1 Introduction

Radar signal detection and parameter estimation technology is one of the most important technologies in modern electronic reconnaissance systems. To increase the survival capability of radar systems without compromising their effectiveness, a wide range of protection technologies has been applied to these systems, including multi-base systems, antenna designs with low side lobes, beam scan controls, power management and control, and low probability of intercept (LPI) radar signals. To reduce the probability of the signals being intercepted and identified, LPI radar [14] generally uses continuous-wave (CW) signals rather than short-pulse signals. At present, linear frequency-modulated continuous-wave (LFMCW) signals [4, 15], which feature high distance/high-speed resolution and provide accurate measurements and good anti-jamming and LPI characteristics, have been widely used in LPI radar and have found extensive applications in small- and microscale detection devices.

To date, various signal processing techniques have been investigated for the detection and estimation of LPI radar signals, including LFMCW signals. Both linear and nonlinear time-frequency analysis methods, including the Wigner–Ville distribution (WVD) [14], the quadrature mirror filters bank (QMFB) [2], the fractional Fourier transform (FRFT) [3, 13, 17], the chirplet transform [12], the Radon-ambiguity transform (RAT) [11, 16], and the Wigner–Hough transform (WHT) [7, 10], were explored for LFM or LFMCW signals. Among these methods, WHT- and FRFT-based techniques perform LFM signal detection and waveform parameter identification by searching for peaks in the corresponding parameter domains. It has been shown that these methods offer performance levels that are equivalent to that of the generalized likelihood ratio test (GLRT) detector and the maximum likelihood estimation (MLE) in the detection and estimation of single-pulse LFM signals [8] which means that they are the best available detection algorithms for single-pulse LFM signals. Theoretically, the ability of reconnaissance receivers to detect LFMCW signals within the observation time should be enhanced and accompanied by an increase in the number of modulation periods. However, unlike LFM signals, the instantaneous frequency of the LFMCW signal changes periodically within the observation time. The output signal-to-noise ratios (SNRs) of both the WHT and FRFT for signal detection cannot be improved by extension of the observation time. The output SNR gains of these methods are confined within a single modulation period. Therefore, WHT- and FRFT-based methods are not optimal detection algorithms for LFMCW signals.

In 2010, Geroleo proposed the periodic WHT for LFMCW signal detection [5, 6]. By taking the cyclic properties of LFMCW signals into account, the transform can realize periodic energy accumulation over the entire observation time and thus effectively improves the detection capability when compared with the conventional WHT [18]. The detection performance of the FRFT is comparable to that of the WHT for single-pulse LFM signals, and thus, a novel signal processing method based on coherent integration of the FRFT, called the periodic FRFT (PFRFT), is proposed in this paper. This method takes advantage of the good LFM signal detection capability of the FRFT and also uses the cyclic properties of LFMCW signals to achieve energy accumulation. The proposed PFRFT significantly outperforms the FRFT for LFMCW signals.

The rest of this paper is organized as follows. In Sect. 2, the PFRFT signal processing method, which is based on the FRFT detection principle for single-pulse LFM signals, is defined. In Sect. 3, the signal processing gain of the discrete PFRFT for LFMCW signals is mathematically analyzed. In Sect. 4, the modulus square detection statistic of the LFMCW signals is investigated in the presence of Gaussian white noise. In Sect. 5, an adaptive threshold detection and parameter estimation algorithm is proposed. Finally, in Sect. 6, the detection and estimation performance of this transform is studied through simulation and theoretical analysis.

2 Periodic Fractional Fourier Transform for LFMCW Signals

The FRFT is a classical detection and estimation method for LFM signals, and the FRFT of a signal x(t) is defined as [17]

$$\begin{aligned} \hbox {FRFT}[x(t)]=\int \nolimits _{-\infty }^\infty {x(t)K_\alpha (t,u)} \hbox {d}t, \end{aligned}$$
(1)

where the kernel function is

$$\begin{aligned} K_\alpha (t,u)=\left\{ { \begin{array}{ll} \sqrt{\frac{1-j\cot \alpha }{2\pi }}\cdot e^{j\left( \frac{1}{2}u^{2}\cot \alpha -ut\csc \alpha +\frac{1}{2}t^{2}\cot \alpha \right) }&{} \\ &{}\alpha \ne n\pi \\ \delta (t+u)\,&{}\alpha =(2n\pm 1)\pi \\ \delta (t-u)\,&{}\alpha =2n\pi \\ \end{array}} \right. , \end{aligned}$$
(2)

Strong energy aggregation can be achieved when using the FRFT for LFM signals because the kernel function matches the LFM signals exactly at an appropriate rotation angle \(\alpha \), and an impulse is formed in the fractional domain \((\alpha ,u)\). When compared with LFM signals, LFMCW signals can be regarded as LFM signals within only one modulation period. The LFMCW signals received at a reconnaissance receiver can be written as

$$\begin{aligned} s(t)=A\exp \left[ j\left( \varphi +2\pi f_{i}t+\pi g\,\hbox {mod}\left( t+\tau ,T\right) ^{2}\right) \right] , \end{aligned}$$
(3)

where A is the amplitude, \(\varphi \) is the random initial phase, \(f_{i}\) is the initial frequency, g is the chirp rate, T is the modulation period, \(\tau \) is the initial time offset, and \(\,\hbox {mod}(\cdot )\) represents the modulo operator.

Application of the FRFT to LFMCW signals shows that the kernel function of the FRFT does not match LFMCW signals over the entire observation time, but only in a single LFM period. Multiple impulses form in fractional domains at a rotation angle \(\alpha \), and the number of peaks is equal to the number of LFM periods contained in the LFMCW signals when the initial time offset \(\tau =0\). To match the LFMCW signals perfectly throughout the entire observation interval, the kernel function must be redesigned. By taking the time-frequency properties of LFMCW signals into account, the kernel function of the PFRFT can be redesigned as

$$\begin{aligned} K_{\tilde{\alpha }} \left( t,\tilde{\tau },\tilde{u},\tilde{T}\right) =\left\{ { \begin{array}{ll} \sqrt{\frac{1-j\cot \tilde{\alpha }}{2\pi }}\exp \left( j\frac{\,\hbox {mod}\left( t+\tilde{\tau },\tilde{T}\right) ^{2}+\tilde{u}^{2}}{2} \right. &{}\\ \left. \cot \tilde{\alpha }-j\left( t+\tilde{\tau }\right) \tilde{u}\csc \quad \tilde{\alpha }\right) &{}\tilde{\alpha }\ne n\pi \\ \sum \nolimits _{n=-\infty }^\infty {\delta \left( t+\tilde{\tau }-\tilde{u}-n\tilde{T}\right) } &{}\tilde{\alpha }=2n\pi \\ \sum \nolimits _{n=-\infty }^\infty {\delta \left( t+\tilde{\tau }+\tilde{u}-n\tilde{T}\right) }&{}\tilde{\alpha }=\left( 2n\pm 1\right) \pi \\ \end{array}} \right. , \end{aligned}$$
(4)

where \(\tilde{\tau }\) represents the initial time offset search parameter and \(\tilde{T}\) represents the modulation period search parameter. The parameter set of the PFRFT kernel function is \(\tilde{\Omega }=\left( \tilde{\alpha },\tilde{u},\tilde{T},\tilde{\tau }\right) \). When compared with the kernel function of the FRFT, the new kernel function has two additional parameters, \(\tilde{\tau }\) and \(\tilde{T}\), which are used to search for unknown parameters of the LFMCW signals. The PFRFT of a signal x(t) is defined as

$$\begin{aligned} \hbox {PFRFT}[x(t)]=\int \nolimits _{-\infty }^{+\infty } {x(t)} K_{\tilde{\alpha }} \left( t,\tilde{\tau },\tilde{u},\tilde{T}\right) \hbox {d}t. \end{aligned}$$
(5)

According to (4), the PFRFT of a signal can also be written as

$$\begin{aligned} \hbox {PFRFT}[x(t)]=\sum \limits _{n}{\int \nolimits _{0}^{\tilde{T}} {x\left( t+n\tilde{T}\right) } K\left( t,\tilde{\tau },\tilde{u}\right) \hbox {d}t} , \end{aligned}$$
(6)

where \(K\left( t,\tilde{\tau },\tilde{u}\right) =\exp \left( j\left( \left( t+\tilde{\tau }\right) ^{2}+\tilde{u}^{2}\right) \cot \tilde{\alpha }/2+j\left( t+\tilde{\tau }\right) \tilde{u}\csc \tilde{\alpha }\right) \). The above equation indicates that the PFRFT of a signal is a coherent accumulation with block \(\tilde{T}\) in the fractional domain.

Let \(A_\alpha =\sqrt{\left( 1-j\cot \tilde{\alpha }\right) /2\pi }\), and then, according to the definition above, the PFRFT of a LFMCW signal can be represented as

$$\begin{aligned} \hbox {PFRFT}\left( \tilde{\alpha },\tilde{u},\tilde{T},\tilde{\tau }\right)= & {} AA_\alpha \exp \left[ j\left( \varphi +\frac{\tilde{u}^{2}}{2}\cot \tilde{\alpha }\right) \right] \int \nolimits _{-\infty }^{+\infty } \exp \nonumber \\&\times \left[ j\left( \pi g\,\hbox {mod}\left( t+\tau ,T\right) ^{2}+\frac{\cot \tilde{\alpha }}{2}\,\hbox {mod}\left( t+\tilde{\tau },\tilde{T}\right) ^{2}\right) \right. \nonumber \\&\left. +j\left( 2\pi f_{i}t-\tilde{u}\right. \csc \tilde{\alpha }\left( t+\tilde{\tau }\right) \right] \hbox {d}t. \end{aligned}$$
(7)

Therefore, when the search parameters \((\tilde{T},\tilde{\tau })\) and \((T,\tau )\) are equal, the PFRFT at the rotation angle \(\tilde{\alpha }=-\hbox {arc}\cot 2\pi g\) is

$$\begin{aligned} \hbox {PFRFT}\left( \tilde{\alpha },\tilde{u},\tilde{T},\tilde{\tau }\right) =AA_\alpha \exp \left[ j\left( \varphi +\frac{\tilde{u}^{2}}{2}\cot \tilde{\alpha }-2\pi f_{i}\tilde{\tau }\right) \right] \delta \left( \tilde{u}-\frac{2\pi f_{i}}{\csc \tilde{\alpha }}\right) . \end{aligned}$$
(8)

Equation (8) demonstrates that the PFRFT of a LFMCW signal reaches its maximum modulus value when \(\tilde{\alpha }=-\hbox {arc}\cot 2\pi g\), \(\tilde{u}=2\pi f_{i}/\csc \tilde{\alpha }\), \(\tilde{T}=T\), and \(\tilde{\tau }=\tau \). This means that the PFRFT can attain the best energy aggregation in the fractional domain for LFMCW signals, similar to the performance of the FRFT for LFM signals. Therefore, the corresponding kernel function \(K_{\tilde{\alpha }} \left( t,\tilde{\tau },\tilde{u},\tilde{T}\right) \) is the matching function of the LFMCW signal. The PFRFT can obtain the modulation parameters of LFMCW signals by searching the peaks in the four-dimensional parameter domain.

As shown in Fig. 1, the PFRFT realizes piecewise accumulation with segmentation \(\tilde{T}\) in the fractional domain. The nature of piecewise coherent accumulation means that the phase of the kernel function of the PFRFT can change adaptively with the period \(\tilde{T}\), in a similar manner to the phases of LFMCW signals. The coherent accumulation process can be interpreted as follows: The transform first accumulates energy by using the FRFT in each LFM period and then coherently adds the FRFT energies from all periods (shown in Fig. 1 as the solid arrows plus dotted arrows). This interpretation is also represented by (6) above and by (11) below. The PFRFT for a LFMCW signal only has one maximum value in the four-dimensional parameter domain. Similarly, the 2D slice in \((\tilde{T},\tilde{\tau })=(T,\tau )\) shows only an impulse rather than the multiple peaks that appear in the FRFT with the energy aggregation in each LFM period (shown in Fig. 1 by the solid arrows).

Fig. 1
figure 1

Detection illustration of the PFRFT and FRFT

If we assume that the sampling frequency is \(f_{s}\) and the number of samples in the observation time is N, then the discrete form of the LFMCW signal can be expressed as:

$$\begin{aligned} s(n)=A\exp \left[ j\left( \varphi +2\pi nf_{i}/f_{s}+\pi g\,\hbox {mod}(n/f_{s}+\tau ,T)^{2}\right) \right] n=0,1,\ldots ,N-1. \end{aligned}$$
(9)

It can easily be shown that the discrete PFRFT of a signal achieves its maximum modulus when \(\tilde{\alpha }=-\hbox {arc}\cot 2\pi g\), \(\tilde{u}=2\pi f_{i}/\csc \tilde{\alpha }\), \(\tilde{T}=T\), and \(\tilde{\tau }=\tau \), and it satisfies

$$\begin{aligned} \left| \hbox {PFRFT} \right| _{\max }^{2}=N^{2}A^{2}\left| {A_\alpha } \right| ^{2}. \end{aligned}$$
(10)

The computational complexity of the existing fast discrete FRFT [1] is \(O(N\log _{2}N)\), and we can use it as the basis for construction of a fast algorithm for the PFRFT. After analysis of the PFRFT, the transform can be given as

$$\begin{aligned} P_{\alpha ,\tau ,T} [x(t)]=\sum \limits _{n}{e^{-ju(nT-\tau )\csc \alpha }F\left[ g_{T}(t-\tau +nT)x(t)\right] } , \end{aligned}$$
(11)

where \(P_{\alpha ,\tau ,T} [\cdot ]\) represents the PFRFT, \(F[\cdot ]\) represents the FRFT, and \(g_{T}(t)\) is a rectangular window function with length T. Therefore, the steps required to calculate the PFRFT using the FRFT are as follows. First, compensate the phase of each block of signals with the length T and then calculate the FRFT of the sum of all the blocks. Only one FRFT operation is needed when using this method.

3 Signal Processing Gain Analysis of PFRFT for LFMCW Signals

Assume that the observed signal is

$$\begin{aligned} x(n)=s(n)+w(n), \quad n=0,1,\ldots ,N-1, \end{aligned}$$
(12)

where s(n) represents the LFMCW signal and w(n) is the zero-mean, stationary, complex white Gaussian noise.

Detection of LFMCW signals using the PFRFT is determined by whether or not there is a modulus maximum in the fractional domain. In general, the FRFT uses the squared modulus test, and the test statistic of the PFRFT can thus be defined as

$$\begin{aligned} I_{x}=\left| {\hbox {PFRFT}_{x}} \right| ^{2}. \end{aligned}$$
(13)

Based on the definition of the SNR, the output SNR after discrete PFRFT processing can be written as

$$\begin{aligned} \hbox {SNR}_\mathrm{out} =\frac{\left| {\hbox {PFRFT}_{s}} \right| ^{4}}{\hbox {var}\left\{ \left| {\hbox {PFRFT}_{x}} \right| ^{2}\right\} }, \end{aligned}$$
(14)

where \(\left| {\hbox {PFRFT}_{s}} \right| \) is the peak of the modulus of signal s(n) after the PFRFT processing and \(\left| {\hbox {PFRFT}_{x}} \right| \) is the peak of the modulus of received signal x(n) after PFRFT processing. The variable \(\hbox {var}\left\{ \left| {\hbox {PFRFT}_{x}} \right| ^{2}\right\} \) can be written as

$$\begin{aligned} \hbox {var}\left\{ \left| {\hbox {PFRFT}_{x}} \right| ^{2}\right\} =E\left[ \left| {\hbox {PFRFT}_{x}} \right| ^{4}\right] -E^{2}\left[ \left| {\hbox {PFRFT}_{x}} \right| ^{2}\right] . \end{aligned}$$
(15)

Because of the non-correlation of the signal and the noise, it can easily be deduced that

$$\begin{aligned} E\left[ \left| {\hbox {PFRFT}_{x}} \right| ^{2}\right]= & {} \sum \limits _{n}{\sum \limits _{k}{E([s(n)+w(n)}} ]\left[ s^{*}(k)+w^{*}(k)\right] )K(n)K^{*}(k) \nonumber \\= & {} N^{2}A^{2}\left| {A_\alpha } \right| ^{2}+N\sigma _{w}^{^{2}} \left| {A_\alpha } \right| ^{2}. \end{aligned}$$
(16)

Based on the definition of the PFRFT, we then have

$$\begin{aligned} E\left[ \left| {\hbox {PFRFT}_{x}} \right| ^{4}\right]= & {} \sum \limits _{n,k,m,i} {E([s(n)+w(n)][s(k)+w(k)]} \nonumber \\&\quad \left[ s^{*}(m)+w^{*}(m)\right] \left[ s^{*}(i)+w^{*}(i)\right] )H_{n,k,m,i}, \end{aligned}$$
(17)

where \(H_{n,k,m,i} =K(n)K(k)K^{*}(m)K^{*}(i)\) and \(K(\cdot )\) is the abbreviated formula for the kernel function of the PFRFT. Because all the third-order moments of the zero-mean white Gaussian noise are equal to zero, we obtain:

$$\begin{aligned} E\left[ \left| {\hbox {PFRFT}_{x}} \right| ^{4}\right] =N^{4}A^{4}\left| {A_\alpha } \right| ^{4}+2N^{2}\sigma _{w}^{4}\left| {A_\alpha } \right| ^{4}+4N^{3}A^{2}\left| {A_\alpha } \right| ^{4}\sigma _{w}^{2}. \end{aligned}$$
(18)

By substituting (16) and (18) into (15), we obtain

$$\begin{aligned} \hbox {var}\left\{ \left| {\hbox {PFRFT}_{x}} \right| ^{2}\right\} =N^{2}\sigma _{w}^{4}\left| {A_\alpha } \right| ^{4}+2N^{3}A^{2}\sigma _{w}^{2}\left| {A_\alpha } \right| ^{4}. \end{aligned}$$
(19)

Therefore, the output SNR after PFRFT processing is

$$\begin{aligned} \hbox {SNR}_\mathrm{out}= & {} \frac{N^{4}A^{4}\left| {A_\alpha } \right| ^{4}}{N^{2}\sigma _{w}^{4}\left| {A_\alpha } \right| ^{4}+2N^{3}A^{2}\sigma _{w}^{2}\left| {A_\alpha } \right| ^{4}} \nonumber \\= & {} \frac{N^{2}\hbox {SNR}_\mathrm{in}^{2}}{2\hbox {NSNR}_\mathrm{in} +1}, \end{aligned}$$
(20)

where \(\hbox {SNR}_\mathrm{in} \) is the input SNR, which is defined as \({A^{2}}/{\sigma _{_{w}}^{2}}\).

Equation (20) indicates that \(\hbox {SNR}_\mathrm{out} \approx \hbox {NSNR}_\mathrm{in} /2\) when the input SNR is comparatively high (\(\hbox {SNR}_\mathrm{in} \gg 1)\). In this case, the sampling number N is used as the multiplier of the output SNR, which indicates that the discrete PFRFT can greatly improve the output SNR by increasing the number of samples and thus can improve the overall detection performance.

Next, we analyze the processing gain advantages of the PFRFT when compared with the FRFT. Suppose that the number of LFM periods in the observation time \(T_\mathrm{obs} \) is M; the number of samples in the observation time is then \(N=\hbox {MTf}_{s}\). Based on the above analysis, the signal processing gain of the PFRFT for LFMCW signals is

$$\begin{aligned} \hbox {SNR}_\mathrm{out}^\mathrm{PFRFT} =\frac{\left( \hbox {MTf}_{s}\right) ^{2}\hbox {SNR}_\mathrm{in}^{2}}{2\hbox {MTf}_{s}\hbox {SNR}_\mathrm{in} +1}. \end{aligned}$$
(21)

Because the FRFT processing is based on only one single LFM period, its processing gain is also limited to a single modulation period. In this case, the number of samples is \(T\cdot f_{s}\). Therefore, the signal processing gain of the FRFT for LFMCW signals is given by

$$\begin{aligned} \hbox {SNR}_\mathrm{out}^\mathrm{FRFT} =\frac{\left( Tf_{s}\right) ^{2}\hbox {SNR}_\mathrm{in}^{2}}{2Tf_{s}\hbox {SNR}_\mathrm{in} +1}. \end{aligned}$$
(22)

It can be found from (22) that the signal processing gain advantage of the PFRFT when compared with the FRFT is

$$\begin{aligned} Adv=\frac{\hbox {SNR}_\mathrm{out}^\mathrm{PFRFT}}{\hbox {SNR}_\mathrm{out}^\mathrm{FRFT}}=\frac{M^{2}\left( 2Tf_{s}\hbox {SNR}_\mathrm{in} +1\right) }{2\hbox {MTf}_{s}\hbox {SNR}_\mathrm{in} +1}. \end{aligned}$$
(23)

Equation (23) shows that the signal processing gain advantage \(\hbox {Adv}\approx M\) when \(\hbox {Tf}_{s}\hbox {SNR}_\mathrm{in} \gg 1\). This means that the signal processing gain advantage is mainly dependent on the number of LFM periods M in the observation period. As the number of periods M increases, then the processing gain advantage also increases. Figure 2 shows the processing gain advantage with different values of input SNR. It can be seen from the figure that the processing gain advantage increases when the input SNR decreases in certain intervals, which indicates that the PFRFT has a more obvious processing gain advantage with lower values of SNR than with higher values. However, it should be noted that the processing gain advantage is insignificant when the input SNR is less than a threshold value because the output SNR is too small for signals to be detected in such a case. In addition, it can be seen from Fig. 2 that \(\hbox {Adv}\approx 4\) when \(\hbox {SNR}_\mathrm{in} \in [-20,0]\). This indicates that a detection performance advantage of 6 dB can be attained using the PFRFT when compared with the FRFT in theory, and this will be verified in the simulation analysis of the detection and estimation process that follows.

In summary, the PFRFT can detect LFMCW signals by increasing the number of periods contained in the received signals, even under lower SNR conditions, but the FRFT cannot detect these signals.

Fig. 2
figure 2

Signal processing gain advantage of the PFRFT when compared with the FRFT; \(T{*}f_{s}=500\), \(M=4\)

4 Probability Density Function Analysis of the Test Statistic

To propose an adaptive threshold detection algorithm for LFMCW signals, it is first essential to know the probability density function (PDF) of the test statistic \(I_{x}\) based on the Neyman–Pearson criterion. Assume that x(n) is a noisy signal; the detection problem is then to decide whether hypothesis \(H_{0}\) or hypothesis \(H_{1}\) is true, and this can be written as

$$\begin{aligned} \left\{ {\begin{array}{l} H_{0}{: } x(n)=w(n) \\ H_{1}{: } x(n)=s(n)+w(n) \\ \end{array}} \right. , \end{aligned}$$
(24)

where w(n) is the zero-mean, stationary, complex white Gaussian noise, i.e., \(N(0,\sigma _\omega ^{2})\), and s(n) is the LFMCW signal, which is independent of the noise.

The test statistic is then defined as follows. First, we deduce the PDF of the test statistic denoted by \(I_{1}\) under the assumption that \(H_{1}\) is true. Define \(\mathbf{X}=[x(0),\ldots ,x(N-1)]^{T}\), \(\mathbf{K}=[k(0),\ldots ,k(N-1)]^{T}\) and \(\mathbf{Y}=\mathbf{X}^{T}\mathbf{K}\); consequently

$$\begin{aligned} I_{1}=\mathbf{YY}^{H}. \end{aligned}$$
(25)

It can be seen that \(\mathbf{X}\) is subject to a Gaussian distribution with mean \(\upmu \) and variance \(\mathbf{C}\), i.e., \(\mathbf{X}\sim N(\mu , \mathbf{C})\), where \(\upmu =[s(0),\ldots ,s(N-1)]^{T}\) and \(\mathbf{C}=\sigma ^{2}\mathbf{I}\). Therefore,

$$\begin{aligned} \mathbf{Y}\sim N\left( \mu ^{T}\mathbf{K},\left| \mathbf{K} \right| ^{2}\sigma ^{2}\right) . \end{aligned}$$
(26)

Based on the mathematical statistics given in [9], we know that if a real random variable \(x=\sum \nolimits _{i=1}^\nu {x_{i}^{2}} \), where all \(x_{i}\) are independent of each other and \(x_{i}\sim N(\mu _{i},1)\), then x corresponds to a non-central Chi-square distribution with degree of freedom \(\nu \) and non-central parameter \(\lambda =\sum \nolimits _{i=1}^\nu {\mu _{i}^{2}} \). Because \(\mathbf{Y}\) is a complex random variable with independent real and imaginary parts, the PDF can be normalized as

$$\begin{aligned} \frac{\mathbf{Y}}{\left| \mathbf{K} \right| \sigma /\sqrt{\hbox {2}}}\sim N\left( \frac{\mu ^{T}\mathbf{K}}{\left| \mathbf{K} \right| \sigma /\sqrt{\hbox {2}}},0.5\right) . \end{aligned}$$
(27)

Therefore,

$$\begin{aligned} \mathbf{YY}^{H}\sim \left| \mathbf{K} \right| ^{2}\frac{\sigma ^{2}}{\hbox {2}}\cdot \chi ^{\prime 2}_{2}\left( \lambda ^{{\prime }}\right) . \end{aligned}$$
(28)

This means that \(I_{1}\) is subject to a non-central Chi-square distribution with degree of freedom \(\nu =2\) and non-central parameter \(\lambda ^{{\prime }}=\hbox {2}\left| \mu \right| ^{2}/\sigma ^{2}\). According to the properties of the non-central Chi-square distribution, we obtain

$$\begin{aligned} E(I_{1})= & {} \left| \mathbf{K} \right| ^{2}\frac{\sigma _{w}^{^{2}}}{\hbox {2}}\left( \nu +\lambda ^{{\prime }}\right) =\left| \mathbf{K} \right| ^{2}\sigma _{w}^{^{2}} \left( \hbox {1}+\frac{\left| \mu \right| ^{2}}{\sigma _{w}^{^{2}}}\right) , \end{aligned}$$
(29)
$$\begin{aligned} \hbox {var}(I_{1})= & {} \left| \mathbf{K} \right| ^{4}\frac{\sigma _{w}^{4}}{\hbox {4}}\left( 2\nu +4\lambda ^{{\prime }}\right) =\left| \mathbf{K} \right| ^{4}\sigma _{w}^{^{4}} \left( \hbox {1}+\frac{\hbox {2}\left| \mu \right| ^{2}}{\sigma _{w}^{^{2}}}\right) . \end{aligned}$$
(30)

It is easy to verify that (30) is equivalent to (19) by simple variable substitution, which in turn verifies that the PDF deduction is correct here. Similarly, it can also be deduced that \(I_{x}\) corresponds to a central Chi-square distribution with degree of freedom \(\nu =2\) under the assumption that \(H_{0}\) is true, i.e.,

$$\begin{aligned} I_{0}\sim \left| \mathbf{K} \right| ^{2}\frac{\sigma ^{2}}{\hbox {2}}\cdot \upchi _{2}^{2}. \end{aligned}$$
(31)

Under the Neyman–Pearson criterion, the false alarm probability \(P_{f}\) and the detection probability \(P_{d}\) are determined by

$$\begin{aligned} P_{f}= & {} \int \nolimits _{T}^\infty {p(I_{x}\left| {H_{0}} \right. )} \hbox {d}I, \end{aligned}$$
(32)
$$\begin{aligned} P_{d}= & {} \int \nolimits _{T}^\infty {p(I_{x}\left| {H_{1}} \right. )} \hbox {d}I, \end{aligned}$$
(33)

where T is the adaptive threshold that is determined by (32) for a specific false alarm probability.

Fig. 3
figure 3

\(p(I_{x}|H_{0})\) and \(p(I_{x}|H_{1})\), where \(T{*}f_{s}=500\), and \(M=4\). a SNR\(=-\)20 dB. b SNR\(=-\)25 dB

The PDFs of LFMCW signals that contain four periods at different input SNRs under the assumptions that \(H_{0}\) and \(H_{1}\) are true are shown in Fig. 3, which illustrates the adaptive nature of the threshold. The figure shows that \(p(I_{x}\left| {H_{0}} \right. )\) is invariant under various input SNR values, whereas \(p(I_{x}\left| {H_{1}} \right. )\) shifts to the left at lower input SNRs because of the reduction in the non-central parameter \(\lambda ^{{\prime }}\). Therefore, the reduction in \(\lambda ^{{\prime }}\) at lower input SNR values leads in turn to reduction in the detection probability for a given false alarm probability.

In addition, the receiver operating characteristic (ROC) curves for LFMCW signals containing four LFM periods with different Gaussian noise levels are given in Fig. 4. The figure shows that the PFRFT has a good detection performance for LFMCW signals when \(SNR\ge -20\) dB.

Fig. 4
figure 4

ROC curves for the PFRFT-based detection, where \(T{*}f_{s}=500\), and \(M=4\)

5 Detection and Estimation Algorithm for LFMCW Signals

Based on the analysis above, we propose a new LFMCW signal detection algorithm based on the PFRFT. As shown in Fig. 5, the required steps are as follows:

  1. Step 1:

    set the search intervals for the parameters and compute the squared modulus of the PFRFT, \(\left| {\hbox {PFRFT}\left( \tilde{\alpha },\tilde{u},\tilde{T},\tilde{\tau }\right) } \right| ^{2}\), for the intercepted signals.

  2. Step 2:

    search the peak value \(\left| {\hbox {PFRFT}(\alpha ,u,T,\tau )} \right| _{\max }^{^{2}} \) in the domain \(\Omega \!=\!\left( \tilde{\alpha },\tilde{u},\tilde{T},\tilde{\tau }\right) \).

  3. Step 3:

    determine the detection threshold for a given false alarm probability using (31) and (32).

  4. Step 4:

    if the peak value surpasses the determined threshold and the peak position is \((\alpha ,u,T,\tau )\), then the modulation period T and the initial time offset \(\tau \) can be estimated directly, and the carrier frequency \(f_{i}\) and the frequency modulation rate g can be estimated using (34):

    $$\begin{aligned} \left. {\begin{array}{l} g=-\cot \alpha /2\pi \\ f_{i}=u\csc \alpha /2\pi \\ \end{array}} \right\} . \end{aligned}$$
    (34)

    To verify the detection and estimation effects of the proposed transform, we performed a simulation under the following conditions: the number of modulation periods of the LFMCW signals \(M=4\), the modulation period \(T=50\,\upmu \hbox {s}\), the initial frequency \(f_{i}=500\) kHz, the frequency modulation rate \(g=20\) MHz/ms, and the sampling rate \(f_{s}=10\) MHz, and 2000 sampling points occur within the observation time of 200 \(\upmu \hbox {s}\).

Fig. 5
figure 5

Adaptive threshold detection and estimation procedure based on the PFRFT for LFMCW signals

Fig. 6
figure 6

SPWVD of the LFMCW signal, where \(\hbox {SNR}=0\) dB, and \(\tau =0\)

Fig. 7
figure 7

The PFRFT for the LFMCW signal, where \(\hbox {SNR}=0\) dB, and \(\tau =T/5\)

Fig. 8
figure 8

The FRFT for the LFMCW signal, where \(\hbox {SNR}=0\) dB, and \(\tau =T/5\)

Figure 6 shows the smoothed pseudo-Wigner distribution (SPWVD) of the signal with \(\hbox {SNR}=0\) dB, in which we can see that a noisy LFMCW signal contains four concatenated LFM signals. Figure 7 shows the normalized modulus of the PFRFT for the LFMCW signal with initial time offset \(\tau =T/5\) (the 2D slice at the actual modulation period and the initial time offset), in which a single peak appears at the corresponding position that was determined using (34). Figure 8 shows the normalized modulus of the FRFT for the LFMCW signal with initial time offset \(\tau =0\). The figure shows that four peaks are present. The first peak that occurs can be used to estimate the initial frequency and the frequency modulation rate. Note that, under the influence of the initial time delay, the peak position does not appear at \((\tilde{\alpha },\tilde{u})\), which corresponds to the actual initial frequency \(f_{i}\) and the frequency modulation rate g when the initial time offset \(\tau \ne 0\), and is analogous to the WHT in [6]. Comparison of Fig. 7 with Fig. 8 shows that the output SNR of the PFRFT is significantly higher than that of the FRFT. This demonstrates that the PFRFT can coherently accumulate the LFM energy of each of the modulation periods and automatically eliminate the effects of the nonzero initial time offset.

6 Performance Analysis of Detection and Parameter Estimation

6.1 Detection Performance Analysis

Figure 9 shows the detection probabilities of the PFRFT and FRFT for LFMCW signals at various input SNRs. In this case, the false alarm probability is set to \(P_{f}=10^{-5}\), the theoretical detection probabilities of the PFRFT and FRFT are calculated directly using (32) and (33), and the simulation results for the detection probability can be obtained using the adaptive detection and estimation algorithm that was proposed in Sect. 5 by performing 1000 Monte Carlo tests. We can see that:

  1. (a)

    the theoretical and simulated detection curves for the PFRFT and the FRFT are very similar;

  2. (b)

    when compared with the FRFT, the PFRFT for LFMCW signals with four periods attains a processing gain advantage of approximately 6 dB. The input SNR requirements for 100 % detection probability using these two methods are \(-\)17 and \(-\)11 dB, respectively, and this conforms to the values from the theoretical analysis above.

Fig. 9
figure 9

Detection probabilities of LFMCW signals at various input SNRs

6.2 Relationship Between Parameter Estimations Based on PFRFT and MLE

It was determined in [6] that the maximum likelihood estimation (MLE) for LFMCW signals is

$$\begin{aligned} \left( f_{i},g,T,\tau \right) =\arg \mathop {\max }\limits _{\tilde{f},\tilde{g},\tilde{T},\tilde{\tau }} \left| {\sum \limits _{n}{x(n)\exp (j2\pi \tilde{f}_{i}n/f_{s}+}} \right. \left. {j\pi \tilde{g}\,\hbox {mod}(n/f_{s}+\tilde{\tau },T))} \right| ^{2}. \end{aligned}$$
(35)

The test statistic for the PFRFT can be deduced from the definition of the PFRFT, which is

$$\begin{aligned} I= & {} \left| {\hbox {PFRFT}_{x}} \right| ^{2} \nonumber \\= & {} \left| {\sum \limits _{n}{x(n)} \exp \left( j\frac{\,\hbox {mod}(n/f_{s}+\tilde{\tau },\tilde{T})^{2}}{2}\cot \tilde{\alpha }-j\tilde{u}\csc \tilde{\alpha }n/f_{s}\right) } \right| ^{2}. \end{aligned}$$
(36)

LFMCW signal parameter estimation based on the PFRFT given in this paper is performed by searching \(\left( \tilde{\alpha },\tilde{u},\tilde{T},\tilde{\tau }\right) \), which must satisfy the following condition:

$$\begin{aligned} \left( \alpha ,u,T,\tau \right) =\arg \mathop {\max }\limits _{\tilde{f},\tilde{g},\tilde{T},\tilde{\tau }} \left| \sum \limits _{n}{x(n)} \exp \left( j\frac{\,\hbox {mod}\left( n/f_{s}+\tilde{\tau },\tilde{T}\right) ^{2}}{2} \cot \tilde{\alpha } -j\tilde{u}\csc \tilde{\alpha }n/f_{s}\right) \right| ^{2}. \end{aligned}$$
(37)

Based on (34), we know that (35) is equivalent to (37). Therefore, the parameter estimation process based on the PFRFT is equivalent to performing MLE for the LFMCW signals.

6.3 Simulation of Parameter Estimation

To test the parameter estimation performance of the proposed transform, the normalized root mean square error (NRMSE) of the parameter \((f_{i},g,T,\tau )\), which is based on the PFRFT and FRFT, is analyzed in Fig. 10. The simulation parameters are the same as those used in the simulation above.

Fig. 10
figure 10

Estimation performances for LFMCW signals at various input SNRs. a PFRFT. b FRFT

Figure 10 shows that:

  1. (a)

    the PFRFT can estimate the initial frequency, the chirp rate, the time offset and the modulation period by searching the peak values in the four-dimensional parameter domain, whereas the FRFT cannot estimate the initial time offset.

  2. (b)

    the NRMSE of LFMCW signal parameter estimation based on the PFRFT and the FRFT is better than \(10^{-2}\) when the SNR is higher than \(-\)18 and \(-\)12 dB, respectively. If the SNR is reduced any further, then the parameter estimation performance drops dramatically. In general, the PFRFT has a parameter estimation performance advantage of about 6 dB when compared with the FRFT, which is consistent with the detection performance result described above.

7 Conclusion

Following the development of LPI radar technology, CW radar signals can be used to detect targets by multi-pulse accumulation through specific waveform design. Simultaneously, this technology can reduce the peak power of its signals to avoid being intercepted by radar reconnaissance receivers. Conventional signal analysis methods that are tailored for single-pulse processing divide the CW signals into multiple pulses and ignore the periodic modulation of the CW signals.

The PFRFT proposed in this paper can achieve multi-period coherent accumulation for LFMCW signals and outperforms the FRFT in terms of detection performance. In addition, the signal detection capability of the PFRFT improves proportionally with increases in the observation time and the number of available pulses. Therefore, the PFRFT provides asymptotically optimal estimation of LFMCW signals. When compared with the FRFT, the computational complexity of the PFRFT is rather high, but this complexity can be reduced with prior knowledge of the modulation period. For practical application of the PFRFT, faster transforms and more efficient peak search algorithms are required.

Only single-pulse LFMCW signal detection and estimation is considered in this paper. Our future work will focus on studies of the detection, estimation, and separation of multi-component LFMCW signals. In addition, the PFRFT can also be applied to other signal types (including the symmetrical triangular LFMCW signal and polyphase coded signals), and this offers further opportunities to extend our research.