## 2 Fast Telemetry and Communication Method Based on Doppler Diversity Reception
### Doppler Diversity Reception Method Based on Artificial Frequency Shift  
The Doppler diversity reception method based on artificial frequency shift is realized by shifting frequency offset differently through the received time-domain signal as well as generating and combining different signals of frequency shift branches. Interference signals of each branches after frequency shift are unrelated or weakly correlated. Different branches can be viewed as independently  
Figure 2: Parallel fast acquisition process based on multiple channels.  
fading channels and diversity gains can be acquired from combining signals of branches. The reception framework is shown in Figure 3.  
Here, \(Q\) is the number of shifting branches. \(\left\{f_{x1},f_{x2},...,f_{xQ}\right\}\) is the artificial shift frequency in different branches. \(k\) is the sequence number of signals after sampling. \(\eta\) is the weighting coefficient at combination of signals. \(H\) is the frequency domain channel response after time-frequency transformation. \(N\) is the number of data symbols in one FFT processing.  
It can be seen from Figure 1 that the discrete modulation signal \(d_{n}\) can be viewed as frequency-domain signal and it is transformed to time domain through IFFT. At this moment, transmitting signals in discrete domain can be expressed as:  
\[s(k)=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}d_{n}e^{j\frac{2\pi}{N}kn} \tag{4}\]  
Under this circumstance, channel can be equivalent to:  
\[h(t,\tau)=\sum_{p=1}^{N_{p}}a_{p}\delta(\tau-\tau_{p})e^{j2\pi f_{rp}t} \tag{5}\]  
where \(a_{p}\) and \(\tau_{p}\) is the channel relative gain and time delay of the \(p\)th multipath. \(f_{rp}\) is the residual Doppler frequency offset on the pathway \(p\). \(f_{rp}\) conforms to uniform distribution in the range of \([-F_{D}]\). \(F_{D}\) is the maximum value residual Doppler frequency offset \(f_{r}\) and the continuous time of symbol is recorded as \(T\). Without loss of generality, the normalized Doppler frequency offset and normalized artificial shift frequency are applied to analyze system performance. \(F_{D}T\) is defined a normalized Doppler frequency offset and its value is in the range of \([0,1]\). \(f_{x}T\) is defined a normalized artificial shift frequency.  
The equivalent signal after compensation in rapid coarse acquisition can be expressed as:  
\[y_{r}(k)=\frac{1}{\sqrt{N_{p}N}}\sum_{p=1}^{N_{p}}a_{p}e^{j2\pi f_{rp}k}\sum_{ n=0}^{N-1}d_{n}e^{j\frac{2\pi}{N}n(k-\tau_{p})}+v(k) \tag{6}\]  
Figure 3: Structure of Doppler diversity reception.  
The equivalent signal is multiplied with \(e^{-j2\pi f_{x}k}\) to shifting different artificial shift frequency \(f_{x}\), thus generating several independent signals of shifting branches. On the shifting branch with a frequency offset of \(f_{x}\), FFT of the \(i\) receiving signal can be expressed as:  
\[Y(i,f_{x})=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}y_{r}(k)e^{-j2\pi f_{x}k}e^{j\frac {2\pi}{N}ik} \tag{7}\]  
Finally, the combined output signal is:  
\[Z=\sum_{q=1}^{Q}\eta_{q}H^{*}(i,f_{x_{q}})Y \tag{8}\]  
where \(\eta_{q}\) is the combined weighted coefficient of the branch \(q\) (\(q\) = 1, 2,... \(Q\)) and \(H^{*}\) is the gain coefficient of frequency-domain channel which is estimated within sub-channels. Finally, Z is demodulated.  
For the selection of artificial shift frequency, the authors of [8] decomposed Equation (7) into useful signals and interference signal. Correlation coefficient of interference signals between two branches is analyzed. It concludes that the correlation coefficient of interference signals of branches is almost 0 when the normalized Doppler frequency offset is higher than 0.25, the normalized artificial shift frequency within 0.75-0.8, and the signal to interference ratio increases with the increase of normalized Doppler frequency offset. Therefore, it is believed that the best normalized artificial shift frequency is in the interval of [0.75, 0.8].