Demodulation, range gating and A/D conversion

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1 Narrowband receiver signal

The narrow-band RWP signal at the output port of the low noise amplifier can be written as

S_{rx}(t)=A(t)\cos \left[\omega_c t + \Phi(t)\right].

All available information about the scattering process is contained in the amplitude and phase modulation of the received signal S_{rx}(t). It is technically difficult to sample such a signal, therefore a demodulation step is performed first, which essentially removes the irrelevant carrier frequency \omega_c while the modulation information contained in the instantaneous amplitude A(t) and the instantaneous phase \Phi(t) remains unchanged.

2 Demodulation: Analytic signal

For demodulation, a new signal S^{+}(t) with the Fourier spectrum \hat{S}^{+} is created:

\hat{S}^{+}(\omega) = \hat{S}(\omega) + \sgn\left[\omega\right] \hat{S}(\omega).

This operation removes the negative part of the original signal spectrum. The signal S^{+}(t) is called the analytic signal or pre-envelope of S(t), see McDonough and Whalen (1995) .

In the time domain, it is formed as

S^{+}(t) = S(t) + i \mathcal{H}\left[S(t)\right],

where the operator \mathcal{H} denotes the Hilbert transform.

The analytic narrowband signal can be written as

S^{+}(t) = \left[\tilde{S}(t)e^{i\omega_c t} \right],

where \tilde{S}(t) is the complex envelope of the original signal. Multiplication of S^{+}(t) with e^{-i \omega_c t} removes the carrier and gives the complex envelope

\tilde{S}(t) = S^{+}(t) e^{-i \omega_c t} = (S(t) + i \mathcal{H}\left[S(t)\right]) e^{-i\omega_c t} = I(t) + i Q(t),

where the real part of the complex envelope is the so-called in-phase I(t) and the imaginary part Q(t) the quadrature phase of the signal. The Hilbert transform is not easily implemented in real systems. Instead, the in-phase and quadrature-phase components are determined using a quadrature demodulator. Details depend on the receiver architecture of the RWP.

3 Range gating

For a fixed beam direction, RWP transmit a series of short electromagnetic pulses, each one separated by a time interval \Delta T. For a single pulse, the sampling in time allows the determination of the radial distance of the measurement using the well-known propagation speed of the wave group. The maximum distance for unambiguously determining the measurement distance is limited by the pulse separation or inter-pulse-period \Delta T and h_{max} = c \Delta T
/2 is called the maximum unambiguous range. \Delta T has to be set sufficiently high to prevent range aliasing problems, that is arrival of backscattering signals from the preceding pulse after the next pulse is transmitted. For a typical wind profiler it is if the order of 10^{-4}s.

Range gating is usually done in the A/D process using sample and hold circuitry. The sample strobe required for range gating and pulse repetition is provided by the radar controller. If the range sampling frequency is given by 1/\Delta t and N_h is an integer denoting the number of range gates with \Delta T < N_h \Delta t, then signal \tilde{S}(t) is obtained at the discrete grid

\tilde{S}[j,n] = \tilde{S}(t_0 + j \Delta t + n \Delta T)~, j = 0,\ldots ,N_h -1,~~ n = 0, \ldots, N_T-1.

For each range gate j, that is for the height c/2 \cdot j \cdot \Delta t, a discrete time series of the complex envelope of the signal with a sampling interval of \Delta T is obtained (Hardware effects like the group delay of the pulse in the radar electronics are ignored for simplicity).

4 RWP raw data

For every range gate, a complex time series is obtained as

S[n] = S_I[n] + i S_Q[n], \qquad n=0,...,N_T-1.

Note that the range gate index and the tilde denoting the complex envelope are suppressed for convenience.

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