Demodulation, range gating and A/D conversion

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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