## 3 Dual-PRF Waveform Design and Signal Processing
### Signal Processing
#### 3.4.1 DFT-Based Signal Processing  
Without loss of generality, the designed waveform can work with common signal processing methods such as the DFT-based method and the sparse recovery method. As we know, the designed dual-PRF waveform includes two uniform waveforms with different PRFs. So we can extract the two uniform waveforms from the overall sensing waveform and apply the DFT-based method to realize the parameter estimation. The DFT-based signal processing framework is illustrated in Figure 8.  
First, it needs to extract \(2 Tr\) and \(3 Tr\) pulses data (\(\mathbf{Y}_{2 Tr}\), \(\mathbf{Y}_{3 Tr}\)) according the sensing time from the nonuniform sensing receiving data. Taking the \(2 Tr\) pulses data for example, we apply DFT along each column of the data to obtain the range spectrum \(\mathbf{\tilde{Y}}_{2 Tr}=[\mathbf{\tilde{y}}_{2 Tr}(0),\dots,\mathbf{\tilde{y}}_{2 Tr}(N_{c} -1)]\), where \(\mathbf{\tilde{y}}_{2 Tr}(n)\) is the \(n\)th range cell. Then, we apply the static clutter suppression along the slow-time dimension to each range cell, as given by  
\[\mathbf{\tilde{y}}_{2 Tr}(n)=\mathbf{\tilde{y}}_{2 Tr}(n)-\frac{1}{N_{s}}\sum\mathbf{\tilde{y }}_{2 Tr}(n), \tag{24}\]  
where \(\sum\mathbf{\tilde{y}}_{2 Tr}(n)\) denotes the element summation of \(\mathbf{\tilde{y}}_{2 Tr}(n)\). After that, we estimate velocity \(v\) along the slow-time dimension by the DFT method in order to obtain the range-velocity septrum \(\mathbf{\tilde{Y}}_{2 Tr}\) under uplink-flexible mode. Specially, we use the non-uniform DFT (NUDFT) method [34] to estimate the velocity of the dual-PRF sensing pulse under uplink-fixed mode because its extracting pulse is not strictly uniform timing. And the function of NUDFT is  
\[X(\omega_{n})=\sum_{k=0}^{K-1}x(t_{k})e^{-j\frac{2\pi}{K}nt_{k}}, \tag{25}\]  
where \(x(t_{k})\) is the \(k\)th sample data, and \(X(\omega_{n})\) is the \(n\)th NUDFT value. Above all, the IDFT method, the DFT method, and the NUDFT method are all based on DFT theory, so we call them all the DFT-based method. Next, applying the OS-CFAR detector [35] to \(\mathbf{\tilde{Y}}_{2 Tr}\), it can detect the reflected objects. And the steps of the OS-CFAR detector are as follows: (1) select some protection cells around the \(\mathbf{\tilde{Y}}_{2 Tr}(n,k)\) cell to be detected; (2) select reference cells next to the protection cells and sort them by power, and select the middle one as the power of clutter and noise \(P_{\text{CN}}\). (3) compute the threshold \(  
that means it is an object cell, otherwise, it is a noise cell. Applying the same steps above to all \(3 Tr\) pulses data, we can obtain the range-velocity spectrum of it. Finally, we solve the velocity ambiguity using the CRT method [36] and computing the velocity using the frequency-time phase regression (FTPR) [37] method. According to the theory analysis and the ambiguity function of the designed waveform, we know that the velocity estimation of the \(2 Tr\) and \(3 Tr\) sensing pulses exhibit ambiguity values. We write their velocity values as \(\left\{v_{2 Tr}^{p}\right\}_{p=0}^{P-1}\) and \(\left\{v_{3 Tr}^{q}\right\}_{q=0}^{Q-1}\) including \(P\) and \(Q\) ambiguity values, respectively. Then we calculate the absolute differences of the two groups of ambiguity values  
\[\Delta\{v_{na}\}_{na=0}^{PQ-1}=\left|\left\{v_{2 Tr}^{p}\right\}_{p=0}^{P-1}- \left\{v_{3 Tr}^{q}\right\}_{q=0}^{Q-1}\right|. \tag{26}\]  
Obviously, \(PQ\) is the total number of absolute differences and the unambiguity velocity is the group which has the minimum absolute differences. Considering the \(\hat{\mathbf{Y}}_{n,k}\) cell is an object cell, we then extract the spectrum of the \(n\)th range cell \(\hat{\boldsymbol{y}}(n)\) and set the spectrum zeros except for the \(\{\text{{kth}}-1,\text{{kth}},\text{{kth}}+1\}\) cells.