## 3 Principles of Speed Estimation for Inter-Symbol Processing
### Signal Demodulation  
The relative radial motion between the high-speed maneuvering target and the receiving terminal causes a serious Doppler effect, which generates Doppler shift, such as in the typical hypersonic flight conditions, Ka communication band Doppler shift up to hundreds of kHz. The communication radiated source signals under the motion condition and the demodulation under the stationary transmission condition will be different due to the existence of the Doppler effect, which results in the phase change of the system and superimposed on the phase of the modulation signal of the digital phase modulation system, resulting in the demodulation phase judgment error. The Doppler effect causes the phase of the system to change and superimpose on the phase of the modulating signal of the digital phase modulation system, resulting in an error in the judgment of the demodulation phase. Carrier phase synchronization [9] is performed on the signal before demodulation, and the estimated carrier phase deviation is compensated for each symbol. In this paper, we assume that the source of communication radiation is a QPSK signal. QPSK is a digital modulation method that modulates two bits as a group so that each symbol can carry more information. In the demodulation system, it is assumed that the symbol synchronization has been completed and each symbol picks a normalized (symbol energy of 1) optimal sampling point of:  
\[r[K]=s[K]\exp[j(2\pi\Delta fK+\Delta\theta)]+n[K] \tag{4}\]  
where \(s[K]\) is the modulation symbol, which is a point in the constellation diagram; \(\Delta f\) is the carrier frequency bias; \(\Delta\theta\) is the carrier phase bias; and \(n[K]\) is the additive complex noise. If the modulation symbol \(s[K]\) and the complex noise power \(\sigma^{2}\) are known, then the conditional probability of \(r[K]\) with respect to the phase deviation \(\Delta\theta\)\((\Delta\theta\in[-\pi,\pi))\) is expressed as follows:  
\[P(r[K]|\Delta\theta,s[K])=\frac{1}{\pi\sigma^{2}}\exp\left(-\frac{|r[K]-s[K] \exp\left(j\Delta\theta\right)|^{2}}{\sigma^{2}}\right) \tag{5}\]  
Since it is a blind estimation, the receiver does not know the true value of \(s[K]\), but it can be reasonably assumed that the modulation data are a priori isochronous, i.e., the a  
Figure 2: Schematic diagram of the signal segmentation principle. priori probability of \(s[K]\) selecting any constellation point in QPSK is \(1/4\), then the irrelevant parameter \(s[K]\) can be eliminated by averaging to obtain the following expression:  
\[ \begin{array}{ll}P(r[K]|\Delta\theta)&=\frac{1}{4}\sum\limits_{i=0}^{3}P(r[K]| \Delta\theta,s[K]=c_{i})\\ &=\frac{1}{4\pi\sigma^{2}}\sum\limits_{i=0}^{3}\exp\left(-\frac{ \left|r[K]-c_{i}\exp(j\Delta\theta)\right|^{2}}{\sigma^{2}}\right)\end{array}  \tag{6}\]  
According to the MMSE guidelines, there are  
\[\Delta\hat{\theta}=\int_{-\pi}^{\pi}\Delta\theta P(\Delta\theta|r[K])\,\mathrm{ d}\Delta\theta \tag{7}\]  
The MMSE estimate can be obtained as  
\[\Delta\hat{\theta}(r[K])=\frac{\int_{-\pi}^{\pi}\Delta\theta\sum\limits_{i=0}^ {3}\exp\left(-\frac{\left|r[K]-c_{i}e^{j\Delta\theta}\right|^{2}}{\sigma^{2} }\right)\mathrm{d}\Delta\theta}{\int_{-\pi}^{\pi}\sum\limits_{i=0}^{3}\exp \left(-\frac{\left|r[K]-c_{i}e^{j\Delta\theta}\right|^{2}}{\sigma^{2}} \right)\mathrm{d}\Delta\theta}. \tag{8}\]  
In QPSK modulation, there are four possible modulation phases: \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), and \(270^{\circ}\) or \(45^{\circ}\), \(135^{\circ}\), \(225^{\circ}\), and \(315^{\circ}\). These phases correspond to the bit combinations 00, 01, 11, and 10, respectively. QPSK demodulation is the inverse process of QPSK modulation, and its goal is to recover the original digital information from the received modulated signal.