# Sensing Fractional Power Spectrum of Nonstationary Signals with Coprime Filter Banks

Authors: Xiaomin Li; Huali Wang; Wanghan Lv; Haichao Luo
Journal: Complexity
 (2020)

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## Abstract

The coprime discrete Fourier transform (DFT) filter banks provide an effective scheme of spectral sensing for wide-sense stationary (WSS) signals in case that the sampling rate is far lower than the Nyquist sampling rate. And the resolution of the coprime DFT filter banks in the Fourier domain (FD) is2π/MN, where M and N are coprime. In this work, a digital fractional Fourier transform- (DFrFT-) based coprime filter banks spectrum sensing method is suggested. Our proposed method has the same sampling principle as the coprime DFT filter banks but has some advantages compared to the coprime DFT filter banks. Firstly, the fractional power spectrum of the chirp-stationary signals which are nonstationary in the FD can be sensed effectively by the coprime DFrFT filter banks because of the linear time-invariant (LTI) property of the proposed system in discrete-time Fourier domain (DTFD), while the coprime DFT filter banks can only sense the power spectrum of the WSS signals. Secondly, the coprime DFrFT filter banks improve the resolution from 2π/MN to 2πsinα/MN by combining the fractional filter banks theory with the coprime theory. Simulation results confirm the obtained analytical results.


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## Body

## 1. Introduction

Power spectrum plays an important role in signal processing such as array processing [1–4], spectral estimation [5–7], signal detection and estimation [8–12], and so on. As the traditional methods of sensing power spectrum are associated with wide-sense stationary (WSS) signals whose second-order expectations remain unchanged over time, the conventional methods are under Fourier analysis [13–19], and all these developments have led to satisfactory results.Compared with the WSS signals, the nonstationary signals are characterized by being oriented along an oblique axis in the time-frequency domain while the stationary signals are projected in a horizontal direction. And as a spot treatment in the frequency domain, the conventional Fourier analysis does not allow associating events in the time domain with the frequency domain, so Fourier analysis is not a useful tool for processing nonstationary signals.As a generalized form of the Fourier transform (FT), the fractional Fourier transform (FrFT) [20, 21] allows the signals in the time-frequency domain to be projected onto a line of arbitrary angle. In contrast to the standard Fourier analysis, the fractional Fourier analysis has a notable potential in the treatment of nonstationary signals due to an additional degree of freedom. The analysis of nonstationary signals by means of the fractional Fourier transformation has been reported [22–24]. And many signal processing theories in the Fourier domain (FD) have been extended to the fractional Fourier domain (FrFD) based on the relationship between FrFT and FT, such as the filters theory [25–29], the correlation theory [30–32], and the power spectral density theory [33, 34].Meanwhile, it is necessary to sense a wide band of power spectrum, leading to prohibitively high Nyquist rates which will exceed the specifications of best commercial analog-to-digital converters (ADCs). Therefore, designing a sub-Nyquist sampling scheme which can effectively sense the power spectrum of nonstationary signals is a challenging task. Coprime theory [35, 36], which is well suited for analyzing sparsely sampled signals in case that sampling rate is far lower than the Nyquist sampling rate, has gained increasing attention in recent years. Coprime theory can be well coupled with DFT filter banks theory in temporal domain to sense the power spectrum of WSS signals [35]. In [35], the WSS premise of the input signals ensures the decimate operation in cross-correlation that can be coupled with coprimality well (see (13)); therefore, when the inputs are nonstationary, i.e., the second-order expectations of the input sequence change over time [37–40], the decimate operation cannot be performed to introduce the coprime theory in, resulting in inefficient sensing of the power spectrum for nonstationary signals. Thus, the coprime DFT filter banks [35] is not suitable to process nonstationary signals.Recently, several research efforts based on the coprime DFT filter banks [35] are developed from different aspects. Huang et al. [41] applied two simple operations on each low-rate data channel of the original coprime DFT filter banks to form a 0.5 frequency resolution shifted analyzer and synthesized it and the original coprime DFT filter banks to remove all the annoying spurious peaks. However, many decision thresholds were needed to set in the modified analyzer, so the modified analyzer is not suitable for engineering applications. In an attempt to completely remove the spurious-peak side effect, Xiangdong Huang and Yuxuan [42] proposed a high-performance coprime spectral analysis method based on paralleled all-phase point-pass filtering; the proposed method can not only completely remove the spurious-peak side effect, but also can be implemented easily in the real-world application. Vaidyanathan and Pal [43] extended one-dimensional coprime DFT filter banks to multiple dimensions. Wu and Liang [44] used short-time Fourier transform to segment the nonstationary signals into piecewise stationary signals. However, the window function in short-time Fourier transform has the trade-off between temporal resolution and frequency resolution, resulting in low estimation accuracy for the power spectrum.This paper aims to develop a coprime digital fractional Fourier transform (DFrFT) filter banks theory for sensing the fractional power spectrum of the chirp stationary signals, which are nonstationary in the usual sense or in the FD [22, 33]. We describe how to construct a sub-Nyquist system by using two low-speed coprime sampling ADCs and two DFrFT filter banks and derive the mathematical expression of the output, which shows that the proposed system can effectively sense the fractional power spectrum of nonstationary signals without loss of any information.First, following the coprime premise and the property of the low-pass filter in discrete-time fractional Fourier domain (DTFrFD), we construct the coprime pair of DFrFT filter banks and prove the uniqueness of its passband in corresponding DTFD (with its argument scaled bycscα) (see Section 3.3 for details). Second, we prove that the coprime pair of DFrFT filter banks is linear time invariant (LTI) in DTFD based on the low-pass filter theory in DTFrFD, as the corresponding chirp modulated forms of the nonstationary signals are stationary, the decimate operation can be performed in the cross-correlation analysis between the outputs of the coprime pair of DFrFT filter banks, and the conventional power spectrum of the stationary signals can be acquired based on the uniqueness of the passband (see section 3.4 for details). Third, in terms of the polyphase representation of the filters in DTSFrFD, the decimator at the output of the filters can be moved to the left of the polyphase subfilters, resulting in an efficient coprime DFrFT filter banks (see Section 3.5 for details). Further, based on the fractional power spectrum theory, the final fractional power spectrum of the nonstationary signals can be acquired by the proposed coprime DFrFT filter banks. Besides, when the input is deterministic signal, the cross-correlations give the sensing of the spectrum in DTFrFD based on the fractional convolution theory (see Section 4.1 for details). The modified architecture not only senses the nonstationary signals but also has an accurate resolution 2πsinα/MN in DTFD.The outline of this paper is as follows. In Section2, the problem formulation is introduced based on the basic preliminaries. In Section 3, the coprime DFrFT filter banks for sensing the fractional power spectrum of nonstationary signals is elaborated. In Section 4, the special cases for deterministic signals are discussed, and the performance of the proposed method is analyzed in terms of the resolution and the fractional spectrum estimation.


## 2. Preliminaries

### 2.1. DTSFrFT

The FrFT is a generalization of the FT, which essentially allows the signals in the time-frequency domain to be projected onto a line of arbitrary angle [21, 22]. Simplified fractional Fourier transform (SFrFT) [45] has the same effect as FrFT of order α for filter design, but for digital implementation, it is simpler than the original FrFT. In digital signal processing systems, the signals used are digital signals sampled from the analog signals; their representations in simplified fractional Fourier domain (SFrFD) should be obtained by discrete-time simplified fractional Fourier transform (DTSFrFT). The DTSFrFT of xn is defined as follows [45]:(1)X¯αω=Aα∑n=−∞∞xnTej1/2nT2cotα−jωncscα,where ω=uT is the digital frequency of u, which is the variable in SFrFD, and T is the sample spacing in temporal domain.And we have(2)X¯αω=e−jω2/2T2cotαX˜αω,where X¯αω is the DTSFrFT of xn and X˜αω is the DTFrFT of xn.


### 2.2. The Low-Pass Filter in DTSFrFD

Supposehn,n=0,1,…,N−1 is a low-pass filter with length N, and its frequency response is H˜ω˜, i.e.,(3)H˜ω˜=1,ω˜<ω˜c,0,otherwise,where ω˜ is the digital frequency in discrete-time Fourier domain (DTFD) and ω˜c is the cutoff frequency. Let h0n=hne−j1/2n2Δt2cotα, and its fractional frequency response is H¯αω, H¯αω=Aα∑n=−∞∞h0nej1/2n2Δt2cotα−jωncscα=AαH˜ωcscα [26], i.e.,(4)H¯αω=AαH˜ω˜cscα,where ω is the digital frequency in DTSFrFD and H˜ω˜cscα is the frequency response of hn with its argument scaled by cscα. H¯αω is defined as the fractional frequency response of low-pass filter in DTSFrFD. Equation (4) shows that H¯αω, the low-pass filter in DTSFrFD, is equivalent to the discrete-time Fourier transform (DTFT) of hn, n=0,1,…,N−1 with its argument scaled by cscα. So, H¯αω is linear time invariant (LTI) in DTFD. And this relation is the instrument for the signal polyphase representation and filter bank theorems in DTSFrFD as will be discussed later.


### 2.3. Theαth Chirp-Stationary Signal

Definition 1.
(theαth chirp-stationary signal). For a nonstationary random signal xt, if the chirp modulated form x˜t=xtexpjt2/2cotα of xt is stationary, i.e., Rx˜x˜t1,t2=Rx˜x˜ττ=t1−t2, where Rx˜x˜• is the autocorrelation of x˜t, the signal xt is called the αth chirp-stationary signal.
And we have an important result forxt.Theorem 1.
(see [34]). The relationship between the αth fractional power spectrum of xt and the conventional power spectrum of x˜t is(5)P˜xucscα=11+jcotαPxxαu,where P˜xucscα is the conventional power spectrum of stationary signal x˜t and Pxxαu is the αth fractional power spectrum of nonstationary signal xt.


### 2.4. Convolution Theorem in DTSFrFD

For any discrete-time sequencesxnT and hnT, the discrete-time fractional Fourier convolution is as follows [32]:(6)ynT=xnT∗αhnT=Aαe−j1/2n2T2cotαxnTej1/2n2T2cotα∗hnTej1/2n2T2cotα,where ∗α denotes the fractional convolution operator. ∗ denotes the traditional convolution operator. T is the sample spacing in temporal domain. The discrete-time fractional Fourier transform (DTFrFT) of ynT can be denoted as follows:(7)Y˜αω=e−jω2/2T2cotαX˜αωH˜αω,where Y˜αω, X˜αω, and H˜αω are the DTFrFT of ynT, xnT, and hnT. And the DTSFrFT of ynT can be denoted as follows:(8)Y¯αω=X¯αωH¯αω,where Y¯αω, X¯αω, and H¯αω are the DTSFrFT of ynT, xnT, and hnT. Combining equations (6) and (8) illustrates that the discrete-time fractional Fourier convolution of xnT and hnT is corresponding to the product of X¯αω and H¯αω in DTSFrFD. Similarly, suppose ynT=e−j1/2n2T2cotαxnTej1/2n2T2cotα∗hnT.Then,(9)Y˜αω=2πX˜αωH¯αω,where Y˜αω, X˜αω are the DTFrFT of ynT, xnT and H¯αω is the DTSFrFT of hnT.


### 2.5. Original Coprime DFT Filter Banks

The coprime pair of DFT filter banks introduced in [35] which is illustrated in Figure 1 has a unique passband with the width Δω˜ and the center iΔω˜ in the Fourier domain (FD) (see [35] for details). As a result, any stimulus component near iΔω˜, i∈Z+, can be uniquely identified by the product filter banks.Figure 1
A coprime pair of DFT filter banks. Here,M and N are coprime.In Figure1, the cross-correlation between ukn and vln is(10)Euknvl∗n=∫02πSxxejω˜Gl∗ejω˜Hkejω˜dω˜2π,where Sxxejω˜ is the power spectrum of xn which is the common WSS input of the two filter banks and Glejω˜ and Hkejω˜ are the frequency response of filters Glz and Hkz, respectively, which are acquired through applying zooming and shifting operations on the transfer curve of ideal lowpass filters Hz (or Gz) using the shift parameter k (or l). In terms of the properties of the unique passband in the FD, equation (10) can be approximately written as(11)Euknvl∗n≈cSxxejωi˜,where c is a constant which depends on the exact passband shapes and on the coprime pair M and N and ω˜i=2πi/MN. Further, based on the wide-sense stationary (WSS) premise of input xn, time domain averages can be used to estimate correlations:(12)1L∑n=0L−1Euknvl∗n≈cSxxejω˜i,and since ukn and vln are outputs of LTI systems with a common WSS input xn, they are jointly WSS, so(13)1L∑n=0L−1EuknMNvl∗nMN≈cSxxejω˜i,where M and N are coprime. That is, we can simply multiply decimated versions of ukn and vln and perform the averaging. Then, based on the polyphase forms of filters Hkz and Glz , the decimator ↓MN can be combined with Hkz and Glz to realize the coprime decimate for input signal xn as shown in Figure 2. Besides, in Figure 2, xun and xvn can be, respectively, acquired by employing two analog-to-digital converters (ADCs) with the sampling rates fs1=1/MT, fs2=1/NT to discretize a continuous signal xt.Figure 2
Coprime DFT filter banks.


## 2.1. DTSFrFT

The FrFT is a generalization of the FT, which essentially allows the signals in the time-frequency domain to be projected onto a line of arbitrary angle [21, 22]. Simplified fractional Fourier transform (SFrFT) [45] has the same effect as FrFT of order α for filter design, but for digital implementation, it is simpler than the original FrFT. In digital signal processing systems, the signals used are digital signals sampled from the analog signals; their representations in simplified fractional Fourier domain (SFrFD) should be obtained by discrete-time simplified fractional Fourier transform (DTSFrFT). The DTSFrFT of xn is defined as follows [45]:(1)X¯αω=Aα∑n=−∞∞xnTej1/2nT2cotα−jωncscα,where ω=uT is the digital frequency of u, which is the variable in SFrFD, and T is the sample spacing in temporal domain.And we have(2)X¯αω=e−jω2/2T2cotαX˜αω,where X¯αω is the DTSFrFT of xn and X˜αω is the DTFrFT of xn.


## 2.2. The Low-Pass Filter in DTSFrFD

Supposehn,n=0,1,…,N−1 is a low-pass filter with length N, and its frequency response is H˜ω˜, i.e.,(3)H˜ω˜=1,ω˜<ω˜c,0,otherwise,where ω˜ is the digital frequency in discrete-time Fourier domain (DTFD) and ω˜c is the cutoff frequency. Let h0n=hne−j1/2n2Δt2cotα, and its fractional frequency response is H¯αω, H¯αω=Aα∑n=−∞∞h0nej1/2n2Δt2cotα−jωncscα=AαH˜ωcscα [26], i.e.,(4)H¯αω=AαH˜ω˜cscα,where ω is the digital frequency in DTSFrFD and H˜ω˜cscα is the frequency response of hn with its argument scaled by cscα. H¯αω is defined as the fractional frequency response of low-pass filter in DTSFrFD. Equation (4) shows that H¯αω, the low-pass filter in DTSFrFD, is equivalent to the discrete-time Fourier transform (DTFT) of hn, n=0,1,…,N−1 with its argument scaled by cscα. So, H¯αω is linear time invariant (LTI) in DTFD. And this relation is the instrument for the signal polyphase representation and filter bank theorems in DTSFrFD as will be discussed later.


## 2.3. Theαth Chirp-Stationary Signal

Definition 1.
(theαth chirp-stationary signal). For a nonstationary random signal xt, if the chirp modulated form x˜t=xtexpjt2/2cotα of xt is stationary, i.e., Rx˜x˜t1,t2=Rx˜x˜ττ=t1−t2, where Rx˜x˜• is the autocorrelation of x˜t, the signal xt is called the αth chirp-stationary signal.
And we have an important result forxt.Theorem 1.
(see [34]). The relationship between the αth fractional power spectrum of xt and the conventional power spectrum of x˜t is(5)P˜xucscα=11+jcotαPxxαu,where P˜xucscα is the conventional power spectrum of stationary signal x˜t and Pxxαu is the αth fractional power spectrum of nonstationary signal xt.


## 2.4. Convolution Theorem in DTSFrFD

For any discrete-time sequencesxnT and hnT, the discrete-time fractional Fourier convolution is as follows [32]:(6)ynT=xnT∗αhnT=Aαe−j1/2n2T2cotαxnTej1/2n2T2cotα∗hnTej1/2n2T2cotα,where ∗α denotes the fractional convolution operator. ∗ denotes the traditional convolution operator. T is the sample spacing in temporal domain. The discrete-time fractional Fourier transform (DTFrFT) of ynT can be denoted as follows:(7)Y˜αω=e−jω2/2T2cotαX˜αωH˜αω,where Y˜αω, X˜αω, and H˜αω are the DTFrFT of ynT, xnT, and hnT. And the DTSFrFT of ynT can be denoted as follows:(8)Y¯αω=X¯αωH¯αω,where Y¯αω, X¯αω, and H¯αω are the DTSFrFT of ynT, xnT, and hnT. Combining equations (6) and (8) illustrates that the discrete-time fractional Fourier convolution of xnT and hnT is corresponding to the product of X¯αω and H¯αω in DTSFrFD. Similarly, suppose ynT=e−j1/2n2T2cotαxnTej1/2n2T2cotα∗hnT.Then,(9)Y˜αω=2πX˜αωH¯αω,where Y˜αω, X˜αω are the DTFrFT of ynT, xnT and H¯αω is the DTSFrFT of hnT.


## 2.5. Original Coprime DFT Filter Banks

The coprime pair of DFT filter banks introduced in [35] which is illustrated in Figure 1 has a unique passband with the width Δω˜ and the center iΔω˜ in the Fourier domain (FD) (see [35] for details). As a result, any stimulus component near iΔω˜, i∈Z+, can be uniquely identified by the product filter banks.Figure 1
A coprime pair of DFT filter banks. Here,M and N are coprime.In Figure1, the cross-correlation between ukn and vln is(10)Euknvl∗n=∫02πSxxejω˜Gl∗ejω˜Hkejω˜dω˜2π,where Sxxejω˜ is the power spectrum of xn which is the common WSS input of the two filter banks and Glejω˜ and Hkejω˜ are the frequency response of filters Glz and Hkz, respectively, which are acquired through applying zooming and shifting operations on the transfer curve of ideal lowpass filters Hz (or Gz) using the shift parameter k (or l). In terms of the properties of the unique passband in the FD, equation (10) can be approximately written as(11)Euknvl∗n≈cSxxejωi˜,where c is a constant which depends on the exact passband shapes and on the coprime pair M and N and ω˜i=2πi/MN. Further, based on the wide-sense stationary (WSS) premise of input xn, time domain averages can be used to estimate correlations:(12)1L∑n=0L−1Euknvl∗n≈cSxxejω˜i,and since ukn and vln are outputs of LTI systems with a common WSS input xn, they are jointly WSS, so(13)1L∑n=0L−1EuknMNvl∗nMN≈cSxxejω˜i,where M and N are coprime. That is, we can simply multiply decimated versions of ukn and vln and perform the averaging. Then, based on the polyphase forms of filters Hkz and Glz , the decimator ↓MN can be combined with Hkz and Glz to realize the coprime decimate for input signal xn as shown in Figure 2. Besides, in Figure 2, xun and xvn can be, respectively, acquired by employing two analog-to-digital converters (ADCs) with the sampling rates fs1=1/MT, fs2=1/NT to discretize a continuous signal xt.Figure 2
Coprime DFT filter banks.


## 3. The Proposed Architecture

### 3.1. Problem Formulation

From the above description in Section2.5, the coprime DFT filter banks architecture [35] as shown in Figure 2 which can sense the power spectrum of stationary signals effectively is constructed based on two points: the passband uniqueness and WSS premise. The passband uniqueness ensures that any stimulus component near the passband can be uniquely identified by the product filter banks, i.e., the equivalence relationship in equation (11). And the WSS premise guarantees the decimated operation in equation (13) which can be combined with the polyphase representation of the filters to achieve the coprime sampling structure as shown in Figure 2. Accordingly, when the input signal xn is not WSS, the decimated operation cannot be performed so that the coprime theory cannot be coupled with DFT filtering bank theory to derive the architecture in Figure 2.


### 3.2. System Description

In this paper, we introduce the fractional Fourier analysis theory (the fractional filters theory and the fractional power spectrum theory) into the original coprime DFT filter banks to derive a modified architecture for sensing the fractional power spectrum of the chirp-stationary signals. First, we construct the coprime pair of DFrFT filter banks which is shown in Figure3 and prove the uniqueness of their passband in DTFD (see Section 3.3 for details). Second, we implement the cross-correlation between the outputs of the two filter banks; then based on the properties of the low-pass filter in DTSFrFD introduced in Section 2.2 and the uniqueness of the passband, the cross-correlation analysis can be converted from DTFrFD into DTFD. As the coprime pair of DSFrFT filter banks is LTI in corresponding DTFD (see Section 3.4 for details) and the input signal is stationary when it is modulated by a corresponding chirp signal, the decimated operation can be performed successfully. Third, in terms of the polyphase representation of the filters in DTSFrFD (see Section 3.5 for details), the decimator at the output of the filters can be moved to the left of the polyphase subfilters, resulting in an efficient polyphase implementation. Further, based on Theorem 4 (see Section 3.6 for details), the final fractional power spectrum of the nonstationary signals can be acquired. And the proposed coprime filter banks is shown in Figure 4.Figure 3
The coprime pair of DFrFT filter banks.Figure 4
The polyphase representation of the coprime pair of DFrFT filter banks.


### 3.3. The Coprime Pair of DFrFT Filter Banks and the Uniqueness of Their Passband

We specifically take advantage of the coprimality ofM and N to construct a coprime pair of DFrFT filter banks by combining an M band DFrFT filter bank and an N band DFrFT filter bank. And the proposal is illustrated in Figure 3. In this system, the two filter banks are composed of M and N subfilters H¯k,α2ω,k=0,…,M−1, G¯l,α2ω,l=0,…,N−1, respectively, which can be acquired through applying zooming and shifting operations on the transfer curve of the ideal low-pass filter H¯α2ω, G¯α2ω in DTSFrFD. And we will prove in the following that the product filter F¯lk,α2ω has a unique passband with the width Δω and the center iΔω.Assume thatH¯α2ω=Aα2H˜ωcscα2 is the ideal low-pass filter in DTSFrFD.(14)H¯α2ω=Aα2,ω˜≤πsinα2M,0,otherwise.Andω=uT is the digital frequency in DTSFrFD; ω˜ is the digital frequency in DTFD. Thus, H¯αNω has N passbands in DTFD, with each passband having width 2πsinα2/MN. H¯k,α2ω=H¯α2ejω˜Ncscα2UMk, k=0,…,M−1, where UM≜e−j2πsinα2/M, can be rewritten as(15)H¯k,α2ω=H¯α2ejω˜Ncscα2UMk=H¯α2ejω˜cscα2−2πksinα2/MNN,k=0,…,M−1.These are shifted version ofH¯α2Nω, in increments of 2πsinα2/MN, for each k in 0≤k≤M−1, H¯k,α2ω has N passbands, each passband has width 2πsinα2/MN, and the N passbands are centered at 2πnsinα2/N+2πksinα2/MN=Mn+k2πsinα2/MN,0≤n≤N−1.Similarly, assume thatG¯α2ω=Aα2G˜ω˜cscα2 is the ideal low-pass filter in DTSFrFD.(16)G¯α2ω=Aα2,ω˜≤πsinα2N,0,otherwise.Thus,G¯α2Mω has M passbands in DTFD, with each passband having width 2πsinα2/MN. G¯l,α2ω=G¯α2ejω˜Mcscα2UNl, l=0,…,N−1, where UN≜e−j2πsinα2/N, can be rewritten as(17)G¯l,α2ω=G¯α2ejω˜Mcscα2UNl=G¯α2ejω˜cscα2−2πlsinα2/MNM,l=0,…,N−1,these are shifted version of G¯α2Mω, in increments of 2πsinα2/MN, for each l in 0≤l≤N−1, G¯l,α2ω has M passbands, each passband has width 2πsinα2/MN, and the M passbands are centered at 2πmsinα2/M+2πlsinα2/MN=Nm+l2πsinα2/MN,0≤m≤M−1.For the two filter banksH¯k,α2ω, k=0,…,M−1, G¯l,α2ω, l=0,…,N−1, now consider the product of the kth and lth responses H¯k,α2ω and G¯l,α2ω:(18)F¯kl,α2ω=H¯k,α2ωG¯l,α2ω=H¯α2ejω˜Ncscα2UMkG¯α2ejω˜Mcscα2UNl,and there are two important theorems as follows:Theorem 2.
Given any integeri in 0≤l≤MN−1, there is a unique F¯kl,αω, i.e., a unique k,l pair, with passband centered at 2πisinα/MN.Theorem 3.
F¯kl,αω has precisely one passband, and it has width 2πsinα/MN. That is, there is only one overlapping band among the N bands of H¯k,αω and M bands of G¯l,αω. This overlapping passband is centered at 2πisinα/MN for some integer i in 0≤l≤MN−1. The proofs are in Appendix A and Appendix B.
Figure5 demonstrates how the filters work for M=4 and N=3. As shown in Figure 5, the filter H¯α2Nω has three passbands, and each has width 2πsinα2/12. There are four distinct shifted versions of H¯α2Nω, H¯k,α2ω=H¯α2ejω˜Ncscα2U4k,0≤k≤3. The filter G¯α2Mω has four passbands, and each has width 2πsinα2/12. There are three distinct shifted versions of G¯α2Mω, G¯l,α2ω=G¯α2ejω˜Mcscα2U3l,0≤l≤2. Each shifted version G¯l,α2ω overlaps with any shifted version H¯k,α2ω in precisely one passband. Therefore, the product F¯kl,α2ω is bandpass with a single passband, having width 2πsinα2/12. Furthermore, the twelve combinations of k,l produce twelve distinct filters F¯kl,α2ω, covering 0≤ω<2πsinα2. Therefore, we get the effect of an MN-band filter bank in DTFD by combining one M-band filter bank with one N-band filter bank.Figure 5
The filtersH¯α2Nω and G¯α2Mω , and their shifted versions, demonstrated for M=4 and N=3.


### 3.4. The Cross-Correlation Analysis

In this part, we will analyze the cross-correlation between the outputs of the two filter banksH¯k,α2ω and G¯l,α2ω in Figure 3 to acquire their corresponding decimated versions, which will be combined with the polyphase representation of DSFrFT filters in Section 3.5 to form the structure in Figure 4.Assume the two filter banksH¯k,α2ω and G¯l,α2ω in Figure 3 have a common α1th chirp-stationary random sequence xn as the input, with sample spacing T; let x˜n=xnejn2T2/2cotα1; as known in Definition 2 in Section 3.6, x˜n is the traditional WSS sequence with autocorrelation R˜xxa, R˜xxa=Ex˜kx˜k+a. The two filter banks H¯k,α2ω, k=0,…,M−1, and G¯l,α2ω, l=0,…,N−1, have M, N outputs, respectively. Now take the kth output u¯kn of the filter bank H¯k,α2ω and the lth output v¯ln of the filter bank G¯l,α2ω. The cross-correlation between u¯kn and v¯ln is(19)Eu¯knv¯l∗n=Ex˜n∗hknx˜∗n∗gl∗n=E∑i1∑i2x˜i1x˜∗i2hkn−i1gl∗n−i2=∑i1∑i2Ex˜i1x˜∗i2hkn−i1gl∗n−i2.Lettinga=i1−i2, we have Eu¯knv¯l∗n=∑i1∑i2R˜xxahkn−i1gl∗n−i2=∑aR˜xxagkl∗a, where R˜xxa is the autocorrelation of x˜n, gkla=∑nhkngl∗n+a. Since G¯lk,α2ω˜cscα2=H¯k,α2ω˜cscα2G¯l,α2∗ω˜cscα2,Parseval’s relation yields(20)Eu¯knv¯l∗n=12π∫02πP˜xω˜cscα2H¯k,α2ω˜cscα2⋅G¯l,α2∗ω˜cscα2dω˜,where P˜xω˜cscα2 is the power spectrum of x˜n. Since M and N are coprime, the product filter H¯k,α2ω˜cscα2G¯l,α2ω˜cscα2 has a single passband, with width 2πsinα2/MN, and center at 2πisinα2/MN, for some integer i in 0≤l≤MN−1. For large MN, the product H¯k,α2ω˜cscα2G¯l,α2ω˜cscα2 is a very narrow band filter, and so(21)Eu¯knv¯l∗n≈AP˜xω˜cscα2,and A is a constant which depends on the exact passband shapes, M and N. Assuming the WSS process of x˜n is ergodic, we use time domain averages to estimate correlations:(22)1L∑n=0L−1u¯knv¯l∗n≈AP˜xω˜cscα2,where L refers to the number of consumed snapshots. As can be seen from Section 2.2, for the low-pass filter H¯α2ω in DTSFrFD, there is H¯α2ω=Aα2H˜ω˜cscα2, where H˜ω˜cscα2 is the DTFT of hn. Thus, H¯α2ω, the low-pass filter in DTSFrFD, is equivalent to H˜ω˜cscα2, which is the low-pass filter in DTFD. Accordingly, H¯α2ω is LTI in DTFD. And filter banks H¯k,α2ω and G¯l,α2ω are LTI in DTFD. Therefore, u¯kn and v¯ln are outputs of LTI systems with a common WSS input x˜n; they are jointly WSS, so(23)1L∑n=0L−1u¯knKv¯l∗nK≈AP˜xω˜cscα2,for any integer K. That is, we can simply multiply decimated versions of u¯kn and v¯ln and perform the averaging.


### 3.5. The Polyphase Representation of DSFrFT Filters

SupposeE˜qMω˜cscα2 is the qth-type I polyphase component of hn which has been defined in Section 2.2,i.e.,(24)H˜ω˜cscα2=∑q=0M−1e−jqω˜cscα2E˜qMω˜cscα2,where E˜qMω˜cscα2=∑n=0+∞h0Mn+qe−jMω˜ncscα2.Substituting (24) into (4),(25)H¯0,α2ω=Aα2∑q=0M−1e−jqω˜cscα2E˜qMω˜cscα2.Equation (25) is the equivalence polyphase representation of H¯α2ω in DTFD. In most applications, the decimator is preceded by a low-pass digital filter called the decimation filter. And according to equation (25), the equivalent polyphase implementation of low-pass decimation filter in DTFD is illustrated in Figure 6.Figure 6
The equivalent polyphase implementation of low-pass decimation filter in DTFD.Accordingly, since(26)H¯α2ω−2πksinα2M=Aα2∑q=0M−1e−jNqω˜cscα2UM−kqE˜qMNω˜cscα2,(27)H¯α2Nω−2πksinα2MN=Aα2∑q=0M−1e−jNqω˜cscα2UM−kqE˜qMNω˜cscα2,0≤k≤M−1.So, we can represent the filtersH¯0,α2ω in appropriate polyphase forms:(28)H¯0,α2ω=Aα2∑q=0M−1e−jNqω˜cscα2UM−kqEq˜MNω˜cscα2,0≤k≤M−1.Figure7 shows the polyphase forms of the filters H¯0,α2ω in DTFD. Similarly, the polyphase forms of the filters G¯l,α2ω are(29)G¯l,α2ω=Aα2∑q=0M−1e−jMqω˜cscα2UM−lqF˜qMNω˜cscα2,0≤l≤N−1,and its polyphase forms in DTFD are shown in Figure 8. And we have the noble identities of decimator [46] which is shown in Figure 9. Such interconnections arise when we try to use the polyphase representation for decimation filters. And we can redraw Figure 6 as Figure 10. Accordingly, combining with the analysis in Section 3.4, when decimation ratio K=MN, an efficient polyphase implementation coprime DSFrFT filter banks under coprime sampling is shown in Figure 4. Besides, in Figure 4, x0n and x1n can be, respectively, acquired by employing two ADCs with the sampling rates fs1=1/MT, fs2=1/NT to discretize a continuous signal xtej1/2t2cotα1.Figure 7
The polyphase representation ofH¯0,α2ω.Figure 8
The polyphase representation ofG¯l,α2ω.Figure 9
The noble identities of decimators.Figure 10
Moving the polyphase components.


### 3.6. The Fractional Power Spectrum of Nonstationary Signals in DTSFrFD

In this section, we will prove that the output of the architecture in Figure4 is the fractional power spectrum of nonstationary signals xn. To make it more clear, we first define the chirp stationarity random sequence.Definition 2 (theαth chirp-stationary random sequence).
For nonstationary random sequencexn, if x˜n=xnejn2T2/2cotα is a stationary random sequence, we call xn the αth chirp-stationary random sequence.
And a useful conclusion associated with the fractional power spectrum ofxn is shown here for convenience.Theorem 4.
The relationship between theαth fractional power spectrum of the αth chirp-stationary random sequence xn and the conventional power spectrum of the stationary random sequence x˜n is(30)P˜xω˜cscα=11+jcotαPxxαω,in which P˜xω˜cscα is the conventional power spectrum of x˜n in DTFD and Pxxαω is the αth fractional power spectrum of xn in DTSFrFD. See Appendix C for the certification process.Therefore, as can be seen from equation (30), except for a constant coefficient, P˜xω˜cscα has the same spectral shape with Pxxαω. So, a traditional spectral estimate of the chirp-stationary sequence x˜n is transformed into the simplified fractional spectral estimation of the nonstationary sequence xn. So, equation (23) can be rewritten as(31)1L∑n=0L−1uknKvlnK≈A11+jcotα2Pxxα2ω,where A is a constant which depends on the exact passband shapes, M and N. That is, in DTSFrFD, the output of the system as shown in Figure 4 is the α2th fractional power spectrum of xn multiplied by a constant.


### 3.7. Choice of the Transform Ordersα1 and α2

As shown in Figure4, there are two transform orders α1 and α2 to be determined. The transform order α1 is used to convert nonstationary signals into stationary signals, while α2 decides the resolution of the proposed system which partially influences the sensing accuracy.The receivedα1th chirp-stationary signal is composed of a modulated signal and noise. Its model is given by(32)xn=sn+ηn,where xn and sn are received signal and modulated signal, respectively. ηn is assumed to be additive white Gaussian noise.The modulated signalsn is given by sn=Aexp−j2πfcn/fNYQ+ϕn+ϕ0, where A is the amplitude, and fc and ϕ0 are the carrier frequency and the initial uniformly distributed random phase, respectively. fNYQ is the Nyquist sampling rate. ϕn is the phase function, which determines the modulation type of the signal. For simplicity and without loss of generality, we assume that A is an invariant constant.In Definition1, we know that for the α1th chirp-stationary signal xt, its chirp modulated form x˜t=xtexpjt2/2cotα1 is stationary, i.e., Rx˜x˜t1,t2=Rx˜x˜ττ=t1−t2; therefore, the transform order α1=arccot−∂ϕn/∂n.In the fractional Fourier domain (FrFD), support of signals’ change is associated with the transform order, and there exists an optimum transform order in which the energy of signals is maximally concentrated [30–32]. When a signal is transformed by FrFT at its optimum order, transform kernel acts as a matched filter. Therefore, the transform order α2 is equal to the optimal transform which can maximize the absolute amplitude, i.e., α2=αopt.Furthermore, since the energy of Gaussian noise signals cannot effectively concentrate in FrFD, the Gaussian noise can be suppressed effectively.The optimal transformation orderαopt corresponding to maximum magnitude obtained from the FrFT is given by αopt=maxαFαt,u2,WhereFαt,u is the FrFT of the stationary signal x˜t=xtexpjt2/2cotα1 in the α-order FrFD. And αopt will directly affect the sensing performance; hence, the method of searching the optimal order is important. The traditional method to get the optimal order in the DTSFrFD is peak sweeping method [32, 33], which is an easy method to realize. And obviously, the search-based algorithms require numerous extra calculations and have the contradiction between estimation performance and complexity.In this section, we introduce the normalized second-order central moment (NSOCM) calculation method [47] to directly obtain the optimal transformation order αopt in DTSFrFD. Compared with the search-based algorithms, the NSOCM approach has higher computational efficiency because of its nonergodic search mechanism.According to [47], the optimal order αopt is normally given by(33)αopt=argminαTBPXαu,0≤α<π,where TBPXαu is the time-bandwidth product for Xαu, and the NSOCM pα of Xαu is defined by(34)pα=∫−∞∞u−mα2Xαu2du/∫−∞∞Xαu2du=ωα−mα2,where mα=∫−∞∞uXαu2du/∫−∞∞Xαu2du is the normalized first-order origin moment of Xαu and ωα=∫−∞∞u2Xαu2du/∫−∞∞Xαu2du is the normalized second-order origin moment of Xαu. The NSOCM pα,pα+1 represent the time width and frequency width of Xαu, respectively. Hence, equation (33) becomes(35)αopt=argminαpα⋅pα+1,0≤α<π2.The NSOCM product is given by(36)pα⋅pα+1=p0⋅p1+14p0−p12−4μ02sin2απ+12μ0p0−p1sin2απ,where μ0=ω0+ω1/2+m0m1−ω0.5 is the mixed second-order moment. Setting the first derivative of pα⋅pα+1 with respect to the order α equal to zero, we obtain(37)tan2αeπ=4μ0p0−p14μ02−p0−p12.For this case whereα is equal to the extreme point αe, the product pα⋅pα+1 reaches the extremum values. This result demonstrates that when α satisfies equation (38) as follows, the product pα⋅pα+1 reaches its minimum.(38)0≤α≤π2,α=14arctan4μ0p0−p14μ02−p0−p12+nπ4,μ0p0−p1sin2απ>0.Based on the theoretical analysis above, the calculation process of the optimal transform order can be summarized into the specific procedures as follows:(1)
Take the 0.5th- and 1st-order DTSFrFT of signal xn to obtain X0.5, X1(2)
Calculate the normalized first-order origin momentsm0 and m1, the normalized second-order origin moments ω0,ω0.5, and ω1, the mixed second-order moment μ0, and NSOCM p0 and p1 in accordance with the definition(3)
Obtain the optimal orderαopt of α by using (38) in the range of 0,π/2.And the flowchart is shown in Figure11.Figure 11
Flowchart of the optimal transformation order calculation.


## 3.1. Problem Formulation

From the above description in Section2.5, the coprime DFT filter banks architecture [35] as shown in Figure 2 which can sense the power spectrum of stationary signals effectively is constructed based on two points: the passband uniqueness and WSS premise. The passband uniqueness ensures that any stimulus component near the passband can be uniquely identified by the product filter banks, i.e., the equivalence relationship in equation (11). And the WSS premise guarantees the decimated operation in equation (13) which can be combined with the polyphase representation of the filters to achieve the coprime sampling structure as shown in Figure 2. Accordingly, when the input signal xn is not WSS, the decimated operation cannot be performed so that the coprime theory cannot be coupled with DFT filtering bank theory to derive the architecture in Figure 2.


## 3.2. System Description

In this paper, we introduce the fractional Fourier analysis theory (the fractional filters theory and the fractional power spectrum theory) into the original coprime DFT filter banks to derive a modified architecture for sensing the fractional power spectrum of the chirp-stationary signals. First, we construct the coprime pair of DFrFT filter banks which is shown in Figure3 and prove the uniqueness of their passband in DTFD (see Section 3.3 for details). Second, we implement the cross-correlation between the outputs of the two filter banks; then based on the properties of the low-pass filter in DTSFrFD introduced in Section 2.2 and the uniqueness of the passband, the cross-correlation analysis can be converted from DTFrFD into DTFD. As the coprime pair of DSFrFT filter banks is LTI in corresponding DTFD (see Section 3.4 for details) and the input signal is stationary when it is modulated by a corresponding chirp signal, the decimated operation can be performed successfully. Third, in terms of the polyphase representation of the filters in DTSFrFD (see Section 3.5 for details), the decimator at the output of the filters can be moved to the left of the polyphase subfilters, resulting in an efficient polyphase implementation. Further, based on Theorem 4 (see Section 3.6 for details), the final fractional power spectrum of the nonstationary signals can be acquired. And the proposed coprime filter banks is shown in Figure 4.Figure 3
The coprime pair of DFrFT filter banks.Figure 4
The polyphase representation of the coprime pair of DFrFT filter banks.


## 3.3. The Coprime Pair of DFrFT Filter Banks and the Uniqueness of Their Passband

We specifically take advantage of the coprimality ofM and N to construct a coprime pair of DFrFT filter banks by combining an M band DFrFT filter bank and an N band DFrFT filter bank. And the proposal is illustrated in Figure 3. In this system, the two filter banks are composed of M and N subfilters H¯k,α2ω,k=0,…,M−1, G¯l,α2ω,l=0,…,N−1, respectively, which can be acquired through applying zooming and shifting operations on the transfer curve of the ideal low-pass filter H¯α2ω, G¯α2ω in DTSFrFD. And we will prove in the following that the product filter F¯lk,α2ω has a unique passband with the width Δω and the center iΔω.Assume thatH¯α2ω=Aα2H˜ωcscα2 is the ideal low-pass filter in DTSFrFD.(14)H¯α2ω=Aα2,ω˜≤πsinα2M,0,otherwise.Andω=uT is the digital frequency in DTSFrFD; ω˜ is the digital frequency in DTFD. Thus, H¯αNω has N passbands in DTFD, with each passband having width 2πsinα2/MN. H¯k,α2ω=H¯α2ejω˜Ncscα2UMk, k=0,…,M−1, where UM≜e−j2πsinα2/M, can be rewritten as(15)H¯k,α2ω=H¯α2ejω˜Ncscα2UMk=H¯α2ejω˜cscα2−2πksinα2/MNN,k=0,…,M−1.These are shifted version ofH¯α2Nω, in increments of 2πsinα2/MN, for each k in 0≤k≤M−1, H¯k,α2ω has N passbands, each passband has width 2πsinα2/MN, and the N passbands are centered at 2πnsinα2/N+2πksinα2/MN=Mn+k2πsinα2/MN,0≤n≤N−1.Similarly, assume thatG¯α2ω=Aα2G˜ω˜cscα2 is the ideal low-pass filter in DTSFrFD.(16)G¯α2ω=Aα2,ω˜≤πsinα2N,0,otherwise.Thus,G¯α2Mω has M passbands in DTFD, with each passband having width 2πsinα2/MN. G¯l,α2ω=G¯α2ejω˜Mcscα2UNl, l=0,…,N−1, where UN≜e−j2πsinα2/N, can be rewritten as(17)G¯l,α2ω=G¯α2ejω˜Mcscα2UNl=G¯α2ejω˜cscα2−2πlsinα2/MNM,l=0,…,N−1,these are shifted version of G¯α2Mω, in increments of 2πsinα2/MN, for each l in 0≤l≤N−1, G¯l,α2ω has M passbands, each passband has width 2πsinα2/MN, and the M passbands are centered at 2πmsinα2/M+2πlsinα2/MN=Nm+l2πsinα2/MN,0≤m≤M−1.For the two filter banksH¯k,α2ω, k=0,…,M−1, G¯l,α2ω, l=0,…,N−1, now consider the product of the kth and lth responses H¯k,α2ω and G¯l,α2ω:(18)F¯kl,α2ω=H¯k,α2ωG¯l,α2ω=H¯α2ejω˜Ncscα2UMkG¯α2ejω˜Mcscα2UNl,and there are two important theorems as follows:Theorem 2.
Given any integeri in 0≤l≤MN−1, there is a unique F¯kl,αω, i.e., a unique k,l pair, with passband centered at 2πisinα/MN.Theorem 3.
F¯kl,αω has precisely one passband, and it has width 2πsinα/MN. That is, there is only one overlapping band among the N bands of H¯k,αω and M bands of G¯l,αω. This overlapping passband is centered at 2πisinα/MN for some integer i in 0≤l≤MN−1. The proofs are in Appendix A and Appendix B.
Figure5 demonstrates how the filters work for M=4 and N=3. As shown in Figure 5, the filter H¯α2Nω has three passbands, and each has width 2πsinα2/12. There are four distinct shifted versions of H¯α2Nω, H¯k,α2ω=H¯α2ejω˜Ncscα2U4k,0≤k≤3. The filter G¯α2Mω has four passbands, and each has width 2πsinα2/12. There are three distinct shifted versions of G¯α2Mω, G¯l,α2ω=G¯α2ejω˜Mcscα2U3l,0≤l≤2. Each shifted version G¯l,α2ω overlaps with any shifted version H¯k,α2ω in precisely one passband. Therefore, the product F¯kl,α2ω is bandpass with a single passband, having width 2πsinα2/12. Furthermore, the twelve combinations of k,l produce twelve distinct filters F¯kl,α2ω, covering 0≤ω<2πsinα2. Therefore, we get the effect of an MN-band filter bank in DTFD by combining one M-band filter bank with one N-band filter bank.Figure 5
The filtersH¯α2Nω and G¯α2Mω , and their shifted versions, demonstrated for M=4 and N=3.


## 3.4. The Cross-Correlation Analysis

In this part, we will analyze the cross-correlation between the outputs of the two filter banksH¯k,α2ω and G¯l,α2ω in Figure 3 to acquire their corresponding decimated versions, which will be combined with the polyphase representation of DSFrFT filters in Section 3.5 to form the structure in Figure 4.Assume the two filter banksH¯k,α2ω and G¯l,α2ω in Figure 3 have a common α1th chirp-stationary random sequence xn as the input, with sample spacing T; let x˜n=xnejn2T2/2cotα1; as known in Definition 2 in Section 3.6, x˜n is the traditional WSS sequence with autocorrelation R˜xxa, R˜xxa=Ex˜kx˜k+a. The two filter banks H¯k,α2ω, k=0,…,M−1, and G¯l,α2ω, l=0,…,N−1, have M, N outputs, respectively. Now take the kth output u¯kn of the filter bank H¯k,α2ω and the lth output v¯ln of the filter bank G¯l,α2ω. The cross-correlation between u¯kn and v¯ln is(19)Eu¯knv¯l∗n=Ex˜n∗hknx˜∗n∗gl∗n=E∑i1∑i2x˜i1x˜∗i2hkn−i1gl∗n−i2=∑i1∑i2Ex˜i1x˜∗i2hkn−i1gl∗n−i2.Lettinga=i1−i2, we have Eu¯knv¯l∗n=∑i1∑i2R˜xxahkn−i1gl∗n−i2=∑aR˜xxagkl∗a, where R˜xxa is the autocorrelation of x˜n, gkla=∑nhkngl∗n+a. Since G¯lk,α2ω˜cscα2=H¯k,α2ω˜cscα2G¯l,α2∗ω˜cscα2,Parseval’s relation yields(20)Eu¯knv¯l∗n=12π∫02πP˜xω˜cscα2H¯k,α2ω˜cscα2⋅G¯l,α2∗ω˜cscα2dω˜,where P˜xω˜cscα2 is the power spectrum of x˜n. Since M and N are coprime, the product filter H¯k,α2ω˜cscα2G¯l,α2ω˜cscα2 has a single passband, with width 2πsinα2/MN, and center at 2πisinα2/MN, for some integer i in 0≤l≤MN−1. For large MN, the product H¯k,α2ω˜cscα2G¯l,α2ω˜cscα2 is a very narrow band filter, and so(21)Eu¯knv¯l∗n≈AP˜xω˜cscα2,and A is a constant which depends on the exact passband shapes, M and N. Assuming the WSS process of x˜n is ergodic, we use time domain averages to estimate correlations:(22)1L∑n=0L−1u¯knv¯l∗n≈AP˜xω˜cscα2,where L refers to the number of consumed snapshots. As can be seen from Section 2.2, for the low-pass filter H¯α2ω in DTSFrFD, there is H¯α2ω=Aα2H˜ω˜cscα2, where H˜ω˜cscα2 is the DTFT of hn. Thus, H¯α2ω, the low-pass filter in DTSFrFD, is equivalent to H˜ω˜cscα2, which is the low-pass filter in DTFD. Accordingly, H¯α2ω is LTI in DTFD. And filter banks H¯k,α2ω and G¯l,α2ω are LTI in DTFD. Therefore, u¯kn and v¯ln are outputs of LTI systems with a common WSS input x˜n; they are jointly WSS, so(23)1L∑n=0L−1u¯knKv¯l∗nK≈AP˜xω˜cscα2,for any integer K. That is, we can simply multiply decimated versions of u¯kn and v¯ln and perform the averaging.


## 3.5. The Polyphase Representation of DSFrFT Filters

SupposeE˜qMω˜cscα2 is the qth-type I polyphase component of hn which has been defined in Section 2.2,i.e.,(24)H˜ω˜cscα2=∑q=0M−1e−jqω˜cscα2E˜qMω˜cscα2,where E˜qMω˜cscα2=∑n=0+∞h0Mn+qe−jMω˜ncscα2.Substituting (24) into (4),(25)H¯0,α2ω=Aα2∑q=0M−1e−jqω˜cscα2E˜qMω˜cscα2.Equation (25) is the equivalence polyphase representation of H¯α2ω in DTFD. In most applications, the decimator is preceded by a low-pass digital filter called the decimation filter. And according to equation (25), the equivalent polyphase implementation of low-pass decimation filter in DTFD is illustrated in Figure 6.Figure 6
The equivalent polyphase implementation of low-pass decimation filter in DTFD.Accordingly, since(26)H¯α2ω−2πksinα2M=Aα2∑q=0M−1e−jNqω˜cscα2UM−kqE˜qMNω˜cscα2,(27)H¯α2Nω−2πksinα2MN=Aα2∑q=0M−1e−jNqω˜cscα2UM−kqE˜qMNω˜cscα2,0≤k≤M−1.So, we can represent the filtersH¯0,α2ω in appropriate polyphase forms:(28)H¯0,α2ω=Aα2∑q=0M−1e−jNqω˜cscα2UM−kqEq˜MNω˜cscα2,0≤k≤M−1.Figure7 shows the polyphase forms of the filters H¯0,α2ω in DTFD. Similarly, the polyphase forms of the filters G¯l,α2ω are(29)G¯l,α2ω=Aα2∑q=0M−1e−jMqω˜cscα2UM−lqF˜qMNω˜cscα2,0≤l≤N−1,and its polyphase forms in DTFD are shown in Figure 8. And we have the noble identities of decimator [46] which is shown in Figure 9. Such interconnections arise when we try to use the polyphase representation for decimation filters. And we can redraw Figure 6 as Figure 10. Accordingly, combining with the analysis in Section 3.4, when decimation ratio K=MN, an efficient polyphase implementation coprime DSFrFT filter banks under coprime sampling is shown in Figure 4. Besides, in Figure 4, x0n and x1n can be, respectively, acquired by employing two ADCs with the sampling rates fs1=1/MT, fs2=1/NT to discretize a continuous signal xtej1/2t2cotα1.Figure 7
The polyphase representation ofH¯0,α2ω.Figure 8
The polyphase representation ofG¯l,α2ω.Figure 9
The noble identities of decimators.Figure 10
Moving the polyphase components.


## 3.6. The Fractional Power Spectrum of Nonstationary Signals in DTSFrFD

In this section, we will prove that the output of the architecture in Figure4 is the fractional power spectrum of nonstationary signals xn. To make it more clear, we first define the chirp stationarity random sequence.Definition 2 (theαth chirp-stationary random sequence).
For nonstationary random sequencexn, if x˜n=xnejn2T2/2cotα is a stationary random sequence, we call xn the αth chirp-stationary random sequence.
And a useful conclusion associated with the fractional power spectrum ofxn is shown here for convenience.Theorem 4.
The relationship between theαth fractional power spectrum of the αth chirp-stationary random sequence xn and the conventional power spectrum of the stationary random sequence x˜n is(30)P˜xω˜cscα=11+jcotαPxxαω,in which P˜xω˜cscα is the conventional power spectrum of x˜n in DTFD and Pxxαω is the αth fractional power spectrum of xn in DTSFrFD. See Appendix C for the certification process.Therefore, as can be seen from equation (30), except for a constant coefficient, P˜xω˜cscα has the same spectral shape with Pxxαω. So, a traditional spectral estimate of the chirp-stationary sequence x˜n is transformed into the simplified fractional spectral estimation of the nonstationary sequence xn. So, equation (23) can be rewritten as(31)1L∑n=0L−1uknKvlnK≈A11+jcotα2Pxxα2ω,where A is a constant which depends on the exact passband shapes, M and N. That is, in DTSFrFD, the output of the system as shown in Figure 4 is the α2th fractional power spectrum of xn multiplied by a constant.


## 3.7. Choice of the Transform Ordersα1 and α2

As shown in Figure4, there are two transform orders α1 and α2 to be determined. The transform order α1 is used to convert nonstationary signals into stationary signals, while α2 decides the resolution of the proposed system which partially influences the sensing accuracy.The receivedα1th chirp-stationary signal is composed of a modulated signal and noise. Its model is given by(32)xn=sn+ηn,where xn and sn are received signal and modulated signal, respectively. ηn is assumed to be additive white Gaussian noise.The modulated signalsn is given by sn=Aexp−j2πfcn/fNYQ+ϕn+ϕ0, where A is the amplitude, and fc and ϕ0 are the carrier frequency and the initial uniformly distributed random phase, respectively. fNYQ is the Nyquist sampling rate. ϕn is the phase function, which determines the modulation type of the signal. For simplicity and without loss of generality, we assume that A is an invariant constant.In Definition1, we know that for the α1th chirp-stationary signal xt, its chirp modulated form x˜t=xtexpjt2/2cotα1 is stationary, i.e., Rx˜x˜t1,t2=Rx˜x˜ττ=t1−t2; therefore, the transform order α1=arccot−∂ϕn/∂n.In the fractional Fourier domain (FrFD), support of signals’ change is associated with the transform order, and there exists an optimum transform order in which the energy of signals is maximally concentrated [30–32]. When a signal is transformed by FrFT at its optimum order, transform kernel acts as a matched filter. Therefore, the transform order α2 is equal to the optimal transform which can maximize the absolute amplitude, i.e., α2=αopt.Furthermore, since the energy of Gaussian noise signals cannot effectively concentrate in FrFD, the Gaussian noise can be suppressed effectively.The optimal transformation orderαopt corresponding to maximum magnitude obtained from the FrFT is given by αopt=maxαFαt,u2,WhereFαt,u is the FrFT of the stationary signal x˜t=xtexpjt2/2cotα1 in the α-order FrFD. And αopt will directly affect the sensing performance; hence, the method of searching the optimal order is important. The traditional method to get the optimal order in the DTSFrFD is peak sweeping method [32, 33], which is an easy method to realize. And obviously, the search-based algorithms require numerous extra calculations and have the contradiction between estimation performance and complexity.In this section, we introduce the normalized second-order central moment (NSOCM) calculation method [47] to directly obtain the optimal transformation order αopt in DTSFrFD. Compared with the search-based algorithms, the NSOCM approach has higher computational efficiency because of its nonergodic search mechanism.According to [47], the optimal order αopt is normally given by(33)αopt=argminαTBPXαu,0≤α<π,where TBPXαu is the time-bandwidth product for Xαu, and the NSOCM pα of Xαu is defined by(34)pα=∫−∞∞u−mα2Xαu2du/∫−∞∞Xαu2du=ωα−mα2,where mα=∫−∞∞uXαu2du/∫−∞∞Xαu2du is the normalized first-order origin moment of Xαu and ωα=∫−∞∞u2Xαu2du/∫−∞∞Xαu2du is the normalized second-order origin moment of Xαu. The NSOCM pα,pα+1 represent the time width and frequency width of Xαu, respectively. Hence, equation (33) becomes(35)αopt=argminαpα⋅pα+1,0≤α<π2.The NSOCM product is given by(36)pα⋅pα+1=p0⋅p1+14p0−p12−4μ02sin2απ+12μ0p0−p1sin2απ,where μ0=ω0+ω1/2+m0m1−ω0.5 is the mixed second-order moment. Setting the first derivative of pα⋅pα+1 with respect to the order α equal to zero, we obtain(37)tan2αeπ=4μ0p0−p14μ02−p0−p12.For this case whereα is equal to the extreme point αe, the product pα⋅pα+1 reaches the extremum values. This result demonstrates that when α satisfies equation (38) as follows, the product pα⋅pα+1 reaches its minimum.(38)0≤α≤π2,α=14arctan4μ0p0−p14μ02−p0−p12+nπ4,μ0p0−p1sin2απ>0.Based on the theoretical analysis above, the calculation process of the optimal transform order can be summarized into the specific procedures as follows:(1)
Take the 0.5th- and 1st-order DTSFrFT of signal xn to obtain X0.5, X1(2)
Calculate the normalized first-order origin momentsm0 and m1, the normalized second-order origin moments ω0,ω0.5, and ω1, the mixed second-order moment μ0, and NSOCM p0 and p1 in accordance with the definition(3)
Obtain the optimal orderαopt of α by using (38) in the range of 0,π/2.And the flowchart is shown in Figure11.Figure 11
Flowchart of the optimal transformation order calculation.


## 4. Discussion

### 4.1. Special Cases for Deterministic Signals

The above system as shown in Figure4 is suitable for the chirp-stationary random signals which have random characters. We have analyzed the output of the system and obtained the conclusion that the proposed system can sense the fractional power spectrum of the chirp-stationary random signals effectively. When the input signals of the system are deterministic signals which have compact support in the FrFD, the cross-correlation analysis is equivalent to the convolution theorem in the sense of FrFT. And special forms of the outputs occur.Whenxn is a deterministic signal, suppose that u¯kMNn is the output of filter bank H¯k,α2ω, k=0,…,M−1,v¯lMNn is the output of filter bank G¯l,α2ω, l=0,…,N−1, and the two filter banks have a common input xn. Then, the product of the multiband filters H¯k,α2ω and G¯l,α2ω is(39)F¯kl,α2ω=H¯k,α2ωG¯l,α2ω=H¯α2ejω˜Ncscα2UMkG¯α2ejω˜Mcscα2UNl,(40)u¯kMNn=xnej1/2n2T2cotα1∗h0n=xnej1/2n2T2cotα1∗h0ne−j1/2n2T2cotα2ej1/2n2T2cotα2.According to the fractional convolution theorem in Section2.4, the spectrum of xnej1/2n2T2cotα∗h0ne−j1/2n2T2cotα2 in FrFD is(41)2πX˜α2ωH¯α2ω,(42)v¯lMNn=xnej1/2n2T2cotα1∗g0n=xnej1/2n2T2cotα1∗g0ne−j1/2n2T2cotα2ej1/2n2T2cotα2.According to the convolution theorem in FrFD, the spectrum ofxnej1/2n2T2cotα1∗g0ne−j1/2n2T2cotα2 in FrFD is(43)2πX˜α2ωG¯α2ω.So, the final output of the system is(44)yn=u¯kMNn∗v¯1MNn,where ∗ denotes the traditional convolution operator.The spectrum ofyn in FrFD is(45)Y˜ω=e−jω2/2T2cotα22πX˜α2ωH¯α2ω2πX˜α2ωG¯α2ω.According to (2), there is(46)Y˜ω=e−jω2/2T2cotα22πX˜α2ωH¯α2ω2πX˜α2ωG¯α2ω=2πejω2/2T2cotα2X¯α2ω2H¯α2ωG¯α2ω,and since Y¯α2ω=e−jω2/2T2cotα2Y˜α2ω, we have(47)Y¯ω=2πX¯α2ω2H¯α2ωG¯α2ω,that is, in DTSFrFD, the system output which is shown in Figure 12 is the square of the fractional spectrum of deterministic signal xn. And this conclusion can be used to estimate the fractional spectrum X¯α2ω.Figure 12
Polyphase representation of the coprime DFrFT filter banks for deterministic signals.


### 4.2. Simulation

We demonstrate the simulations for three different cases: resolution performance, the sensing accuracy, and the influence of the orderα.


#### 4.2.1. Resolution Performance

In this experiment, we will examine the resolution performance of our proposed method. The traditional DFT coprime filter banks structure [35] and the conventional polyphase filter banks [48] are given for comparison. The original multiband signal is denoted by zt. And the noisy signal is zt+nt, where nt is white Gaussian noise. And zt is given by the following:(48)zt=∑i=12zit=∑i=12Eirectt−τisiejπKit2cos2πfit.The choices of the parameters are listed in Table1.Table 1
Choices of parameters for deterministic signals.
Meaningfs1GHzSampling frequencyK10.2×109The signal modulation rate ofz1tK20.18×109The signal modulation rate ofz2ts110μsSignal duration ofz1ts210μsSignal duration ofz2tf190MHzFrequency carrier ofz1tf2105MHzFrequency carrier ofz2tThe optimal orders [34] of z1t and z2t are αz1=arccot−K1=−5.0×10−9, αz2=arccot−K2=−5.6×10−9; z1t and z2t have the best energy concentration at orders αz1 and αz2, respectively. For the system shown in Figure 4, we set M=11 and N=7. And the sampling frequency is N=7. The individual filter banks could only have resolved separations 2πsinα2/11=0.5712sinα2 and 2πsinα2/N=0.8976sinα2, respectively. However, the coprime DFrFT filter banks could have resolved separations 2πsinα2/MN=0.0816sinα2, which is more effective than the individual filter banks. The output of the original system [35] is shown in Figure 13(a); the fractional spectrum estimation in the απ/2≈−9.26×10−9 FrFD is plotted in Figure 13(b). And it can be seen from Figure 13 that the outputs of the original system are overlapped in FD while the outputs of the proposed system are separated from each other in DTSFrFD with order απ/2≈−9.26×10−9.Figure 13
Output of the coprime DFT filter banks and the coprime DSFrFT filter banks.
(a)(b)Figure14 shows the resolution for different numbers of channels. In Figure 14, the number of channels (i.e., M+N) varies from 10 to 50 with a step of 10. And appropriate M and N which are coprime are selected corresponding to every number of channels to reach the maximum resolution. For example, when the number of channels is 10, there are three coprime pairs: (1, 9), (3, 7), and (5, 5), and we set M=5 and N=5 to get the maximum resolution. The maximum resolution of the proposed system is 2πsinα2/MN while the coprime DFT filter banks is 2π/MN. It is observed that in Figure 14, the proposed system has the best performance because the parameter sinα is introduced into the system by DTSFrFT as analyzed in Section 3.3. Figure 14 also shows that our method has better performance when the number of channels increases. Thus, in practical applications, the proposed system is more cost-effective than the traditional coprime DFT filter banks.Figure 14
Resolution vs. channel number.


#### 4.2.2. Spectrum Sensing Accuracy

In the second experiment, we test the spectrum sensing accuracy of our proposed coprime DFrFT filter banks, compared with traditional coprime DFT filter banks [35], spectrum reconstruction technique [49], and non-undersampling method [50].To evaluate the sensing accuracy of the proposed system as shown in Figure4, a chirp-stationary random signal xt with the initial uniformly distributed random phase φ is considered. And xt is given by the following:(49)xt=E·rectt−τsexpjπKt2+j2πft+jφ,where E is the amplitude of the signal which could be fixed. s is the time scale factor which determines the signal duration. K is the signal modulation rate. f is the frequency carrier. rect· is a rectangular time-window denoted by rectt=0, if t>1/2; else, rectt=1. The choices of the parameters for xt are listed in Table 2.Table 2
Choices of the parameters for a chirp-stationary random signal.
Meaningfs1GHzSampling frequencyK2×109The signal modulation rate ofxts10μsSignal duration ofxtf90MHzFrequency carrier ofxtWe specify two coprime integersM=11 and N=7, and based on the definition of αth chirp-stationary signal, we can calculate the fractional order α1=arccot−2Kπ=−8×10−10 and the optimal transformation order α2=−7×10−10. The SNR ranges from 0 dB to 20 dB with step size 2 dB. Here, we define the spectrum sensing accuracy as(50)ε=1N∑n=1Nfnl⌢−fnl2+fnu⌢−fnu2,where fnl⌢ and fnu⌢ are the estimated lower boundary and upper boundary in the jth Monte Carlo simulation, respectively. fnl and fnu are the true lower boundary and upper boundary. Here, 200 Monte Carlo simulations are used.The original input signalxt in the temporal domain is shown in Figure 15(a), and the outputs in the α2=−7×10−10 FrFD are plotted in Figure 15(b). From the results in Figure 15, the fractional powers spectrum of the αth chirp-stationary signal xt can be sensed effectively by the proposed method. And the simulation results are consistent with the analysis in Section 3.Figure 15
Input signal and its fractional power spectrum.
(a)(b)Figure16 shows the sensing accuracy for different SNRs. In Figure 16, non-undersampling method has the best performance because it uses all Nyquist samples in signal processing. However, the results do not preclude the usefulness of the proposed method because the sampling rate of our method is far lower than the Nyquist sampling rate which will greatly reduce the pressure of ADC. Figure 16 also shows that our method has better performance for sensing chirp-stationary random signal than the traditional DFT coprime filter banks. This is because the traditional DFT coprime filter banks is suitable for sensing the power spectrum of wide-sense stationary (WSS) signals, while our proposed system has the ability of sensing the power spectrum of nonstationary signals. The sensing accuracy of the spectrum reconstruction technique improves gradually with the increase in SNR. However, As plotted in Figure 16, our proposed coprime DFrFT filter banks outperform the spectrum reconstruction technique. The explanation for this phenomenon is that cross-interference during reconstruction is so severe that the sensing accuracy is greatly affected by it.Figure 16
Sensing accuracy vs. SNR.


#### 4.2.3. Simulation of Influence of the Orderα

According to the analysis in Section3.7, the αth-order SFrFT can be regarded as the projection on the rotated frequency axis u; consequently, the spectrum distribution of the signal directly depends on the transform order of the SFrFT. When the transform order α2 is equal to the optimal transformation order αopt , the energy of signals are concentrated maximally, and the fractional power spectrum is a pulse, which maximizes the sensing accuracy. Correspondingly, an inaccurate order may lead to changes in the spectral width. If the bandwidth of the real spectrum is wider than the bandwidth of the theoretical order, then the real maximum bandwidth will be bigger than the sampling rate. This will result in spectrum aliasing. The spectrum sensing accuracy will rapidly decrease.In the last experiments, we demonstrate the influence of the orderα2 on the spectrum sensing accuracy. The tested signal is the same as that in Section 4.2.2, and we show the balance between the sensing accuracy and the transform order α2 in the noise-free and noisy cases. Each simulation has 300 trials to ensure statistically stable results.As the optimal transformation order of the tested signal isαopt=−7×10−10 , we investigate the sensing accuracy under different α2 where α2 varies from −0.80×10−9 to −0.60×10−9 with a step of 0.01×10−9 , the SNR is 5,15 dB. It can be seen from Figure 17 that the closer to −0.70×10−9 the transform order α2 is, the smaller the estimation error is. Conversely, the closer to −0.80×10−9 or −0.60×10−9 , the larger the error is. Furthermore, the sensing accuracy in noise-free case is the best because that the signal is not polluted by the noise.Figure 17
The relationship between the sensing accuracy and transform orderα in the noisy case and noise-free case.Figure18 depicts the performance in terms of sensing accuracy with different SNRs and fractional order α2 for the proposed method. The SNR varies from 0 dB to 20 dB with a step of 2 dB. The fractional order α2 varies from −0.80×10−9 to −0.60×10−9 with a step of 0.01×10−9. It is observed that the sensing accuracy has the opposite trend as the SNR.Figure 18
Performance of sensing accuracy in 3D space.Furthermore, both Figures17 and 18 indicate that if the transform order α2 is close to the optimal order αopt, the sensing accuracy will be improved particularly fast. Therefore, the proposed system is sensitive to the transform order α2, especially when the value of α2 is near αopt. So, how to enhance the robustness of the system is a problem that needs to be further studied.


## 4.1. Special Cases for Deterministic Signals

The above system as shown in Figure4 is suitable for the chirp-stationary random signals which have random characters. We have analyzed the output of the system and obtained the conclusion that the proposed system can sense the fractional power spectrum of the chirp-stationary random signals effectively. When the input signals of the system are deterministic signals which have compact support in the FrFD, the cross-correlation analysis is equivalent to the convolution theorem in the sense of FrFT. And special forms of the outputs occur.Whenxn is a deterministic signal, suppose that u¯kMNn is the output of filter bank H¯k,α2ω, k=0,…,M−1,v¯lMNn is the output of filter bank G¯l,α2ω, l=0,…,N−1, and the two filter banks have a common input xn. Then, the product of the multiband filters H¯k,α2ω and G¯l,α2ω is(39)F¯kl,α2ω=H¯k,α2ωG¯l,α2ω=H¯α2ejω˜Ncscα2UMkG¯α2ejω˜Mcscα2UNl,(40)u¯kMNn=xnej1/2n2T2cotα1∗h0n=xnej1/2n2T2cotα1∗h0ne−j1/2n2T2cotα2ej1/2n2T2cotα2.According to the fractional convolution theorem in Section2.4, the spectrum of xnej1/2n2T2cotα∗h0ne−j1/2n2T2cotα2 in FrFD is(41)2πX˜α2ωH¯α2ω,(42)v¯lMNn=xnej1/2n2T2cotα1∗g0n=xnej1/2n2T2cotα1∗g0ne−j1/2n2T2cotα2ej1/2n2T2cotα2.According to the convolution theorem in FrFD, the spectrum ofxnej1/2n2T2cotα1∗g0ne−j1/2n2T2cotα2 in FrFD is(43)2πX˜α2ωG¯α2ω.So, the final output of the system is(44)yn=u¯kMNn∗v¯1MNn,where ∗ denotes the traditional convolution operator.The spectrum ofyn in FrFD is(45)Y˜ω=e−jω2/2T2cotα22πX˜α2ωH¯α2ω2πX˜α2ωG¯α2ω.According to (2), there is(46)Y˜ω=e−jω2/2T2cotα22πX˜α2ωH¯α2ω2πX˜α2ωG¯α2ω=2πejω2/2T2cotα2X¯α2ω2H¯α2ωG¯α2ω,and since Y¯α2ω=e−jω2/2T2cotα2Y˜α2ω, we have(47)Y¯ω=2πX¯α2ω2H¯α2ωG¯α2ω,that is, in DTSFrFD, the system output which is shown in Figure 12 is the square of the fractional spectrum of deterministic signal xn. And this conclusion can be used to estimate the fractional spectrum X¯α2ω.Figure 12
Polyphase representation of the coprime DFrFT filter banks for deterministic signals.


## 4.2. Simulation

We demonstrate the simulations for three different cases: resolution performance, the sensing accuracy, and the influence of the orderα.


### 4.2.1. Resolution Performance

In this experiment, we will examine the resolution performance of our proposed method. The traditional DFT coprime filter banks structure [35] and the conventional polyphase filter banks [48] are given for comparison. The original multiband signal is denoted by zt. And the noisy signal is zt+nt, where nt is white Gaussian noise. And zt is given by the following:(48)zt=∑i=12zit=∑i=12Eirectt−τisiejπKit2cos2πfit.The choices of the parameters are listed in Table1.Table 1
Choices of parameters for deterministic signals.
Meaningfs1GHzSampling frequencyK10.2×109The signal modulation rate ofz1tK20.18×109The signal modulation rate ofz2ts110μsSignal duration ofz1ts210μsSignal duration ofz2tf190MHzFrequency carrier ofz1tf2105MHzFrequency carrier ofz2tThe optimal orders [34] of z1t and z2t are αz1=arccot−K1=−5.0×10−9, αz2=arccot−K2=−5.6×10−9; z1t and z2t have the best energy concentration at orders αz1 and αz2, respectively. For the system shown in Figure 4, we set M=11 and N=7. And the sampling frequency is N=7. The individual filter banks could only have resolved separations 2πsinα2/11=0.5712sinα2 and 2πsinα2/N=0.8976sinα2, respectively. However, the coprime DFrFT filter banks could have resolved separations 2πsinα2/MN=0.0816sinα2, which is more effective than the individual filter banks. The output of the original system [35] is shown in Figure 13(a); the fractional spectrum estimation in the απ/2≈−9.26×10−9 FrFD is plotted in Figure 13(b). And it can be seen from Figure 13 that the outputs of the original system are overlapped in FD while the outputs of the proposed system are separated from each other in DTSFrFD with order απ/2≈−9.26×10−9.Figure 13
Output of the coprime DFT filter banks and the coprime DSFrFT filter banks.
(a)(b)Figure14 shows the resolution for different numbers of channels. In Figure 14, the number of channels (i.e., M+N) varies from 10 to 50 with a step of 10. And appropriate M and N which are coprime are selected corresponding to every number of channels to reach the maximum resolution. For example, when the number of channels is 10, there are three coprime pairs: (1, 9), (3, 7), and (5, 5), and we set M=5 and N=5 to get the maximum resolution. The maximum resolution of the proposed system is 2πsinα2/MN while the coprime DFT filter banks is 2π/MN. It is observed that in Figure 14, the proposed system has the best performance because the parameter sinα is introduced into the system by DTSFrFT as analyzed in Section 3.3. Figure 14 also shows that our method has better performance when the number of channels increases. Thus, in practical applications, the proposed system is more cost-effective than the traditional coprime DFT filter banks.Figure 14
Resolution vs. channel number.


### 4.2.2. Spectrum Sensing Accuracy

In the second experiment, we test the spectrum sensing accuracy of our proposed coprime DFrFT filter banks, compared with traditional coprime DFT filter banks [35], spectrum reconstruction technique [49], and non-undersampling method [50].To evaluate the sensing accuracy of the proposed system as shown in Figure4, a chirp-stationary random signal xt with the initial uniformly distributed random phase φ is considered. And xt is given by the following:(49)xt=E·rectt−τsexpjπKt2+j2πft+jφ,where E is the amplitude of the signal which could be fixed. s is the time scale factor which determines the signal duration. K is the signal modulation rate. f is the frequency carrier. rect· is a rectangular time-window denoted by rectt=0, if t>1/2; else, rectt=1. The choices of the parameters for xt are listed in Table 2.Table 2
Choices of the parameters for a chirp-stationary random signal.
Meaningfs1GHzSampling frequencyK2×109The signal modulation rate ofxts10μsSignal duration ofxtf90MHzFrequency carrier ofxtWe specify two coprime integersM=11 and N=7, and based on the definition of αth chirp-stationary signal, we can calculate the fractional order α1=arccot−2Kπ=−8×10−10 and the optimal transformation order α2=−7×10−10. The SNR ranges from 0 dB to 20 dB with step size 2 dB. Here, we define the spectrum sensing accuracy as(50)ε=1N∑n=1Nfnl⌢−fnl2+fnu⌢−fnu2,where fnl⌢ and fnu⌢ are the estimated lower boundary and upper boundary in the jth Monte Carlo simulation, respectively. fnl and fnu are the true lower boundary and upper boundary. Here, 200 Monte Carlo simulations are used.The original input signalxt in the temporal domain is shown in Figure 15(a), and the outputs in the α2=−7×10−10 FrFD are plotted in Figure 15(b). From the results in Figure 15, the fractional powers spectrum of the αth chirp-stationary signal xt can be sensed effectively by the proposed method. And the simulation results are consistent with the analysis in Section 3.Figure 15
Input signal and its fractional power spectrum.
(a)(b)Figure16 shows the sensing accuracy for different SNRs. In Figure 16, non-undersampling method has the best performance because it uses all Nyquist samples in signal processing. However, the results do not preclude the usefulness of the proposed method because the sampling rate of our method is far lower than the Nyquist sampling rate which will greatly reduce the pressure of ADC. Figure 16 also shows that our method has better performance for sensing chirp-stationary random signal than the traditional DFT coprime filter banks. This is because the traditional DFT coprime filter banks is suitable for sensing the power spectrum of wide-sense stationary (WSS) signals, while our proposed system has the ability of sensing the power spectrum of nonstationary signals. The sensing accuracy of the spectrum reconstruction technique improves gradually with the increase in SNR. However, As plotted in Figure 16, our proposed coprime DFrFT filter banks outperform the spectrum reconstruction technique. The explanation for this phenomenon is that cross-interference during reconstruction is so severe that the sensing accuracy is greatly affected by it.Figure 16
Sensing accuracy vs. SNR.


### 4.2.3. Simulation of Influence of the Orderα

According to the analysis in Section3.7, the αth-order SFrFT can be regarded as the projection on the rotated frequency axis u; consequently, the spectrum distribution of the signal directly depends on the transform order of the SFrFT. When the transform order α2 is equal to the optimal transformation order αopt , the energy of signals are concentrated maximally, and the fractional power spectrum is a pulse, which maximizes the sensing accuracy. Correspondingly, an inaccurate order may lead to changes in the spectral width. If the bandwidth of the real spectrum is wider than the bandwidth of the theoretical order, then the real maximum bandwidth will be bigger than the sampling rate. This will result in spectrum aliasing. The spectrum sensing accuracy will rapidly decrease.In the last experiments, we demonstrate the influence of the orderα2 on the spectrum sensing accuracy. The tested signal is the same as that in Section 4.2.2, and we show the balance between the sensing accuracy and the transform order α2 in the noise-free and noisy cases. Each simulation has 300 trials to ensure statistically stable results.As the optimal transformation order of the tested signal isαopt=−7×10−10 , we investigate the sensing accuracy under different α2 where α2 varies from −0.80×10−9 to −0.60×10−9 with a step of 0.01×10−9 , the SNR is 5,15 dB. It can be seen from Figure 17 that the closer to −0.70×10−9 the transform order α2 is, the smaller the estimation error is. Conversely, the closer to −0.80×10−9 or −0.60×10−9 , the larger the error is. Furthermore, the sensing accuracy in noise-free case is the best because that the signal is not polluted by the noise.Figure 17
The relationship between the sensing accuracy and transform orderα in the noisy case and noise-free case.Figure18 depicts the performance in terms of sensing accuracy with different SNRs and fractional order α2 for the proposed method. The SNR varies from 0 dB to 20 dB with a step of 2 dB. The fractional order α2 varies from −0.80×10−9 to −0.60×10−9 with a step of 0.01×10−9. It is observed that the sensing accuracy has the opposite trend as the SNR.Figure 18
Performance of sensing accuracy in 3D space.Furthermore, both Figures17 and 18 indicate that if the transform order α2 is close to the optimal order αopt, the sensing accuracy will be improved particularly fast. Therefore, the proposed system is sensitive to the transform order α2, especially when the value of α2 is near αopt. So, how to enhance the robustness of the system is a problem that needs to be further studied.


## 4.2.1. Resolution Performance

In this experiment, we will examine the resolution performance of our proposed method. The traditional DFT coprime filter banks structure [35] and the conventional polyphase filter banks [48] are given for comparison. The original multiband signal is denoted by zt. And the noisy signal is zt+nt, where nt is white Gaussian noise. And zt is given by the following:(48)zt=∑i=12zit=∑i=12Eirectt−τisiejπKit2cos2πfit.The choices of the parameters are listed in Table1.Table 1
Choices of parameters for deterministic signals.
Meaningfs1GHzSampling frequencyK10.2×109The signal modulation rate ofz1tK20.18×109The signal modulation rate ofz2ts110μsSignal duration ofz1ts210μsSignal duration ofz2tf190MHzFrequency carrier ofz1tf2105MHzFrequency carrier ofz2tThe optimal orders [34] of z1t and z2t are αz1=arccot−K1=−5.0×10−9, αz2=arccot−K2=−5.6×10−9; z1t and z2t have the best energy concentration at orders αz1 and αz2, respectively. For the system shown in Figure 4, we set M=11 and N=7. And the sampling frequency is N=7. The individual filter banks could only have resolved separations 2πsinα2/11=0.5712sinα2 and 2πsinα2/N=0.8976sinα2, respectively. However, the coprime DFrFT filter banks could have resolved separations 2πsinα2/MN=0.0816sinα2, which is more effective than the individual filter banks. The output of the original system [35] is shown in Figure 13(a); the fractional spectrum estimation in the απ/2≈−9.26×10−9 FrFD is plotted in Figure 13(b). And it can be seen from Figure 13 that the outputs of the original system are overlapped in FD while the outputs of the proposed system are separated from each other in DTSFrFD with order απ/2≈−9.26×10−9.Figure 13
Output of the coprime DFT filter banks and the coprime DSFrFT filter banks.
(a)(b)Figure14 shows the resolution for different numbers of channels. In Figure 14, the number of channels (i.e., M+N) varies from 10 to 50 with a step of 10. And appropriate M and N which are coprime are selected corresponding to every number of channels to reach the maximum resolution. For example, when the number of channels is 10, there are three coprime pairs: (1, 9), (3, 7), and (5, 5), and we set M=5 and N=5 to get the maximum resolution. The maximum resolution of the proposed system is 2πsinα2/MN while the coprime DFT filter banks is 2π/MN. It is observed that in Figure 14, the proposed system has the best performance because the parameter sinα is introduced into the system by DTSFrFT as analyzed in Section 3.3. Figure 14 also shows that our method has better performance when the number of channels increases. Thus, in practical applications, the proposed system is more cost-effective than the traditional coprime DFT filter banks.Figure 14
Resolution vs. channel number.


## 4.2.2. Spectrum Sensing Accuracy

In the second experiment, we test the spectrum sensing accuracy of our proposed coprime DFrFT filter banks, compared with traditional coprime DFT filter banks [35], spectrum reconstruction technique [49], and non-undersampling method [50].To evaluate the sensing accuracy of the proposed system as shown in Figure4, a chirp-stationary random signal xt with the initial uniformly distributed random phase φ is considered. And xt is given by the following:(49)xt=E·rectt−τsexpjπKt2+j2πft+jφ,where E is the amplitude of the signal which could be fixed. s is the time scale factor which determines the signal duration. K is the signal modulation rate. f is the frequency carrier. rect· is a rectangular time-window denoted by rectt=0, if t>1/2; else, rectt=1. The choices of the parameters for xt are listed in Table 2.Table 2
Choices of the parameters for a chirp-stationary random signal.
Meaningfs1GHzSampling frequencyK2×109The signal modulation rate ofxts10μsSignal duration ofxtf90MHzFrequency carrier ofxtWe specify two coprime integersM=11 and N=7, and based on the definition of αth chirp-stationary signal, we can calculate the fractional order α1=arccot−2Kπ=−8×10−10 and the optimal transformation order α2=−7×10−10. The SNR ranges from 0 dB to 20 dB with step size 2 dB. Here, we define the spectrum sensing accuracy as(50)ε=1N∑n=1Nfnl⌢−fnl2+fnu⌢−fnu2,where fnl⌢ and fnu⌢ are the estimated lower boundary and upper boundary in the jth Monte Carlo simulation, respectively. fnl and fnu are the true lower boundary and upper boundary. Here, 200 Monte Carlo simulations are used.The original input signalxt in the temporal domain is shown in Figure 15(a), and the outputs in the α2=−7×10−10 FrFD are plotted in Figure 15(b). From the results in Figure 15, the fractional powers spectrum of the αth chirp-stationary signal xt can be sensed effectively by the proposed method. And the simulation results are consistent with the analysis in Section 3.Figure 15
Input signal and its fractional power spectrum.
(a)(b)Figure16 shows the sensing accuracy for different SNRs. In Figure 16, non-undersampling method has the best performance because it uses all Nyquist samples in signal processing. However, the results do not preclude the usefulness of the proposed method because the sampling rate of our method is far lower than the Nyquist sampling rate which will greatly reduce the pressure of ADC. Figure 16 also shows that our method has better performance for sensing chirp-stationary random signal than the traditional DFT coprime filter banks. This is because the traditional DFT coprime filter banks is suitable for sensing the power spectrum of wide-sense stationary (WSS) signals, while our proposed system has the ability of sensing the power spectrum of nonstationary signals. The sensing accuracy of the spectrum reconstruction technique improves gradually with the increase in SNR. However, As plotted in Figure 16, our proposed coprime DFrFT filter banks outperform the spectrum reconstruction technique. The explanation for this phenomenon is that cross-interference during reconstruction is so severe that the sensing accuracy is greatly affected by it.Figure 16
Sensing accuracy vs. SNR.


## 4.2.3. Simulation of Influence of the Orderα

According to the analysis in Section3.7, the αth-order SFrFT can be regarded as the projection on the rotated frequency axis u; consequently, the spectrum distribution of the signal directly depends on the transform order of the SFrFT. When the transform order α2 is equal to the optimal transformation order αopt , the energy of signals are concentrated maximally, and the fractional power spectrum is a pulse, which maximizes the sensing accuracy. Correspondingly, an inaccurate order may lead to changes in the spectral width. If the bandwidth of the real spectrum is wider than the bandwidth of the theoretical order, then the real maximum bandwidth will be bigger than the sampling rate. This will result in spectrum aliasing. The spectrum sensing accuracy will rapidly decrease.In the last experiments, we demonstrate the influence of the orderα2 on the spectrum sensing accuracy. The tested signal is the same as that in Section 4.2.2, and we show the balance between the sensing accuracy and the transform order α2 in the noise-free and noisy cases. Each simulation has 300 trials to ensure statistically stable results.As the optimal transformation order of the tested signal isαopt=−7×10−10 , we investigate the sensing accuracy under different α2 where α2 varies from −0.80×10−9 to −0.60×10−9 with a step of 0.01×10−9 , the SNR is 5,15 dB. It can be seen from Figure 17 that the closer to −0.70×10−9 the transform order α2 is, the smaller the estimation error is. Conversely, the closer to −0.80×10−9 or −0.60×10−9 , the larger the error is. Furthermore, the sensing accuracy in noise-free case is the best because that the signal is not polluted by the noise.Figure 17
The relationship between the sensing accuracy and transform orderα in the noisy case and noise-free case.Figure18 depicts the performance in terms of sensing accuracy with different SNRs and fractional order α2 for the proposed method. The SNR varies from 0 dB to 20 dB with a step of 2 dB. The fractional order α2 varies from −0.80×10−9 to −0.60×10−9 with a step of 0.01×10−9. It is observed that the sensing accuracy has the opposite trend as the SNR.Figure 18
Performance of sensing accuracy in 3D space.Furthermore, both Figures17 and 18 indicate that if the transform order α2 is close to the optimal order αopt, the sensing accuracy will be improved particularly fast. Therefore, the proposed system is sensitive to the transform order α2, especially when the value of α2 is near αopt. So, how to enhance the robustness of the system is a problem that needs to be further studied.


## 5. Conclusion

This paper puts forward coprime DFrFT filter banks for sensing the fractional power spectrum of the nonstationary signals. We gave a deep analysis on the mechanism that the nonstationary signals cannot be effectively sensed by the original coprime DFT filter banks. By means of the relationship between FrFT theory and FT theory, the modified coprime DFrFT filter banks can sense the fractional power spectrum of the nonstationary signals precisely. Furthermore, the proposed coprime DFrFT filter banks obtain a better resolution scaled bysinα than the original coprime DFT filter banks as well. And the special case for deterministic signals is also discussed based on the convolution theorem in DTSFrFD. Simulation results have shown that our proposed method outperforms traditional coprime DFT filter banks in sensing the fractional power spectrum of nonstationary signals. The further work should address the problem of the robustness of the system to the transform order α2.



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*Source: 2608613-2020-07-17.xml*