To address the above adversary scenarios, we introduce BlueFMCW, a radar system that mitigates the adversary signals. Specifically, instead of transmitting a full chirping signal from the lower to upper frequency, BlueFMCW makes a *random-frequency-jump* in the middle of the chirp signal as shown in Fig. 2 (b) and (c). While the TOF of the reflected signals remains the same, the frequency gap with the adversary signal will be randomized. As a result, the false beat frequency does not stay at the same FFT bin. In other words, the energy of the adversary signal will be *randomly* dispersed over various FFT bins, which results in significantly smaller peaks in the spectrum.

### 3.1 Random frequency hopping

Consider a conventional FMCW chirp signal, i.e., linear frequency modulation signal from a starting frequency \(f_c\) for *T* duration. BlueFMCW creates a series of frequency-hopping chirps by (1) dividing the conventional FMCW signal into *N* equal sub-intervals, and then (2) randomly permuting the sub-chirps. Suppose the victim radar receives a reflected signal from a real object and an adversary signal that spawns a ghost target. The bottom figures in Fig. 2a, b compare the victim’s spectrograms of the conventional FMCW and BlueFMCW. The beat frequency from the adversary signal remains constant through *T* in the conventional FMCW. In contrast, BlueFMCW can *hash* the beat frequency thanks to its *randomized* staring frequencies of the sub-chirps. However, the beat frequency from the real object is not hashed and remains constant through *T*. This is because the beat frequency of a true object only depends on the TOF regardless of the starting frequency. BlueFMCW leverages this to mitigate the adversary signal while achieving the same detection capability on the real objects.

To formalize this, we first denote the starting frequency of the *k*th sub-chirp to be \(f_k\). For the random frequency hopping, we can define a random permutation \(\sigma : F \rightarrow F\), where *F* is a finite set of the index 1, 2, ..., *N*. The two-line notation of \(\sigma\) can be written as:

$$\begin{aligned} \begin{pmatrix} 1 &{} 2 &{} ... &{} N\\ \sigma (1) &{} \sigma (2) &{} ... &{} \sigma (N) \end{pmatrix} \end{aligned}$$

(4)

Figure 3a illustrates an example of BlueFMCW sub-chirps with \(N=4\). In this example, the first sub-chirp of the conventional FMCW is permuted to the third slot, the second sub-chirp to the second slot, the third sub-chirp to the first slot, and the fourth sub-chirp to the third slot. In the next section, we will discuss the impact of the phase discontinuity of the beat signal and how we can reconstruct the original beat signal using the inverse permutation.

### 3.2 Reconstruction

#### 3.2.1 Challenges: resolution and discontinuities

By observing the pattern of the BlueFMCW spectrogram, one can speculate which one is the true object or adversary signal. The true object tends to have a consistent peak frequency while the adversary signal jumps from one to another frequency. Using this observation, one can estimate a rough distance profile from the single sub-chirp’s beat signal. However, the distance resolution of a sub-chirp is degraded by *N* because its bandwidth is reduced by *B*/*N*. As discussed in Sect. 2, the distance resolution achieved by a single sub-chirp becomes \(\frac{cN}{2B}\).

A naive attempt to solve the resolution problem is simply concatenating the beat signals of all sub-chirps in time-order such as

$$\begin{aligned} x_{naive}(t) = [x_1(t), x_2(t), ..., x_N(t)] \end{aligned}$$

(5)

where \(x_k(t)\) is the beat signal by *k*-th sub-chirp. By concatenating all *N* beat signals as illustrated in Fig. 3b, the resolution will remain the same with the conventional FMCW, \(\frac{c}{2B}\). It will create, however, spurious frequency components in the frequency domain. To understand why, we need to examine the phase of the beat signal. Recall the beat signal from Eq. 2. For simplicity, we can rewrite the beat signal of the *k*-th sub-chirp with a single reflection:

$$\begin{aligned} x_k(t) = \exp (j2\pi (\alpha \tau t + f_k \tau - \alpha \tau ^2 /2)) \quad ,t \in [0, T/N] \end{aligned}$$

(6)

The phase in frequency is written as:

$$\begin{aligned} \phi = 2\pi (f_k \tau - \alpha \tau ^2 /2) \end{aligned}$$

(7)

Figure 3c shows an example of the conventional FMCW. As can be seen, the conventional FMCW signal is a special case of BlueFMCW where \(f_k\)’s are sorted in ascending order. Hence, we can represent \(x_2(t)\) in two ways; one with Eq. 6 and the other with the *T*/*N* advanced version of Eq. 2.

$$\begin{aligned} x_2(t) = {\left\{ \begin{array}{ll} \exp (j2\pi (\alpha \tau t + f_2 \tau - \alpha \tau ^2 /2)) &{} \text {by Eq.}~6\\ \exp (j2\pi (\alpha \tau (t+T/N) + f_1 \tau - \alpha \tau ^2 /2)) &{} \text {by Eq.}~2 \end{array}\right. } \end{aligned}$$

(8)

For the conventional FMCW, the phases of the two equation are always the same.

$$\begin{aligned} 2\pi (f_2 \tau - \alpha \tau ^2 /2) = 2\pi (\alpha \tau T/N + f_1 \tau - \alpha \tau ^2 /2) \end{aligned}$$

(9)

The two phases are always identical in the conventional FMCW because \(f_2\) is equal to \(f_1 + \alpha T/N\). However, if the sub-chirps are randomly permuted, it is not guaranteed that \(f_{k+1}\) is equal to \(f_k + \alpha T/N\). This will cause phase discontinuities as shown in Fig. 3d. We will address this issue in the following section.

#### 3.2.2 Phase alignment

To eliminate the phase discontinuity due to the frequency hopping, BlueFMCW re-arranges the beat signals by inverting the permutation used in the frequency hopping. For example in Fig. 3a, the first sub-chirp of the conventional FMCW (starting frequency \(f_1\)) was permuted to the fourth time-slot. If we concatenate the beat signals in time-order, the beat signal starting at \(f_1\) comes at the last place, causing the phase discontinuity. As we know how the sub-chirp was permuted (\(\sigma\)), we can invert the permutation to bring the beat signal back to the correct time-slot so that the phase becomes continuous. By the inverse permutation \(\sigma ^{-1}\), the fourth beat signal is permuted to the first time-slot, and so the others (Fig. 3 b).

Formally, the phase aligned beat signal, \(x_{align}\), is achieved by performing the inverse permutation \(\sigma ^{-1}\) on the time-order beat signals, \(x_{naive}\).

$$\begin{aligned} x_{naive}(t)&= [x_1, x_2, ..., x_N ] \end{aligned}$$

(10)

$$\begin{aligned} x_{align}(t)&= [x_{\sigma ^{-1}(1)}, x_{\sigma ^{-1}(2)}, ..., x_{\sigma ^{-1}(N)}] \end{aligned}$$

(11)

Figure 4 shows the impact of the phase alignment in the beat spectrum. In this example, two true objects and one adversary signal are simulated. The conventional FMCW in Fig. 4 (a) images all three components in the spectrum. In Fig. 4b, the spectrum of \(x_{naive}\) is shown. Due to the phase discontinuity, we can observe many spurious signals around the objects and the smaller and unclear peaks for the true objects. After the phase alignment, Fig. 4c shows two clear peaks of the objects. Note that both BlueFMCWs with and without phase alignment can alleviate the adversary signal. This is because random frequency hopping makes random frequency gaps against the adversary, and thus, the result beat frequencies also become randomized whether phase-aligned or not. The mitigated adversary signal, however, did not disappear. Instead of persisting at one frequency, the adversary signal is randomly dispersed over the various frequencies.

### 3.3 Designing BlueFMCW

In the previous section, we showed that BlueFMCW can spread out the adversary signal while holding the phase continuity of the reflected signals from the real objects. We assume the sub-chirps share the same slope (\(\alpha\)) and duration (\(T_b\)) for the simplicity. In addition, the frequency support of each sub-chirp must be mutual exclusive in order to align the phase by the permutation. For example, if a frequency support overlaps with another, the phase cannot be aligned without dropping the overlapping samples. For the opposite case, if there are empty gaps among the frequency supports, the phase alignment again cannot be achieved without interpolating the missing samples. Thus, having the frequency supports to be mutual exclusive each other and to cover the full bandwidth makes the problems easy.

The question remains *how diverse BlueFMCW can spread out the adversary signals*. To maximize the spread, we want to avoid the adversary signals falling in the same FFT bin. By properly choosing the sub-chirp parameters, the down-converted adversary signals fall into many different FFT bins.^{Footnote 1} As a result, the energy of the adversary signal will be widely spread over the bandwidth. In the opposite case, when the adversary signals fall at a few identical FFT bins, they will end up with high-magnitude tones in FFT.

The beat frequency of an adversary signal in FFT can be represented by the starting frequencies of the victim and aggressor (interferer or attacker), the bandwidth of the sub-chirps (\(B_{sub} = \alpha T_b\)), and the sampling rate (\(f_s\)).

$$\begin{aligned} f_{beat}&= mod(\Delta f + k B_{sub}, f_s) \quad , k = 0,1,2...,N-1 \end{aligned}$$

(12)

where \(\Delta f\) is the frequency difference between the transmitted signal and the received adversary signal, and *k* is an integer dependent on the permutation. In the conventional FMCW, *k* is zero for the entire chirp duration which makes the beat frequency constant. \(\Delta f\) is dependent on the distance, the starting frequency of the aggressor, and the delay of the adversary signal. Therefore, BlueFMCW does not have control over \(\Delta f\). However, BlueFMCW can configure \(B_{sub}\) and \(f_s\). Assume they are set to \(B_{sub} = \frac{n}{m} f_s\) where *m* and *n* are the integers. To calculate \(f_{beat}\), consider the argument of the modulo operation in Eq. 12 as

$$\begin{aligned} \frac{\Delta f + k B_{sub}}{f_s}&= \frac{\Delta f}{f_s} + \frac{kn}{m} \end{aligned}$$

(13)

$$\begin{aligned}&= \underbrace{Q}_{\text {Quotient}} + \underbrace{\delta f + \frac{p}{m}}_{\text {remainder}} \end{aligned}$$

(14)

where *Q* is the quotient of the division, and \(\delta f + \frac{p}{m}\) is the remainder where *p* is an integer such that \(-m<p<m\). We can use the above result to rewrite the beat frequency as

$$\begin{aligned} f_{beat}&= \left(\delta f + \frac{p}{m}\right) f_s \end{aligned}$$

(15)

Note that \(\delta f\) is the constant remainder from \(\frac{\Delta f}{f_s}\) that does not impact on the diversity. Therefore, there exists at most *m* different \(f_{beat}\). For the worst case example (\(m=1\)), consider \(N=4\), \(f_s=100MHz\), \(B_{sub} = 2 f_s = 200MHz\), and \(\Delta f = 70MHz\). No matter how we hop the frequency, the remainder is 70/100, and thereby \(f_{beat}\) is always 70MHz. For a better configuration (\(m=3\)), consider \(B_{sub} = \frac{7}{3} f_s\) with the same setup for the rest. For \(k=0,1,2,3\), the remainders are 70/100, \(70/100 + 1/3\), \(70/100 + 2/3\), and 70/100, respectively. Therefore, we can achieve 3 different \(f_{beat}\) as *m* is set to 3.

Figure 5a shows the BlueFMCW spectrogram of the worst case. Clearly, the adversary is not spread even with the random frequency hopping. This is because \(f_s\) and \(B_{sub}\) have the integer relationship with \(m=1\). The red lines in Fig. 5c correspond to the reconstruction result by the worst case. Figure 5b shows the BlueFMCW spectrogram of \(m=131\). Compared to the worst case, the adversary signal is spread on 131 different beat frequencies. As a result, the adversary signal is greatly reduced as shown in the blue lines in Fig. 5c.