## 3 Principles of Speed Estimation for Inter-Symbol Processing
### Spectral Analysis Across the Symbol  
The Doppler effect introduces a linear phase shift between successively modulated symbols of a transmitted signal, a phenomenon that provides the basis for velocity estimation. The velocity estimation is calculated based on the phase difference between neighboring symbols. This is because the value of the change in frequency difference due to motion is very small relative to the carrier frequency, and these changes are not accurately represented directly by the FFT at the peak of the spectrum. The relationship between the Doppler shift \(f_{d}\) and the speed of motion and carrier frequency \(f_{c}\) is \(f_{d}/f_{c}=v/c\), and the difference between the speed of motion \(v\) and the speed of light \(c\) is extremely large, with an order of magnitude difference of \(10^{5}\) to \(10^{7}\), which makes the direct estimation of the Doppler shift highly accurate.  
However, even subtle frequency changes can be clearly demonstrated by changes in phase [10]. The essence of using the phase difference between neighboring symbols for velocity measurements is that this phase difference is caused by the motion of the target object. If the target object is stationary, then the Doppler shift will be zero and there will be no phase difference between neighboring symbols. Conversely, when the target object is moving, the motion causes a Doppler shift, which introduces a phase difference between neighboring symbols.  
Assuming that the target distance is \(R\) when transmitting the carrier frequency is \(f_{c}\), at this time, the wavelength length can be obtained from the relationship between wavelength and frequency \(\lambda=v/f\). The phase of the received signal varies with distance as:  
\[\varphi=\frac{2\pi\Delta R}{\lambda} \tag{14}\]  
This method utilizes the characteristics of phase differences between adjacent symbols during signal modulation, and the velocity of the target object is inferred by measuring these phase differences. Because the Doppler shift is proportional to the velocity of the target object and because frequency variations can be reflected by phase variations, this method is very effective in velocity estimation. This method provides reliable velocity measurements even when the frequency variation is small.  
For applications in real-world environments, it must be expected that the received signals will be frequency-shifted due to the relative speeds between the communication partners or between the radar platform and the reflecting object. When the two platforms are moving at a relative velocity \(v\) between them, the Doppler shift \(f_{d}\) at the receiver is expressed as follows:  
\[f_{d}=\frac{v}{\lambda}=\frac{vf_{c}}{c} \tag{15}\]  
It can be seen that only \(f_{d}\) is needed to know the speed of the target.  
In actual measurements, the extraction of speed by the FFT algorithm is mainly a two-dimensional matrix in the column vector FFT transform. Equation (3) arbitrarily takes one of the columns \(Y^{\prime}=[y^{\prime}_{1}(1),y^{\prime}_{2}(1),\cdots,y^{\prime}_{M}(1)]^{T}\) to perform \(M\)-point FFT, according to the Nyquist sampling theorem, in order to ensure the accurate extraction of speed information, \(f_{c}/N\geq 2f_{d}\). At this time, the frequency resolution is expressed as follows:  
\[\Delta f_{d}=\frac{f_{s}}{M} \tag{16}\]  
where \(f_{s}\) is the sampling frequency; \(M\) is the number of symbols; and the resolution of the velocity is expressed as follows:  
\[v_{res}=\frac{\lambda}{2 MT} \tag{17}\]  
where \(T\) is a symbol period.