According to the analysis in Sect. 3, the OMP algorithm fails to reconstruct the multiple mono-pulse signals in the completely frequency aliasing situation when the modulated sampling scheme uses the SFM LO. Meanwhile, because the atom calculation in the OMP algorithm is equivalent to the NZ index estimation method in [13], this method also fails to estimate the multiple NZ indices when the LO modulation is the SFM. Naturally, from the perspective of the receiving architecture, finding an improved LO modulation type to obtain a better multiple mono-pulse signals reconstruction is necessary to be investigated.

### 4.1 Improved local oscillator using periodic linear frequency modulation and reconstruction

In Sect. 3, the reason of the OMP algorithm failing to reconstruct the multiple mono-pulse signals under the completely frequency aliasing condition is that the SFM spectrum is out of flatness because of the Jacobi component. Considering the requirements of the modulated sampling scheme LO, the LO modulation type should be periodic and it should contain a certain bandwidth. Therefore, combining with the implementation of sample pulse train generation [25], the periodic linear frequency modulation (PLFM) is chosen as the improved LO modulation type and the LO of the modulated sampling scheme using the PLFM in (7) can be rewritten as

$$\theta_{PLFM} \left( m \right) = \pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2}$$

(25)

where \(\mu_{0}\) is the chirp rate, \(T_{LO}\) is the modulation period of the PLFM, \(T_{LO} < T\) and \(\bmod\) means modulo. In (25), the instantaneous frequency is \(2\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)\) and it is linear in each period \(T_{LO}\). According to the multiple intercepted mono-pulse radar signals in (1), the modulated sampling scheme in (2) and the LO modulation in (25), the outputs can be computed as

$$y\left( m \right) = A\sum\limits_{i = 1}^{s} {e^{{2j\pi \left( {f_{i} - k_{i} f_{s} } \right)mT_{ADC} - jk_{i} \pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} }$$

(26)

Then, the relation between the PLFM LO and the multiple mono-pulse signals reconstruction is analyzed. Based on the OMP algorithm, (12) and (25), the recover signal is

$$r_{PLFM} \left( m \right) = e^{{jk\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }}$$

(27)

where \(k = 0,1, \cdots ,K - 1\). The analysis of step 1 and step 2 of the OMP algorithm is like the analysis of (13). Focusing on the *k*th NZ, it yields

$$e^{jk\theta \left( m \right)} {{\varvec{\uppsi}}}_{M}^{H} {\mathbf{y}} = \mathcal{F}_{D} \left[ {Ae^{{jk\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} \sum\limits_{i = 1}^{s} {e^{{2j\pi \left( {f_{i} - k_{i} f_{s} } \right)mT_{ADC} - jk_{i} \pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} } } \right]$$

(28)

Similar to the analysis in (14) and according to (28), the result of \(\left| {\left\langle {{{\varvec{\Phi}}},{\mathbf{y}}} \right\rangle } \right|\) in the \(k^{th}\) NZ is approximately expressed as:

$$\left| {\left\langle {{{\varvec{\Phi}}},{\mathbf{y}}} \right\rangle } \right|\left( k \right) = \left| {\mathcal{F}_{D} \left[ {y\left( m \right)r_{PLFM} \left( m \right)} \right]} \right| \approx \left| {A\sum\limits_{i = 1}^{s} {\left\{ {2\pi \delta \left( {f - f_{ci} } \right)*\mathcal{F}_{D} \left[ {e^{{j\left( {k - k_{i} } \right)\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} } \right]} \right\}} } \right|$$

(29)

where \(f_{ci} = f_{i} - k_{i} f_{s} ,\;i = 1,2, \cdots ,s\). In (29), the DTFT result is also employed to approximate the DFT result, and the PLFM modulation part in (29) should be investigated to guarantee step 2 of the OMP algorithm returns the atom in the correct NZ under the completely frequency aliasing condition.

Focusing on the PLFM modulation part in (29), it can be calculated as

$$\mathcal{F}_{D} \left[ {e^{{j\left( {k - k_{i} } \right)\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} } \right] \approx \sum\limits_{q = 1}^{Q} {\left| {\xi_{q} \left( f \right)} \right|e^{{j\theta_{q} \left( f \right)}} }$$

(30)

where \(\left| {\xi_{q} \left( f \right)} \right| = \frac{{\sqrt {\left[ {C\left( {u_{q1} } \right) + C\left( {u_{q2} } \right)} \right]^{2} + \left[ {S\left( {u_{q1} } \right) + S\left( {u_{q2} } \right)} \right]^{2} } }}{{\sqrt {2\left( {k - k_{i} } \right)} }}\), \(\theta_{q} \left( f \right) = \arctan \left[ {\frac{{S\left( {u_{q1} } \right) + S\left( {u_{q2} } \right)}}{{C\left( {u_{q1} } \right) + C\left( {u_{q2} } \right)}}} \right] - \frac{{\pi f^{2} }}{{\left( {k - k_{i} } \right)\mu_{0} }}\), \(q = 1,2, \cdots ,Q\), \(Q = round\left( {\frac{T}{{T_{LO} }}} \right)\), \(M_{LO} = round\left( {\frac{{T_{LO} }}{{T_{ADC} }}} \right)\), \(u_{q1} = \sqrt {2\left( {k - k_{i} } \right)\mu_{0} } \left[ {M_{LO} T_{ADC} \left( {q - 1} \right) - \frac{f}{{\left( {k - k_{i} } \right)\mu_{0} }}} \right]\), \(u_{q2} = \sqrt {2\left( {k - k_{i} } \right)\mu_{0} } \left[ {M_{LO} T_{ADC} q - \frac{f}{{\left( {k - k_{i} } \right)\mu_{0} }}} \right]\), \(u_{Q2} = \sqrt {2\left( {k - k_{i} } \right)\mu_{0} } \left[ {M_{LO} T_{ADC} - \frac{f}{{\left( {k - k_{i} } \right)\mu_{0} }}} \right]\), \(C\left( \cdot \right)\) and \(S\left( \cdot \right)\) are the Fresnel integral results.

In (30), the PLFM part in the frequency domain contains several Fresnel integral results and it approximates the summation of several rectangles, which means the spectrum of the PLFM has no Jacobi component and it is flat. According to the matched NZ index condition and the spectrum peak in (29), when \(k = k_{z}\) and \(f = f_{ci}\), the result in (29) is

$$\begin{aligned} \left| {\left. {\mathcal{F}_{D} \left[ {y\left( m \right)r_{PLFM} \left( m \right)} \right]} \right|_{{k = k_{z} ,f = f_{ci} }} } \right| & \approx 2\pi A\left| {\sum\limits_{z \ne i,i = 1}^{s} {\left\{ {\delta \left( 0 \right)*\mathcal{F}_{D} \left[ {e^{{j\left( {k - k_{i} } \right)\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} } \right]} \right\} + \delta \left( 0 \right)} } \right| \\ & = 2\pi A\left| {\sum\limits_{z \ne i,i = 1}^{s} {\sum\limits_{q = 1}^{Q} {\left| {\xi_{q} \left( 0 \right)} \right|e^{{j\theta_{q} \left( 0 \right)}} } + \delta \left( 0 \right)} } \right| \\ \end{aligned}$$

(31)

In addition, the unmatched NZ index situation is considered, and the corresponding result in (29) is

$$\begin{aligned} \left| {\left. {\mathcal{F}_{D} \left[ {y\left( m \right)r_{PLFM} \left( m \right)} \right]} \right|_{{f = f_{ci} }} } \right| & \approx 2\pi A\left| {\sum\limits_{i = 1}^{s} {\left\{ {\delta \left( 0 \right)*\mathcal{F}_{D} \left[ {e^{{j\left( {k - k_{i} } \right)\pi \mu_{0} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }} } \right]} \right\}} } \right| \\ & = 2\pi A\left| {\sum\limits_{i = 1}^{s} {\sum\limits_{q = 1}^{Q} {\left| {\xi_{q} \left( 0 \right)} \right|e^{{j\theta_{q} \left( 0 \right)}} } } } \right| \\ \end{aligned}$$

(32)

Obviously, the maximum value in (31) is greater than that in (32) because (31) contains \(\delta \left( 0 \right)\). Therefore, \(\left| {\mathcal{F}_{D} \left[ {y\left( m \right)r_{PLFM} \left( m \right)} \right]} \right|\) in (29) can achieve the maximum when \(k = k_{i}\) and \(f = f_{ci}\).

As a result, step 2 can obtain the support \(\Lambda_{{i_{t} }}\) that is located in the correct NZ. Furthermore, step 3 will demodulate, filter and clean the reconstructed signal. Finally, each intercepted signal can be reconstructed by the OMP algorithm. In the view of frequency analysis, compared with the spectrum of the SFM, the spectrum of the PLFM is flat due to the Fresnel integral, and the result of the matched NZ index in (31) will not be overwhelmed by the result of the unmatched NZ index in (32).

In terms of the parameter estimation or interception performance, when the result in (29) meets its maximum, the estimated NZ index and the folded carrier frequencies can be obtained as

$$\hat{k}_{i} ,\hat{f}_{ci} = \mathop {\arg }\limits_{{k_{i} ,f_{ci} }} \left\{ {\max \left[ {\left| {\left\langle {{{\varvec{\Phi}}},{\mathbf{y}}} \right\rangle } \right|\left( k \right)} \right]} \right\}$$

(33)

where \(i = 1,2, \cdots ,s\). The estimated carrier frequencies of the intercepted multiple mono-pulse signals are computed as \(\hat{f}_{i} = \hat{f}_{ci} + \hat{k}_{i} f_{s}\). Therefore, if the mono-pulse signals are recovered successfully, the parameters of the mono-pulse signals intercepted by the scheme can be estimated.

It is worth noting that the result of the last term in (30) will be no longer a rectangle when the distance between \(u_{q2}\) and \(u_{Q2}\) is very short. In other words, if \(u_{Q2} - u_{q2}\) in (30) is very small, the linear frequency change of the last piece of the PLFM is not obvious and the corresponding spectrum is similar to the spectrum of single tone. Thus, the result in (30) will contain an impulse function, and the summation result in (32) is not flat, which may lead signal reconstruction failure. Since the parameters of the PLFM LO are known and the time domain of the mono-pulse signal can be obtained by signal detection [21], increasing the chirp rate or omitting the last piece of the PLFM in the modulated sampling scheme output can avoid this circumstance.

### 4.2 Modulation parameter setting criterion

Next, based on the improved LO modulation type, the parameter setting criterion is investigated to obtain a better reconstruction performance.

In this subsection, the CS model of the modulated sampling scheme \({\mathbf{y}} = {\mathbf{\Phi X}}\) in (2) is analyzed again. The sensing matrix \({{\varvec{\Phi}}}\) satisfies the restricted isometry property (RIP) of order \(s\) with parameter \(\delta_{s} \in \left[ {0,\;1} \right)\) if

$$\left( {1 - \delta_{s} } \right)\left\| {\mathbf{X}} \right\|_{2}^{2} \le \left\| {{\mathbf{\Phi X}}} \right\|_{2}^{2} \le \left( {1 + \delta_{s} } \right)\left\| {\mathbf{X}} \right\|_{2}^{2}$$

(34)

holds for every s-sparse signal \({\mathbf{X}}\). Based on the relation between the RIP and eigenvalues [26], we have

$$1 - \delta_{s} \le \lambda_{\min } \left[ {G\left( {{{\varvec{\Phi}}}_{\Lambda } } \right)} \right] \le \lambda_{\max } \left[ {G\left( {{{\varvec{\Phi}}}_{\Lambda } } \right)} \right] \le 1 + \delta_{s}$$

(35)

where the Gram matrix \(G\left( {{{\varvec{\Phi}}}_{\Lambda } } \right) = {{\varvec{\Phi}}}_{\Lambda }^{H} {{\varvec{\Phi}}}_{\Lambda }\), and \({{\varvec{\Phi}}}_{\Lambda }\) consists of the columns of \({{\varvec{\Phi}}}\) with indices \(\Lambda \in s\), \(s \subset \left\{ {1,\;2,\; \cdots ,\;N} \right\}\).

Since the eigenvalues of the Gram matrix \(G\left( {{\varvec{\Phi}}} \right)\) contain the eigenvalues information of all sub-matrix \({{\varvec{\Phi}}}_{\Lambda }\), we focus on the Gram matrix \(G\left( {{\varvec{\Phi}}} \right)\) [27] and it can be expressed as

$$G\left( {{\varvec{\Phi}}} \right) = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{M} } & {{\mathbf{G}}_{10} } & \cdots & {{\mathbf{G}}_{{\left( {K - 1} \right)0}} } \\ {{\mathbf{G}}_{01} } & {{\mathbf{I}}_{M} } & \cdots & {{\mathbf{G}}_{{\left( {K - 1} \right)1}} } \\ \vdots & \vdots & \ddots & \vdots \\ {{\mathbf{G}}_{{0\left( {K - 1} \right)}} } & {{\mathbf{G}}_{{1\left( {K - 1} \right)}} } & \cdots & {{\mathbf{I}}_{M} } \\ \end{array} } \right]$$

(36)

In (36), \({\mathbf{G}}_{k1,k2} \in {\mathbb{C}}^{M \times M}\) is the off-diagonal sub-matrix, where the NZ indices are \(k1 = 0,\;1,\; \cdots ,\;K - 1\), and \(k2 = 0,\;1,\; \cdots ,\;K - 1\). According to the Gershgorin circle theorem, each Gershgorin disk radius should be less than \(\delta_{s}\) to guarantee the sensing matrix \({{\varvec{\Phi}}}\) satisfies the RIP because the diagonal elements of \(G\left( {{\varvec{\Phi}}} \right)\) are equal to 1. Thus, the off-diagonal element in the off-diagonal sub-matrix is [17]

$$\left| {\left. {g_{k1,k2,c,r} } \right|_{k1 \ne k2} } \right| < \frac{{\delta_{s} }}{s}$$

(37)

where the column index is \(c = 0,\;1,\; \cdots ,\;M - 1\) and the row index is \(r = 0,\;1,\; \cdots ,\;M - 1\). If the number of s-sparse signals increases, the values of off-diagonal elements should be decreased to guarantee the relation in (37).

Then, the off-diagonal element in (36) can be calculated as

$$g_{k1,k2,c,r} = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {e^{{j\left( {k2 - k1} \right)\theta_{PLFM} \left( m \right)}} e^{{ - j\frac{2\pi }{M}m(r - c)}} }$$

(38)

The off-diagonal element in (38) can be regarded as the DFT result of the signal using the improved LO modulation. Therefore, to recover more s-sparse signals intercepted by the modulated sampling scheme (i.e.. increasing \(s\)), the bandwidth of the LO modulation \(B_{LO}\) should be increased to satisfy (37), which means the chirp rate \(\mu_{0}\) or the modulation period in (25) should be greater.

Moreover, the above criterion can be interpreted in frequency domain as well. According to the Parseval theorem, when the bandwidth of the LO modulation increases, the spectrum magnitude of the LO modulation will decrease and the frequency aliasing impact is less. However, increasing the bandwidth of the LO modulation has a limitation because the modulated sampling scheme requires \(B_{LO} \ll f_{s}\) [23].

Furthermore, since the modulated sampling is in the middle of the uniform sampling and the CS random sampling, the statistical RIP (StRIP) is employed to analyze the parameter setting criterion. The upper bound on general RIP numbers of the modulated sampling scheme is [11]

$$\delta_{s} \le sC\sqrt {\frac{1}{{B_{LO} T}}}$$

(39)

where \(C\) is a constant related to the LO modulation type and the LO modulation bandwidth. Therefore, when the number of the intercepted mono-pulse signals increases (i.e., \(s\) increases), the LO modulation bandwidth \(B_{LO}\) should be increased to guarantee the relation in (39). On the other hand, when \(B_{LO}\) increases, the upper bound in (39) will be decreased and \(\delta_{s}\) can satisfy the recovery requirement \(\delta_{2s} < \sqrt 2 - 1\) [28].

In summary, compared with the typical LO modulation type using the SFM, the OMP algorithm has the ability to reconstruct the multiple mono-pulse radar signals intercepted by the modulated sampling scheme using the PLFM LO under the completely frequency aliasing condition. Besides, increasing the PLFM bandwidth can increase the reconstruction number of the multiple mono-pulse radar signals.

### 4.3 Extended discussion

To further explain the improvement of the PLFM LO, some other modulation types of the LO are discussed. In terms of the LO modulation requirements [11,12,13], the LO modulation should have a bandwidth and be periodic. The LO modulation bandwidth is the information to recover the original signal frequency and its value is required to be distinguishable in different NZs. Moreover, because the sample pulse is generated at the ZCR time, the LO modulation should be periodic for the purpose of sample pulse generation implementation. In addition, the LO modulation is non-random to preserve the intercepted signal structure. Presently, the main non-random periodic modulation types are the phase modulation and the frequency modulation. Hence, the binary phase shift keying (BPSK) and the periodic nonlinear frequency modulation (PNLFM) are considered in this subsection.

When the LO modulation type is the BPSK, the ZCR time is calculated by the summation of several BPSK signals with different carrier frequencies and BPSK codes [29]. Essentially, this LO modulation can be regarded as a receiving structure using multiple channels with different codes like the MWC [7], whereas the merit of the modulated sampling scheme using single channel is vanished and it brings more receiving cost.

Then, the LO using the PNLFM is discussed, and the LO of the modulated sampling scheme using the cubic PNLFM can be expressed as

$$\theta_{PNLFM} \left( m \right) = 2\pi \left[ {\frac{{\mu_{1} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{2} }}{2} + \frac{{\mu_{2} \bmod \left( {mT_{ADC} ,\;T_{LO} } \right)^{3} }}{3}} \right]$$

(40)

where \(\mu_{1}\) and \(\mu_{2}\) are the modulation parameters of the quadratic and cubic terms, respectively. Other parameters in (40) are the same as those in (25).

The cubic term in (40) can be regarded as parts of Maclaurin series of \(\sin\) and \(\cos\) functions with the independent variable \(\mu_{2}\). As a result, the spectrum of the PNLFM is not as flat as the PLFM. Thus, according to the analysis in Sect. 4.1, the multiple signals reconstruction performance using the PNLFM LO will be deteriorated.

Furthermore, compared with the PNLFM, the ZCR time generation and the implementation of the PLFM are easier because of the linear frequency property. Moreover, since the PLFM in one period has an optimum kernel function—fractional Fourier transform (FRFT) [30] and it is a linear transform, the output of the modulated sampling scheme using PLFM LO can be processed easily. Overall, this paper chooses the PLFM as the LO modulation to improve the multiple mono-pulse reconstruction.