2. Radar Basics for Communication Engineers

Why Start With Radar

ISAC asks one radio to do two jobs. The communication job is familiar: send bits, decode bits. The sensing job comes from radar, and it uses a different vocabulary. This note builds that vocabulary from scratch, so the rest of the track reads easily.

A radar does three things with an echo:

• Range: how long the echo took to come back tells how far the object is.
• Velocity: how much the echo's frequency shifted tells how fast the object moves toward or away from the radio.
• Angle: how the echo's phase changes across an antenna array tells which direction it came from.

Then it has to answer a fourth question first: is there an echo at all, or just noise?

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Range: Timing the Echo

A wave travels at the speed of light $c \approx 3\times10^8$ m/s. If the echo arrives $\tau$ seconds after transmission, the wave has covered the distance twice, out and back:

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

• $R$: range to the target in metres.
• $\tau$: round-trip delay in seconds.
• Scale: 1 µs of delay is 150 m of range; 1 ns is 15 cm.

This is the monostatic case: transmitter and echo receiver at the same place. When they are apart (bistatic), the delay measures the total path transmitter → target → receiver, and the target lies somewhere on an ellipse with the two radios as foci.

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Velocity: The Doppler Shift

A target moving toward the radio compresses the wave; moving away stretches it. The echo comes back at a slightly different frequency:

$$f_D = \frac{2 v_r}{\lambda} = \frac{2 v_r f_c}{c}$$

• $f_D$: Doppler shift in hertz, positive for an approaching target.
• $v_r$: radial speed, the part of the velocity along the line of sight.
• $\lambda$, $f_c$: wavelength and carrier frequency.
• Why the 2: the target sees a shifted wave and then re-radiates it while still moving, so the shift happens twice.

At 28 GHz, a pedestrian walking at 1.5 m/s gives about 280 Hz; a car at 30 m/s gives about 5.6 kHz. Only the radial part counts: a car crossing the beam at right angles produces almost no Doppler.

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Angle: Where the Echo Came From

With several antennas spaced half a wavelength apart, an echo from angle $\theta$ reaches neighbouring elements with a phase difference

$$\Delta\phi = \pi \sin\theta .$$

Measuring that phase step across the array gives the direction. The array can tell two directions apart when they differ by roughly one beamwidth, about $2/N$ radians for $N$ elements. This is the same physics as beam steering, used in reverse.

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Why Echoes Are So Weak

A communication signal travels one way, and its power spreads over a sphere: it falls as $1/R^2$. An echo does that twice: once on the way to the target, and again after the target scatters it back. The radar equation captures this:

$$P_r = \frac{P_t\, G_t\, G_r\, \lambda^2\, \sigma}{(4\pi)^3\, R^4}$$

• $P_t$, $G_t$, $G_r$: transmit power and antenna gains.
• $\sigma$: radar cross section (RCS) in m², how strongly the target scatters. A person is roughly 1 m², a car roughly 10–100 m², a small drone roughly 0.01 m².
• $R^4$: the price of the round trip.

Compare a one-way link to a user at the same distance: its received power falls only as $R^2$. Double the distance and the user loses 6 dB, while the echo loses 12 dB. That is why sensing range is usually much shorter than communication range from the same base station, and why ISAC systems care so much about processing gain: integrating many symbols adds the energy back.

Why the echo fades so much faster

Target

Small drone (0.01 m²)Person (1 m²)Car (10 m²)Truck (100 m²)

Carrier

3.5 GHz28 GHz

Transmit power

30 dBm

Antenna gain (TX and RX)

20 dBi

Symbols combined

64

Communication range6032 muser SNR ≥ 0 dB

Sensing range310 mecho ≥ 13 dB after integration

Processing gain18.1 dB64 symbols

Ratio19×comm reach / sensing reach

$$P_r^{\text{comm}} = \frac{P_t G_t G_r \lambda^{2}}{(4\pi)^{2} R^{2}}, \qquad P_r^{\text{echo}} = \frac{P_t G_t G_r \lambda^{2} \sigma}{(4\pi)^{3} R^{4}}, \qquad \mathrm{SNR}_{\text{echo}} = \frac{P_r^{\text{echo}}}{N}\cdot M$$

$$R_{\text{sense}} = \left(\frac{P_t G^{2} \lambda^{2} \sigma M}{(4\pi)^{3} N\, \mathrm{SNR}_{\min}}\right)^{1/4} = 310\ \text{m} \quad (\sigma = 10\ \text{m}^2,\ M = 64)$$

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Resolution Versus Accuracy

Two different questions are easy to mix up:

• Resolution: can I tell two nearby targets apart?
• Accuracy: how precisely can I locate one target?

Range resolution depends only on bandwidth:

$$\Delta R = \frac{c}{2B}$$

100 MHz resolves targets 1.5 m apart; 1 GHz resolves 15 cm. Accuracy for a single target is much finer than that and improves with SNR, roughly as $\Delta R / \sqrt{\mathrm{SNR}}$. A high-SNR radar can locate one car to a few centimetres while still being unable to separate two cars parked half a metre apart.

Resolution: can we see two targets or just one blob?

Gap between the targets

1 m

Bandwidth

150 MHz

Resolution cell c/2B99.9 cm

Target gap1.00 m

VerdictTwo targetsgap ≥ resolution cell

$$\Delta R = \frac{c}{2B} = \frac{3\times10^{8}}{2\times 150\times10^{6}} = 99.9\ \text{cm}, \qquad \text{resolved if } \ |R_2 - R_1| \ge \Delta R$$

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Detection: Is Anything There?

Before measuring anything, the receiver has to decide whether a peak is a target or just noise. It compares each range cell against a threshold:

• Detection probability $P_D$: the chance a real target crosses the threshold.
• False-alarm probability $P_{FA}$: the chance noise alone crosses it.

Raise the threshold and false alarms drop, but weak targets are missed. Lower it and you catch more targets, along with more ghosts. For noise whose power is exponentially distributed (complex Gaussian noise), the false-alarm probability for a threshold $T$ times the noise power is

$$P_{FA} = e^{-T}.$$

Real receivers rarely know the noise power exactly, so they estimate it from neighbouring cells and scale the threshold from that estimate. This is CFAR (constant false-alarm rate) detection, the standard first step in almost every radar and ISAC receiver.

Detection: target or just noise?

Threshold above mean noise

11 dB

Weak target SNR

9 dB

Noise realisation

Noise 1Noise 2Noise 3Noise 4

Targets detected1 of 2

False alarms here0expected 0.00

P_FA per cell3.4e-6

$$P_{FA} = e^{-T} = e^{-10^{11/10}} = 3.41\times10^{-6}, \qquad \mathbb{E}[\text{false alarms}] = (N_{\text{cells}} - 2)\,P_{FA}$$

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.

Takeaway

• Echo to numbers: delay → range, Doppler → radial speed, phase across the array → angle.
• Budget: echoes fade as $R^4$, so sensing range is shorter than communication range and needs processing gain.
• Two different goals: bandwidth sets resolution; SNR sharpens accuracy; a threshold decides what counts as a target.

The next note shows how a normal OFDM communication signal can perform all three measurements without sending any extra radar pulse.

References for this section3

• M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
• M. A. Richards, Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014.
• F. Liu et al., "Integrated Sensing and Communications: Toward Dual-Functional Wireless Networks for 6G and Beyond", IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728-1767, 2022.