Physical parameters of HF radar operation

Table with numerical values of parameters

Electromagnetic wave

L wavelength (m)
F frequency (Hz)
C = 3e8 speed of light (m/s)

L*F = C

Surface gravity wave

l wavelength (m)
g gravity (m/s2)
c phase speed (m/s)
cb phase speed of Bragg waves (m/s)

c = sqrt(g*l/2pi)

Bragg condition:

l = L / 2

cb = sqrt(g*L/4pi)

Frequency of Bragg waves:

fb = sqrt(g*F/(pi*C))

Range mapping

The TX signal is linearly chirped upward. The RX signal is broad spectrum, with echoes at frequencies close to the TX signal for nearby targets, and at increasingly lower frequencies as target distance increases. The Fourier transform of chirp echos thus performs range-mapping. The factor 2 comes from the forward and return path.

B modulation bandwidth (Hz)
d physical range resolution (m)

d = c / (2*B)

Maximum range

Empirical relationship; for ground-wave radars (3 < F < 50 MHz):

k = 1.8e12 constant (m Hz)
D maximum range (m)
m number of range cells

D * F = k
m = D / d

Demodulation bandwidth

The RX signal is complex-demodulated (homodyned) by mixing with a copy of the TX signal in phase (I) and in quadrature (Q). The LF bandwidth is the mapping of the maximum range into the frequency domain. The Nyquist frequency corresponding to the LF A/D converters must be larger than this frequency, and powerful analog/and/or digital low-pass filters must prevent the folding of interferences into the audio band.

t chirp duration (s)
r chirp frequency rate (Hz/s)
R chirp frequency per range cell (Hz)
b low frequency bandwidth (Hz)

r = B / t

R = 1 / t
Did I goof here? should there be a 2?

b = m / t = D / (d * t)

Velocity measurement

The radar is not truly a Doppler radar in the sense that it does not measure radio frequencies to the mHz. In practice, a single chirp gives the complex backscatter (amplitude and phase) as a function of range. Repeating chirps then give time series of amplitudes and phases. These time series contains information of the slow motion of targets in each range cell, as their phase slowly changes. The Fourier transform of these time series is the familiar range-resolved Doppler spectrum.

The maximum velocity corresponds to the Nyquist frequency, or 2pi phase change, or propagation of the scatterers by 1 em wavelength during two chirps. The velocity resolution corresponds to 2pi phase change, or propagation of the scatterer by 1 em wavelength during the acquisition period.

v spectral velocity resolution (m/s)
V maximum (Nyquist) velocity (m/s)
n number of chirps
T sampling period

v = L / ( n * t ) = L / T

V = v * n / 2 = L / ( 2 * t )

Table of optimum parameters for various frequencies