Part 2.1 and 2.2 analyze the signal model of the FDA radar and the amplitude-comparison monopulse reconnaissance system, respectively. In practice, the amplitude-comparison monopulse direction-finding system distributes multiple receivers uniformly in the same position but in different directions. It determines the DOA of signals by comparing the amplitude of the signals received by different receivers. The incoming wave direction determined by the monopulse system is based on the received beampattern, which is affected by the frequency of the incoming wave. The above analysis is based on the fixed incoming wave frequency. However, according to the analysis in Part 2.1, we can know that by introducing a tiny frequency offset between adjacent elements, the FDA can achieve an S-shaped beampattern. Therefore, we can see that the FDA signal received by the direction-finding system does not conform to the direction-finding principle of the amplitude-comparison monopulse. Thus, the FDA may achieve the DOA location deception on the amplitude-comparison monopulse. To research the deception effect, we establish a DOA location deception model based on the FDA to contrast the amplitude-comparison monopulse direction-finding system.

### 3.1 Deception model based on FDA

Assuming that on arrival at the direction-finding system, the signal radiated by the FDA is with incidence angle \(\theta\), and the FDA radar and the amplitude-comparison monopulse direction-finding system are in the same plane XOZ, then the position relationship between the direction-finding system and FDA radar is shown in Fig. 4.

Assuming that the receiver of the direction-finding system is an array antenna containing *M* elements, the signal received by the receiver *n* is

$$s_{n} \left( {t,\theta } \right) = \sum\limits_{m = 1}^{M} {\sum\limits_{q = 1}^{Q} {S_{tqm} (t)} } = \sum\limits_{m = 1}^{M} {\sum\limits_{q = 1}^{Q} {A_{q} A_{m} \exp \left[ {j2\pi f_{q} (t - R_{nqm} /c)} \right]} }$$

(17)

where \(R_{nqm}\) is the range between element *q* of the FDA and element *m* of the receiver *n*, i.e.,

$$R_{nqm} = R - \left( {q - 1} \right)d\sin \theta + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)$$

(18)

Then, (17) can be further rewritten as

$$\begin{aligned} s_{n} \left( {t,\theta } \right) & = \sum\limits_{m = 1}^{M} {\sum\limits_{q = 1}^{Q} {A_{q} A_{m} \exp \left\{ {j2\pi f_{q} \left[ {t - {{\left[ \begin{gathered} R - \left( {q - 1} \right)d\sin \theta \hfill \\ + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right) \hfill \\ \end{gathered} \right]} \mathord{\left/ {\vphantom {{\left[ \begin{gathered} R - \left( {q - 1} \right)d\sin \theta \hfill \\ + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right) \hfill \\ \end{gathered} \right]} c}} \right. \kern-\nulldelimiterspace} c}} \right]} \right\}} } \\ & = \sum\limits_{m = 1}^{M} {\sum\limits_{q = 1}^{Q} {A_{q} A_{m} \exp \left\{ {j2\pi \left( {f_{0} + \Delta f_{q} } \right)\left[ {t - {{\left[ \begin{gathered} R - \left( {q - 1} \right)d\sin \theta \hfill \\ + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right) \hfill \\ \end{gathered} \right]} \mathord{\left/ {\vphantom {{\left[ \begin{gathered} R - \left( {q - 1} \right)d\sin \theta \hfill \\ + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right) \hfill \\ \end{gathered} \right]} c}} \right. \kern-\nulldelimiterspace} c}} \right]} \right\}} } \\ \end{aligned}$$

(19)

Assuming that the amplitude of the radiated signal of each transmitting element and the receiving gain of the receiving element are both considered to be equal, i.e. \(A_{q} = A = 1\), \(A_{m} = A = 1\), (19) can be rewritten as

$$\begin{aligned} s_{n} \left( {t,\theta } \right) & = \sum\limits_{m = 1}^{M} {\sum\limits_{q = 1}^{Q} {\exp \left\{ {j2\pi \left( {f_{0} + \Delta f_{q} } \right)\left[ {t - \frac{{R - \left( {q - 1} \right)d\sin \theta + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)}}{c}} \right]} \right\}} } \\ & = \sum\limits_{q = 1}^{Q} {\sum\limits_{m = 1}^{M} {\exp \left\{ {j2\pi \left( {f_{0} + \Delta f_{q} } \right)\left[ {t - \frac{{R - \left( {q - 1} \right)d\sin \theta + \left( {m - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)}}{c}} \right]} \right\}} } \\ & = \sum\limits_{q = 1}^{Q} {\frac{{{\text{sin}}\left( {M\pi \left( {f_{0} + \Delta f_{q} } \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right)}}{{\sin \left( {\pi \left( {f_{0} + \Delta f_{q} } \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right)}}} \\ & \quad \exp \left\{ {j2\pi \left( {f_{0} + \Delta f_{q} } \right)\left[ {t - \frac{{R - \left( {q - 1} \right)d\sin \theta + {{\left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} \mathord{\left/ {\vphantom {{\left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}}}{c}} \right]} \right\} \\ \end{aligned}$$

(20)

For the phased array signal, i.e. \(\Delta f_{q} = 0\), then

$$s_{{{\text{PA}}\_n}} \left( {t,\theta } \right) = \exp \left\{ {j\phi_{1} } \right\}\frac{{\sin \left[ {M\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}{{\sin \left[ {\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}\frac{{\sin \left( {Q\pi f_{0} d\sin \theta } \right)}}{{\sin \left( {\pi f_{0} d\sin \theta } \right)}}$$

(21)

where \(\phi_{1} = 2\pi f_{0} \left\{ {t - {{\left[ \begin{gathered} R - {{\left( {Q - 1} \right)d\sin \theta } \mathord{\left/ {\vphantom {{\left( {Q - 1} \right)d\sin \theta } 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ + {{\left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} \mathord{\left/ {\vphantom {{\left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ \end{gathered} \right]} \mathord{\left/ {\vphantom {{\left[ \begin{gathered} R - {{\left( {Q - 1} \right)d\sin \theta } \mathord{\left/ {\vphantom {{\left( {Q - 1} \right)d\sin \theta } 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ + {{\left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} \mathord{\left/ {\vphantom {{\left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ \end{gathered} \right]} c}} \right. \kern-\nulldelimiterspace} c}} \right\}\).

Combining (20) and (21), we learn that compared with phased array signal, the amplitude of the FDA signal received by the amplitude-comparison monopulse direction-finding system is coupled with distance, time and angle, due to the introduction of the small frequency offset among the elements. Therefore, the direction-finding system cannot accurately estimate the DOA of the signal. In fact, assuming that the outputs of direction-finding signal processor are \(s_{n} \left( {t,\theta } \right)\) and \(s_{n + 1} \left( {t,\theta } \right)\), the signal can be determined to be between the included angle of the two adjacent receivers. Then, the estimated DOA obtained by the direction-finding system is

$$\hat{\theta } = {\text{solve}}\left[ {\begin{array}{*{20}c} {\frac{{s_{n} \left( {t,\theta } \right)}}{{s_{n + 1} \left( {t,\theta } \right)}} = \frac{{\sin \left[ {\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n + 1} } \right)} \right]}}{{\sin \left[ {M\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n + 1} } \right)} \right]}} \cdot \frac{{\sin \left[ {M\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}{{\sin \left[ {\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}},} & {\theta \in \left[ {\theta_{n} ,\theta_{n + 1} } \right]} \\ \end{array} } \right]$$

(22)

And the location estimation can be expressed as

$$\hat{x} = R\sin \hat{\theta }$$

(23)

The location deviation is

$$\Delta x = R\sin \hat{\theta } - R\sin \theta$$

(24)

Generally, when the measured location deviates from the transmitting antenna array, the deception effect on the direction-finding system is considered better, i.e.

$$\Delta x > 0||\Delta x < - (Q - 1)d$$

(25)

### 3.2 Model verification and performance analysis

#### 3.2.1 SNR analysis

Signal-to-noise ratio (SNR) is an important parameter in the direction-finding system. As the analyses above, we can know that the FDA can generate a range-angle-time-dependent beampattern. Thus, in the constant noise environment, the FDA signal has the time-varying SNR, while the PA signal has the approximately constant SNR. Therefore, we define the instantaneous SNR (ISNR) to express the time-varying SNR:

$${\text{ISNR}}_{{{\text{FDA}}}} \left( t \right) = 10\log \left( {\frac{{P_{{{\text{FDA}}}} \left( t \right)}}{{P_{w} }}} \right)$$

(26)

where \(P_{w} = w_{0} B\) denotes the average power of noise, with \(w_{0}\) and *B* being the unilateral power spectral density of noise and the bandwidth of FDA signal, respectively. Besides the FDA signal power \(P_{{{\text{FDA}}}} \left( t \right)\) can be given by

$$P_{{{\text{FDA}}}} \left( t \right) = \frac{1}{\tau }\int_{t}^{t + \tau } {\left| {E_{{{\text{FDA}}}} \left( t \right)} \right|^{2} {\text{d}}t} = \frac{1}{\tau }\int_{t}^{t + \tau } {\left| {A_{{{\text{BP}}\_{\text{FDA}}}} \left( t \right)} \right|^{2} {\text{d}}t}$$

(27)

where *t* and \(\tau\) denotes the starting time and period of sampling, respectively.

The SNR of PA can be then given by,

$${\text{SNR}}_{{{\text{PA}}}} = {\text{ISNR}}_{{{\text{PA}}}} \left( t \right) = 10\log \left( {\frac{{P_{{{\text{PA}}}} }}{{P_{w} }}} \right)$$

(28)

where

$$P_{{{\text{PA}}}} = \left| {E_{{{\text{PA}}}} } \right|^{2} = \left| {A_{{{\text{BP}}\_{\text{PA}}}} } \right|^{2} = \left| {A_{{{\text{BP}}\_{\text{FDA}}}} \left( t \right)} \right|^{2}_{{\Delta f_{1:Q} = 0}}$$

(29)

As the analysis above, the SNR is mainly affected by the antenna gain of the different signals. Therefore, for the fixed far-field target, the antenna gain ratio of the FDA to the PA can be expressed as

$$\hbar \left( t \right) = \frac{{\left| {E_{{{\text{FDA}}}} \left( t \right)} \right|}}{{\left| {E_{{{\text{PA}}}} } \right|}} = \frac{{\left| {E_{{{\text{FDA}}}} \left( t \right)} \right|}}{{\left| {E_{{{\text{FDA}}}} \left( t \right)} \right|_{{\Delta f_{1:Q} = 0}} }}{ = }\frac{{\left| {A_{{{\text{BP}}\_{\text{FDA}}}} \left( t \right)} \right|}}{{\left| {A_{{{\text{BP}}\_{\text{FDA}}}} \left( t \right)} \right|_{{\Delta f_{1:Q} = 0}} }}$$

(30)

Following the same signal-arriving-time assumption, it holds that \(\left| {E_{{{\text{PA}}}} } \right| = \left| {E_{{{\text{FDA}}}} \left( t \right)} \right|_{{\Delta f_{1:Q} = 0}} = \max \left| {E_{{{\text{FDA}}}} \left( t \right)} \right|\), i.e.,

$$\hbar \left( t \right) = \frac{{\left| {E_{{{\text{FDA}}}} \left( t \right)} \right|}}{{\max \left| {E_{{{\text{FDA}}}} \left( t \right)} \right|}} \le 1$$

(31)

Therefore, following the same noise environment and signal-arriving time assumption, the ISNR of the FDA signal will not exceed the SNR of the PA signal.

#### 3.2.2 Error analysis

Positioning error is an important parameter to describe the positioning accuracy of the direction-finding system. However, one of the primary objectives of our deception method is to increase the measurement error of the direction-finding system. Therefore, to indicate the superiority of our deception method, the root-mean-square error (RMSE) and variation (Var) are derived.

Under the condition of Monte Carlo experiments, the RMSE of angle measurement \(\hat{\theta }\) is given by

$${\text{RMSE}}\left( {\hat{\theta }} \right) = \sqrt {E\left[ {\left( {\hat{\theta } - \theta } \right)^{2} } \right]} = \sqrt {{\text{Var}}\left( {\hat{\theta }} \right) + \left[ {E\left( {\hat{\theta }} \right) - \theta } \right]^{2} }$$

(32)

where \({\text{Var}}\left( {\hat{\theta }} \right)\) represents the variance of \(\hat{\theta }\), and it can be calculated as

$${\text{Var}}\left( {\hat{\theta }} \right) = E\left\{ {\left[ {\hat{\theta } - E\left( {\hat{\theta }} \right)} \right]^{2} } \right\}$$

(33)

Similar to (32), the RMSE of \(\hat{x}\) can be also given by

$${\text{RMSE}}(\hat{x}) = \sqrt {E\left[ {\left( {\hat{x} - x} \right)^{2} } \right]} = \sqrt {{\text{Var}}\left( {\hat{x}} \right) + \left[ {E\left( {\hat{x}} \right) - x} \right]^{2} }$$

(34)

where \({\text{Var}}\left( {\hat{x}} \right) = E\left\{ {\left[ {\hat{x} - E\left( {\hat{x}} \right)} \right]^{2} } \right\}\) denotes the variance of \(\hat{x}\).

Actually, since the array antenna cannot be considered as a point, we generally take the array center into consideration, let \(\hat{x}_{0} = R\sin \hat{\theta }\) and \(x_{0} = x - \left( {Q - 1} \right)d/2\), we have,

$${\text{RMSE}}(\hat{x}_{0} ) = \sqrt {E\left[ {\left( {\hat{x}_{0} - x_{0} } \right)^{2} } \right]} = \sqrt {{\text{Var}}\left( {\hat{x}_{0} } \right) + \left[ {E\left( {\hat{x}_{0} } \right) - x_{0} } \right]^{2} }$$

(35)

where \({\text{Var}}\left( {\hat{x}} \right) = {\text{Var}}\left( {\hat{x}_{0} } \right)\).

#### 3.2.3 CRLB analysis

Cramér–Rao Lower Bound (CRLB) shows the ideal performance that the direction-finding system can achieve. However, different signals actually have different characteristics, meaning that the different signals the system receives, the different performances are. In other words, if the FDA signals have the higher CRLB than other deceptive signals, the FDA signals have a better deception effect. Suppose \(A_{{{\text{FDA}}}} \left( t \right)\) is the transmit beam of the FDA, then we have

$$\begin{aligned} s_{n} \left( {t,\theta } \right) & \approx A_{{{\text{FDA}}}} \left( t \right)\frac{{\sin \left[ {M\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}{{\sin \left[ {\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}\\ &\quad \exp \left\{ {j\pi f_{0} \left[ {2t - \frac{{2R - \left( {Q - 1} \right)d\sin \theta + \left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)}}{c}} \right]} \right\} \\ & = A_{{{\text{FDA}}}} \left( t \right)A_{r} \left( \theta \right)\exp \left\{ {j\pi f_{0} \left[ {2t - \frac{{2R - \left( {Q - 1} \right)d\sin \theta + \left( {M - 1} \right)d_{1} \sin \left( {\theta - \theta_{n} } \right)}}{c}} \right]} \right\} \\ & \approx A_{{{\text{FDA}}}} \left( t \right)A_{r} \left( \theta \right)\exp \left\{ {j\pi f_{0} \left[ {2t - \frac{{2R - \left( {Q - 1} \right)d\sin \theta + \left( {M - 1} \right)d_{1} \left( {\theta - \theta_{n} } \right)}}{c}} \right]} \right\} \\ & = A_{{{\text{FDA}}}} \left( t \right)A_{r} \left( \theta \right)\exp \left\{ {j\left( {\phi \theta_{n} + \psi } \right)} \right\} \\ \end{aligned}$$

(36)

where

$$A_{r} \left( \theta \right) = \frac{{\sin \left[ {M\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}{{\sin \left[ {\pi f_{0} d_{1} \sin \left( {\theta - \theta_{n} } \right)} \right]}}$$

(37-1)

$$\phi = {{\pi f_{0} \left( {M - 1} \right)d_{1} } \mathord{\left/ {\vphantom {{\pi f_{0} \left( {M - 1} \right)d_{1} } c}} \right. \kern-\nulldelimiterspace} c}$$

(37-2)

$$\psi = 2\pi f_{0} \left( {t - {R \mathord{\left/ {\vphantom {R c}} \right. \kern-\nulldelimiterspace} c}} \right) + \pi f_{0} \left[ {{{\left( {Q - 1} \right)d\sin \theta } \mathord{\left/ {\vphantom {{\left( {Q - 1} \right)d\sin \theta } c}} \right. \kern-\nulldelimiterspace} c} - {{\left( {M - 1} \right)d_{1} \theta } \mathord{\left/ {\vphantom {{\left( {M - 1} \right)d_{1} \theta } c}} \right. \kern-\nulldelimiterspace} c}} \right]$$

(37-3)

Then, the general signal sampling processed by the *N*-receiver monopulse direction-finding system can be expressed as

$$\begin{aligned} {\mathbf{x}}\left[ n \right] & = {\mathbf{s}}\left[ n \right] + {\mathbf{w}}\left[ n \right] \\ & = \left| {A\left( {t,\theta } \right)} \right|\exp \left( {j\left( {\phi \theta_{n} { + }\psi } \right)} \right) + {\mathbf{w}}\left[ n \right]\quad n = 0,1, \ldots N - 1 \\ \end{aligned}$$

(38)

where \(A\left( {t,\theta } \right) = A_{{{\text{FDA}}}} \left( t \right)A_{r} \left( \theta \right)\) denotes the received signal amplitude.

Therefore, the CRLBs of the FDA signals for the measured DOA and location error are

$${\text{CRLB}}_{{{\text{FDA}}\_\theta }} = \frac{{\tan^{2} \theta }}{{2N \cdot {\text{ISNR}}_{{{\text{FDA}}}} \cdot \left[ {M\cos \left( {M\varphi } \right) - A_{r} \left( \theta \right)} \right]^{2} }}$$

(39-1)

$${\text{CRLB}}_{{{\text{FDA}}\_x}} = \frac{{R^{2} \sin^{2} \theta }}{{2N \cdot {\text{ISNR}}_{{{\text{FDA}}}} \cdot \left[ {M\cos \left( {M\varphi } \right) - A_{r} \left( \theta \right)} \right]^{2} }}$$

(39-2)

Similarly, the CRLB of PA signals is derived by

$${\text{CRLB}}_{{{\text{PA}}\_\theta }} = \frac{{\tan^{2} \theta }}{{2N \cdot {\text{SNR}}_{{{\text{PA}}}} \cdot \left[ {M\cos \left( {M\varphi } \right) - A_{r} \left( \theta \right)} \right]^{2} }}$$

(40-1)

$${\text{CRLB}}_{{{\text{PA}}\_x}} = \frac{{R^{2} \sin^{2} \theta }}{{2N \cdot {\text{SNR}}_{{{\text{PA}}}} \cdot \left[ {M\cos \left( {M\varphi } \right) - A_{r} \left( \theta \right)} \right]^{2} }}$$

(40-2)

As the analysis above, we can know following the same noise environment and signal-arriving time assumption, the ISNR of the FDA do not exceed that of the PA, meaning that \({\text{CRLB}}_{{{\text{FDA}}}} \ge {\text{CRLB}}_{{{\text{PA}}}}\), i.e., the FDA signals have a better deception effect than the PA signals.