## 2 Fast Telemetry and Communication Method Based on Doppler Diversity Reception
### Fast Coarse Acquisition of Doppler Frequency Offset  
Doppler frequency offset in the hypersonic-speed flight environment can reach as high as 1.5 MHz. Doppler frequency offset changes in a large range. If Doppler frequency offset has to be precipitated at a time, tens of thousands of sample points have to be analyzed to assure spectral analysis accuracy, which consumes abundant analysis time [17; 18]. To address this problem, the parallel fast coarse acquisition method is utilized, as shown in Figure 2. The Doppler frequency offset range is divided into several channels to be processed simultaneously, which can reduce acquisition time significantly [19; 20].  
Figure 1: Overall framework of fast telemetry and communication based on Doppler diversity.  
In Figure 2, \(F_{d}\) and Fa are the variation range of Doppler frequency offset [\(f_{d\mathrm{min}}\),\(f_{d\mathrm{max}}\)] and range of Doppler rate-of-change [\(f_{d\mathrm{min}}\),\(f_{d\mathrm{max}}\)]. The variation range of the whole Doppler frequency offset is divided into several sub-ranges at an equal interval and each sub-range is viewed as one channel. Different initial Doppler frequency offset is set for each channel at equal intervals. A large error between the acquired frequency offset and actual frequency offset caused by change rate of Doppler can lead to error of FFT analysis results. To prevent this error, Doppler rate-of-change in each channel is also divided into several sub-channels at an equal interval. Moreover, different initial Doppler rate-of-change is set with equal interval in each sub-channel.  
Local carrier of the \(K_{j}\) Doppler rate-of-change channel and the Ki Doppler frequency offset channel can be expressed as:  
\[f_{\mathrm{C}K_{j}K_{j}}=e^{-j2\pi(f_{d}k+\frac{1}{2}f_{d}k^{2})} \tag{2}\]  
where \(i\) = 1,2,...,\(N\); \(j\) = 1,2,... \(M\).\(f_{di}\) is the Doppler frequency offset which set by the \(K_{i}\) Doppler frequency offset channel. \(f_{dj}\) is the change rate of Doppler which is set by the \(K_{j}\) Doppler change rate channel.  
In each sub-channel, the received signals are mixed with local frequency \(f_{\mathrm{C}K_{i}K_{j}}\) which is set by \(N\times M\) channels parallel to relieve the effects of Doppler frequency offset in received signals. After high-frequency components are eliminated by a low-pass filter (LPF), FFT spectral analysis on signals in each sub-channel is carried out. Energy extreme point \(V_{k}\) (\(k\) = 1,2,... \(N\times M\)) is searched in the frequency domain. Size of energy extreme point in each sub-channel is compared to recognize the maximum value \(V_{\mathrm{max}}\). \(f_{d}\) and \(f_{a}\) which are set by the channel where the maximum extreme point lies in are used as outcome of the current coarse acquisition, which are recorded as \(F^{\prime}_{d}\) and \(F^{\prime}_{a}\). In this way, coarse acquisition of multiple Doppler frequencies and its rate-of-change is accomplished.  
Based on coarse estimation, the received signals are then compensated and the compensated signal with residual Doppler \(y_{r}(t)\) can be expressed as:  
\[\begin{split} y_{r}(t)&=y(t)e^{-j2\pi(F^{\prime}_{d }t+\frac{1}{2}F^{\prime}_{a}t^{2})}\\ &=h(t)s(t)e^{j2\pi f_{d}t}+v(t)\cdot e^{-j2\pi(F^{\prime}_{d}t+ \frac{1}{2}F^{\prime}_{a}t^{2})}\end{split} \tag{3}\]  
where \(f_{r}=f_{d}-F^{\prime}_{d}+(f_{a}-F^{\prime}_{a})t_{0}\) is the residual Doppler frequency offset after compensation and \(t_{0}\) is the continuous time of data.