Reading — Spectrum & Propagation

By the end of this lesson you should be able to:

1. Place a radar in its frequency band and state the tradeoffs of moving up or down the spectrum.

2. Relate wavelength and frequency through \(\lambda f = c\).

3. Compute free-space path loss (FSPL) in decibels and apply the 6 dB rules for range and frequency.

4. Explain how polarization and antenna gain shape the energy that reaches a target.

The spectrum is the battlefield

Every radar lives at a frequency, and that choice drives everything else: antenna size, resolution, atmospheric loss, and how easy it is to jam. Wavelength and frequency are tied by the speed of light,

\[ \lambda f = c, \qquad c \approx 3\times10^{8}\ \text{m/s}. \]

A handy form for radar work: at \(f\) in GHz, \(\lambda \approx 30/f\) in centimeters. So X-band at 10 GHz has a 3 cm wavelength; L-band at 1.5 GHz is about 20 cm.

Radar bands

The standard IEEE letter bands and their rough character:

Band	Frequency	Wavelength	Typical use / character
L	1–2 GHz	~15–30 cm	Long-range search; low atmospheric loss, large antennas, coarse resolution.
S	2–4 GHz	~7.5–15 cm	Search & moderate-range; weather radars.
C	4–8 GHz	~3.7–7.5 cm	Compromise band; some weather and fire control.
X	8–12 GHz	~2.5–3.7 cm	Fire control, airborne radar; good resolution, manageable antenna size.
Ku / K / Ka	12–40 GHz	~0.75–2.5 cm	Seekers, high-resolution imaging; small antennas, higher atmospheric loss.

The general trend: lower frequency gives longer range and less weather loss but needs a big antenna and gives poor resolution; higher frequency gives fine resolution and small apertures but suffers more atmospheric attenuation. Fire-control radars cluster around X-band because it balances these — which is why the B-21 threat picture is dominated by X-band fire control.

Key Concept

Frequency choice is a chain of consequences. Pick \(f\) and you have implicitly chosen wavelength, antenna size for a given gain, resolution, atmospheric loss, and your exposure to particular jamming techniques.

Free-space path loss

Energy radiated into free space spreads over an expanding sphere. By the time it reaches range \(R\), the power density has fallen as \(1/R^2\) (one-way). Free-space path loss captures how much a signal weakens between two isotropic antennas. In decibels, with range in kilometers and frequency in GHz, the working formula is

\[ \boxed{\,L_\text{fs,dB} = 20\log_{10}(R_\text{km}) + 20\log_{10}(f_\text{GHz}) + 92.45\,} \]

Two things to notice. First, loss grows with both range and frequency. Second, both appear as \(20\log_{10}(\cdot)\) — so each behaves identically in dB.

The 6 dB rules

Because \(20\log_{10}(2) \approx 6.02\) dB:

• Double the range → add 6 dB of loss.

• Double the frequency → add 6 dB of loss.

These are worth memorizing because they let you reason about link budgets without a calculator. Going from 50 km to 100 km costs 6 dB. Moving a sensor from X-band (10 GHz) to Ku (20 GHz) costs another 6 dB of one-way path loss before you account for anything else.

Quick Exercise

An X-band radar (\(f = 10\) GHz) sees a target at 50 km. Using the FSPL formula and the 6 dB rules:

1. What is the one-way FSPL at 50 km?

2. The target flies out to 200 km. How much additional loss?

3. A second radar at 20 GHz looks at the same 200 km target. How does its FSPL compare to the 10 GHz radar at 200 km?

Solution

1. \(L = 20\log_{10}(50) + 20\log_{10}(10) + 92.45 = 33.98 + 20 + 92.45 \approx 146.4\) dB.

2. 50 → 200 km is two doublings (×4), so \(+12\) dB.

3. Doubling frequency adds one more 6 dB, so the 20 GHz radar sees about 6 dB more one-way path loss than the 10 GHz radar at the same range.

Polarization

A radar’s electric field oscillates in a particular orientation — its polarization (horizontal, vertical, or circular). The receiving antenna is matched to a polarization too. When transmit and receive polarizations disagree, the captured power drops. A full cross-polarization mismatch (horizontal into a vertical antenna) can cost 20–30 dB.

This cuts both ways in EW. A target’s RCS depends on polarization, so an adversary may choose a polarization that maximizes return. And polarization is a lever for EP and EA: polarization-agile radars resist some jammers, while a jammer that does not match the victim’s polarization wastes most of its power.

Antennas: turning power into gain

An antenna concentrates radiated power into a beam. Its gain over isotropic is set by its effective aperture \(A_e\) and the wavelength:

\[ G \approx \frac{4\pi A_e}{\lambda^2}. \]

For a fixed physical aperture, shorter wavelength → higher gain (and a narrower beam). The half-power beamwidth of an aperture of dimension \(D\) is roughly

\[ \theta_\text{BW} \approx \frac{\lambda}{D}. \]

So the same dish gives a tighter beam at higher frequency. This is the other half of the frequency tradeoff: high frequency buys gain and angular resolution from a small aperture, which is exactly why missile seekers — where size is at a premium — push into Ku/Ka band.

Key Concept

Antenna gain and beamwidth are two sides of the same coin: \(G \approx 4\pi A_e/\lambda^2\) and \(\theta_\text{BW} \approx \lambda/D\). Concentrating energy into a narrow beam is what makes a radar’s effective transmitted power far exceed what an isotropic radiator would deliver — and it is the \(G_t\) and \(G_r\) terms we meet next.

Wrap-Up

Frequency sets a radar’s whole personality through \(\lambda f = c\) and the band tradeoffs. Free-space path loss, \(L_\text{fs,dB} = 20\log_{10}(R_\text{km}) + 20\log_{10}(f_\text{GHz}) + 92.45\), thins the signal with both range and frequency, and the 6 dB rules let you reason about it in your head. Polarization mismatch can cost 20–30 dB, and antenna gain \(G \approx 4\pi A_e/\lambda^2\) concentrates power into a beam whose width is \(\theta_\text{BW} \approx \lambda/D\).

These are the pieces of the constant \(K\) from Lesson 1. The next lesson assembles them — transmit power, two antenna gains, target RCS, and the two-way spread — into the radar range equation, and shows where the fourth-power law comes from.