## 2 System Overview
### Digital Modulation Schemes  
As stated, this radar system can support a variety of operating modes, all of which will have the BPS input logic follow \(b=\sim d\). For this paper, the focus is on the PMCW modes through the introduction of digital codes by switching the phase of the carrier between two available phase states. The CW mode is enabled when the BPS inputs are constant. The main use for this mode is to recover unmodulated Doppler frequency responses, such as target motion, respiration, and hand gestures. In CW mode, the radar power consumption, transmitted power, and received power can also be measured and \(F_{\text{C}}\) can be tuned. By introducing a repeating binary string of \(\{01\}\) to the BPS, the output waveform becomes a phase-shift keyed (PSK) signal. This mode convolves a square wave onto the carrier frequency \(F_{\text{C}}\) at the binary modulation frequency \(F_{\text{C}lk}\) which gets reflected off the subject while encoded. The phase of the introduced PSK wave can be described as  
\[\theta_{PSK}(t)=\theta_{0}+\frac{\Delta\theta}{2}\times\text{sign}(\sin(2\pi F _{\text{C}lk}t)) \tag{2}\]  
where the phase difference \(\Delta\theta\) is the absolute difference between binary states of the BPS. For the lowest PSK bit error rate using a BPS, the ideal \(\Delta\theta\) should be 180\({}^{\circ}\). A PRBS is used for PMCW operation, which acts similarly to the PSK mode while having the recovered waveform also contains the binary sequence. This can be used to isolate the recovered signal from other PMCW radar systems operating in the same area and frequency band, be used in joint-radar communication systems, or be a TX/RX channel in a CDMA bistatic MIMO radar system. For this system, an MLS was chosen as the PMCW modulation scheme. These sequences have desirable orthogonality properties, being balanced and having a single point of autocorrelation following  
\[\theta_{x,x}(0)=N\text{ and }\theta_{x,x}(l)=-1\text{ for }1\leq l<N\text{ and }l=l\text{ mod }N, \tag{3}\]  
where \(N\) is the period of the sequence and \(l\) is the cyclic shift amount of the original sequence [41]. There are a finite number of sequences for each degree of \(m\) with a length of \(N=2^{m}-1\). Each binary MLS also has a polynomial interpretation over GF(2), a finite or Galois field of order 2, which must be a primitive polynomial of degree \(m\). These polynomials take the form  
\[P(x)=x^{m}+a_{m-i}x^{m-i}+\cdots+a_{0}\text{ for }m>i \tag{4}\]  
and can be directly converted to the tap arrangement in a linear feedback shift register (LFSR). The sequences can be dynamically generated with a digital LSFR if the registers have any starting value other than all zeros, which will never advance from this initial state as the feedback will always be zero. Alternatively, since the cyclic binary states are deterministic, they can be pre-generated to ensure accuracy and loaded into a memory register on the FPGA, which then increments a pointer through the MLS before looping around to the start of the sequence. An example primitive polynomial of degree 4 is  
\[P(x)=x^{4}+x^{1}+1, \tag{5}\]  
its corresponding LSFR arrangement is shown in Figure 4.  
The equation for the phase of the PMCW waveform is similar to (2), but the sign of the phase offset is obtained through a piecewise equation that corresponds to the pseudo-random equation of the sequence. For an MLS, the sign can be defined as the equation \(s(t,m)\) with the number of intervals \(m\) equaling the order of the primitive polynomial used to make the sequence. Using the MLS described in (5), \(s(t,m)\) can be defined as  
\[s(t,m=4)=2a_{0}-1\text{ where }\left\{ \begin{array}{rcl}a_{3}[n+1]=&mod(a_{0}[n]+a_ {3}[n],2)\\ a_{2}[n+1]=&a_{3}[n]\\ a_{1}[n+1]=&a_{2}[n]\\ a_{0}[n+1]=&a_{1}[n]\end{array} \right\} \tag{6}\] \[\text{ for }n=mod(\lfloor F_{Clk}\times(t)\rfloor+n_{0},2^{m}-1) \text{andnonzeroinitialconditions},\]  
where \(n_{0}\) is the initial bit of the MLS and \(F_{Clk}\) is the clock speed used to cycle through states.