Range-Doppler Map

Common Mistake: Confusing Resolution with Maximum Unambiguous Range/Velocity

Quick Check

Definition:
Ambiguity Function

Definition:
Range-Doppler Map