# An Efficient Digital Channelized Receiver for Low SNR and Wideband Chirp Signals Detection
## 2 Methodology
### Proposed Algorithm Description
#### 2.2.3 IFM and Channel Arbitration  
A functional block diagram of IFM and channel arbitration is shown in Figure 6. The magnitude \(A_{k}[m]\) and phase \(\phi_{k}[m]\) of the channel output signal are extracted by a coordinate rotation digital computer (CORDIC) algorithm. The output signal after the channelized structure is the complex signal. Define \(y_{k}[m]=I_{k}[m]+jQ_{k}[m]\); the instantaneous magnitude and phase are shown below [25]. \[A_{k}[m]=\sqrt{I_{k}^{2}[m]+Q_{k}^{2}[m]} \tag{16}\]  
\[\phi_{k}[m]=\arctan\frac{I_{k}[m]}{Q_{k}[m]} \tag{17}\]  
The instantaneous magnitudes are routed to the signal detection module and the instantaneous phases to the IFM module. The detection module provides a detection gate to the IFM module, which determines the channel where the valid signal is located. Considering the influence of the \"rabbit-ear-effect\", it is determined that there is a valid signal only when \(Q\) consecutive sampling points exceed the detection threshold. The minimum pulse width, sampling rate, and channel number determine the value of \(Q\). The phase value obtained by the CORDIC algorithm is \([-\pi,\pi]\). As the number of sample points increases, the actual phase of the signal will span periods, resulting in phase ambiguity. Therefore, the IFM needs to unwrap the phase before the frequency calculation. Let \(C_{m}\) be the phase compensation parameter at time \(m\) and initialize \(C_{1}=0\). The formula for phase deblurring is as follows [25]:  
\[C_{m}= \begin{cases}C_{m-1}-2\pi,&\phi_{k}[m]-\phi_{k}[m-1]\geq\pi\\ C_{m-1}+2\pi,&\phi_{k}[m]-\phi_{k}[m-1]\leq-\pi\\ C_{m-1},&\text{otherwise}\end{cases}  \tag{18}\]  
Figure 6: Channel detection, arbitration, and parameter estimation block diagram. \[\phi_{k}^{\prime}[m]=\phi_{k}[m]+C_{m} \tag{19}\]  
The frequency output can be expressed as  
\[\hat{f}_{k}^{i}[m]=\frac{f_{ch}}{2\pi}\Delta\phi_{k}^{\prime}[m]=\frac{f_{s}}{2 \pi\cdot D}\Delta\phi_{k}^{\prime}[m] \tag{20}\]  
The IFM error is significant with low SNR signals, which cannot meet the requirements. This study proposes a local flat mean algorithm, which eliminates frequency outliers by finding instantaneous frequency points that satisfy the frequency flat condition. The judgment condition of frequency flatness is that the adjacent frequency differences of 5 sampling points are all within the error range \(\Delta f\), that is  
\[ \begin{cases}\left|\hat{f}_{k}^{i}[m+1]-\hat{f}_{k}^{i}[m]\right|<\Delta f\\ \left|\hat{f}_{k}^{i}[m+2]-\hat{f}_{k}^{i}[m+1]\right|<\Delta f\\ \left|\hat{f}_{k}^{i}[m+3]-\hat{f}_{k}^{i}[m+2]\right|<\Delta f\\ \left|\hat{f}_{k}^{i}[m+4]-\hat{f}_{k}^{i}[m+3]\right|<\Delta f\\ \end{cases}  \tag{21}\]  
The setting of \(\Delta f\) is related to the frequency interval between two adjacent sampling points. If the frequency modulation slope is \(\gamma\) for a chirp signal, the frequency interval between two adjacent sampling points is \(f_{step}=\gamma/f_{ch}\). The value of \(\Delta f\) should be greater than \(f_{step}\). In addition, the value of \(\Delta f\) is related to the input SNR. When the SNR is low, the measured instantaneous frequency is more uneven, and the value of \(\Delta f\) should be increased at this time [32]. The value of \(\Delta f\) under different SNRs can be determined by experiment. A flat frequency vector \(\vec{f}_{k}^{avg}\) can be generated by taking the local average of the frequency points satisfying the frequency flat condition. The signal bandwidth and center frequency are calculated by finding the maximum and minimum values of the flat frequency vector. The pulse frequency estimates \(\vec{f}_{k}^{avg}\) are then sent to the channel arbitration logic which determines whether adjacent channel signals belong to the same signal. The channel arbitration logic is shown in Equation (22), where \(\Delta B\) is determined by the prototype filter response.