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Radlab /
Physical parameters of HF Doppler Radio operationTable with numerical values of the 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 (downward). The RX signal is broad spectrum, with echoes at frequencies close to the TX signal for nearby targets, and at increasingly lower (higher) frequencies as target distance increases. The Fourier transform of chirp echoes thus performs range-mapping. The factor 2 comes from the forward and return path. rc = range cell B modulation bandwidth (Hz) d physical range resolution (m/rc) d = C / (2*B) Maximum range Empirical relationship; for ground-wave over sea water (3 < F < 50 MHz): k = 1.8e12 constant (m Hz) D maximum range (m) m number of range cells (rc) 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/rc) b low frequency bandwidth (Hz) r = B / t R = 1 / t b = m * R = m / t = D / (d * t) (low frequency sample rate should be > 2 * b). Frequency-to-distance mapping (m/HZ) c / (2 * B * R) = c * t / (2 * B) Velocity measurement The radio is not truly a Doppler radar in the sense that it does not measure radio frequencies to the mHz. In practice, a single chirp demodulated by orthogonal sin and cos local oscillators, gives the complex backscatter (amplitude and phase) as a function of range. Repeating chirps then give time series of amplitudes and phases in each range cell. These time series contains information of the slow motion of targets in each range cell, as their phases slowly evolve. The complex Fourier transform of these time series is the familiar range-resolved Doppler spectrum. The maximum velocity corresponds to the Nyquist frequency, or 2 pi phase change, or propagation of the scatterers by 1 em wavelength L during two chirps. The velocity resolution v corresponds to 2 pi phase change, or propagation of the image of the source by the scatterers by 1 em wavelength L, or propagation of the scatterers by 1/2 em wavelength, during the acquisition period T. v spectral velocity resolution (m/s) V maximum (Nyquist) velocity (m/s) n number of chirps T acquisition period v = L / ( n * t ) = L / T V = v * n / 2 = L / ( 2 * t ) Table of optimum parameters for various frequencies A few assumptions: velocity resolution is imposed to be 2 cm/s; modulation bandwidth is 1% of frequency; Doppler spectral width = 6x Bragg velocity to ensure look-alike spectra; phased array is 12 antennas; VOP is 0.66. Broadening by windowing (i.e. Blackman against rectangular) not taken into account for the computation of the range and velocity resolutions. Computation sequence: given F, compute L, l and cb. Given B linked to F, compute d. Given F, compute D and m. Given v and L, compute T. Given cb, compute V. Given V and L, compute t. Given t and T, compute n. Given t and m, compute b. Param 12 MHz 15 MHz 30 MHz 50 MHz 150 MHz unit L 25 20 10 6 2 m cb 4.42 3.95 2.79 2.16 1.25 m/s B 120 150 300 500 1,500 kHz d 1.25 1.0 0.5 0.3 0.1 km D 150 120 60 36 12 km m 120 120 120 120 120 # cells v 2 2 2 2 2 cm/s T 1250 1000 500 300 100 s V 26.5 23.7 16.7 13 7.5 m/s t 0.471 0.422 0.299 0.231 0.133 s 1/t 2.12 2.37 3.34 4.33 7.52 Hz/RC b 255 284 401 520 902 Hz n 2654 2370 1672 1299 752 # chirps L/2 (air) 12.5 10 5 3 1 m 11*L/2 (air) 137.5 110 55 33 11 m L/4 (cable) 4.125 3.30 1.65 0.99 0.33 m |