## 3 Low-Complexity Optimization of the Algorithm
### Multiplex FIR Filter  
Based on the above analysis, the MSD algorithm needs to perform the correlation operation between the local reference signals and the received signal, which is essentially equivalent to the calculation process of the matching filter. Consequently, we use the structure of an FIR filter to implement the MSD algorithm on the FPGA platform. (a) Filter structure  
The FIR filter consists of three components: adders, multipliers, and delay blocks [19; 20]. Figure 5 shows a typical structure of a single-channel and single-coefficient FIR filter. Here, \(h(i)\) represents the filter coefficients. According to Equations (12)-(14), during the correlation operation, the real and imaginary parts are computed separately. Therefore, the value represented by \(h(i)\) is the quantized value of either the real or imaginary part, and the value of \(h(i)\) is equivalent to the discrete form of \(\overset{\sim}{L}(t)\) in Equation (11). \(Z^{-1}\) is the delay unit, M is the filter order, \(x(t)\) is the input data, and \(y(t)\) is the output data. In this paper, the FIR filter for processing a single observation interval signal with a single-channel and single-coefficient is represented as follows:  
\[r(i)=\sum\limits_{Ti=0}^{(N+2)N_{s}-1}\widetilde{L}(Ti)s(i-Ti) \tag{15}\]  
where \(s(i)\) is the received signals within the observation interval, \(Ti\) represents the sampling time point, \(N_{s}\) is the oversampling factor, and \(r(i)\) is the output data of the FIR filter. (b) Filter multiplexing  
Figure 5: Single-coefficient FIR filter structure diagram. In a typical PCM/FM signal demodulation system, it is necessary to fix the sampling rate to a lower value, typically in the range of tens of MHz, before demodulation. However, the clock frequency of the FIR filter structure can reach several hundred MHz [21; 22]. Assume that the clock frequency of the FIR filter structure is \(f_{1}\) and that the sampling rate of the signals is \(f_{2}\). The parameter \(q\) is defined as \(q=f_{1}/f_{2}\). For the traditional single-channel FIR filter, if the processing of one sampled data can be completed within \(q\) clock cycles, the FIR filter will meet the performance requirements. However, the FIR filter structure in this configuration exhibits relatively low resource utilization, since the processing of the sampled data can be completed in a single clock cycle, but during the rest of the \(q-1\) clock cycle, the MSD algorithm does not perform any operation but waits. Especially for multi-channel receiver systems, the single-channel FIR filter will cause a large waste of resources and increase the cost of hardware implementation. In this paper, we increase the input signal's sampling frequency of the FIR filter and design a multi-channel and multi-coefficient FIR filter structure instead of a single-channel FIR filter to enhance resource utilization. The mathematical expression for the multi-channel and multi-coefficient is as follows:  
\[r_{pk}(i)=\sum_{Ti=0}^{(N+2)N_{b}-1}\widetilde{L}_{k}(Ti)s[i-(4 Ti+p-1)] \tag{16}\]  
where \(k\) denotes the \(k\)-th set of filter coefficients, and \(p\) is the sequence number of the filter channels. Considering that the PCM/FM signal demodulation method in this paper is orthogonal baseband demodulation, as shown in Equation (7), each data computation involves four types of convolution operations, including the convolutions of the real and imaginary parts of the observed signals and the real and imaginary parts of the local reference signals. Therefore, by modifying the FIR filter into a four-channel, dual-coefficient array structure, four filter resources can be simplified into a single filter resource, as shown in the module diagram in Figure 6.  
In Figure 6, \(r(p)\) represents the output data of the \(p\)-th channel; \(P\) is the total number of channels; and \(K\) is the total number of coefficients.