Research Article | Open Access
Xiaomin Li, Huali Wang, Wanghan Lv, Haichao Luo, "Sensing Fractional Power Spectrum of Nonstationary Signals with Coprime Filter Banks", Complexity, vol. 2020, Article ID 2608613, 17 pages, 2020. https://doi.org/10.1155/2020/2608613
Sensing Fractional Power Spectrum of Nonstationary Signals with Coprime Filter Banks
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) is , where and 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 to by combining the fractional filter banks theory with the coprime theory. Simulation results confirm the obtained analytical results.
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 . In , 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  is not suitable to process nonstationary signals.
Recently, several research efforts based on the coprime DFT filter banks  are developed from different aspects. Huang et al.  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  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  extended one-dimensional coprime DFT filter banks to multiple dimensions. Wu and Liang  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 by ) (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 in DTFD.
The outline of this paper is as follows. In Section 2, 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.
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)  has the same effect as FrFT of order 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 is defined as follows :where is the digital frequency of , which is the variable in SFrFD, and is the sample spacing in temporal domain.
And we havewhere is the DTSFrFT of and is the DTFrFT of .
2.2. The Low-Pass Filter in DTSFrFD
Suppose is a low-pass filter with length , and its frequency response is , i.e.,where is the digital frequency in discrete-time Fourier domain (DTFD) and is the cutoff frequency. Let , and its fractional frequency response is , , i.e.,where is the digital frequency in DTSFrFD and is the frequency response of with its argument scaled by . is defined as the fractional frequency response of low-pass filter in DTSFrFD. Equation (4) shows that , the low-pass filter in DTSFrFD, is equivalent to the discrete-time Fourier transform (DTFT) of , with its argument scaled by . So, 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. The Chirp-Stationary Signal
Definition 1. (the chirp-stationary signal). For a nonstationary random signal , if the chirp modulated form of is stationary, i.e., , where is the autocorrelation of , the signal is called the chirp-stationary signal.
And we have an important result for .
Theorem 1. (see ). The relationship between the fractional power spectrum of and the conventional power spectrum of iswhere is the conventional power spectrum of stationary signal and is the fractional power spectrum of nonstationary signal .
2.4. Convolution Theorem in DTSFrFD
For any discrete-time sequences and , the discrete-time fractional Fourier convolution is as follows :where denotes the fractional convolution operator. denotes the traditional convolution operator. is the sample spacing in temporal domain. The discrete-time fractional Fourier transform (DTFrFT) of can be denoted as follows:where , , and are the DTFrFT of , , and . And the DTSFrFT of can be denoted as follows:where , , and are the DTSFrFT of , , and . Combining equations (6) and (8) illustrates that the discrete-time fractional Fourier convolution of and is corresponding to the product of and in DTSFrFD. Similarly, suppose .
Then,where , are the DTFrFT of , and is the DTSFrFT of .
2.5. Original Coprime DFT Filter Banks
The coprime pair of DFT filter banks introduced in  which is illustrated in Figure 1 has a unique passband with the width and the center in the Fourier domain (FD) (see  for details). As a result, any stimulus component near , , can be uniquely identified by the product filter banks.
In Figure 1, the cross-correlation between and iswhere is the power spectrum of which is the common WSS input of the two filter banks and and are the frequency response of filters and , respectively, which are acquired through applying zooming and shifting operations on the transfer curve of ideal lowpass filters (or ) using the shift parameter (or ). In terms of the properties of the unique passband in the FD, equation (10) can be approximately written aswhere is a constant which depends on the exact passband shapes and on the coprime pair and and . Further, based on the wide-sense stationary (WSS) premise of input , time domain averages can be used to estimate correlations:and since and are outputs of LTI systems with a common WSS input , they are jointly WSS, sowhere and are coprime. That is, we can simply multiply decimated versions of and and perform the averaging. Then, based on the polyphase forms of filters and , the decimator can be combined with and to realize the coprime decimate for input signal as shown in Figure 2. Besides, in Figure 2, and can be, respectively, acquired by employing two analog-to-digital converters (ADCs) with the sampling rates , to discretize a continuous signal .
3. The Proposed Architecture
3.1. Problem Formulation
From the above description in Section 2.5, the coprime DFT filter banks architecture  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 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 Figure 3 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.
3.3. The Coprime Pair of DFrFT Filter Banks and the Uniqueness of Their Passband
We specifically take advantage of the coprimality of and to construct a coprime pair of DFrFT filter banks by combining an band DFrFT filter bank and an band DFrFT filter bank. And the proposal is illustrated in Figure 3. In this system, the two filter banks are composed of and subfilters , , respectively, which can be acquired through applying zooming and shifting operations on the transfer curve of the ideal low-pass filter , in DTSFrFD. And we will prove in the following that the product filter has a unique passband with the width and the center .
Assume that is the ideal low-pass filter in DTSFrFD.
And is the digital frequency in DTSFrFD; is the digital frequency in DTFD. Thus, has passbands in DTFD, with each passband having width . , , where , can be rewritten as
These are shifted version of , in increments of , for each in , has passbands, each passband has width , and the passbands are centered at .
Similarly, assume that is the ideal low-pass filter in DTSFrFD.
Thus, has passbands in DTFD, with each passband having width . , , where , can be rewritten asthese are shifted version of , in increments of , for each in , has passbands, each passband has width , and the passbands are centered at .
For the two filter banks , , , , now consider the product of the and responses and :and there are two important theorems as follows:
Theorem 2. Given any integer in , there is a unique , i.e., a unique pair, with passband centered at .
Theorem 3. has precisely one passband, and it has width . That is, there is only one overlapping band among the bands of and bands of . This overlapping passband is centered at for some integer in . The proofs are in Appendix A and Appendix B.
Figure 5 demonstrates how the filters work for and . As shown in Figure 5, the filter has three passbands, and each has width . There are four distinct shifted versions of , . The filter has four passbands, and each has width . There are three distinct shifted versions of , . Each shifted version overlaps with any shifted version in precisely one passband. Therefore, the product is bandpass with a single passband, having width . Furthermore, the twelve combinations of produce twelve distinct filters , covering . Therefore, we get the effect of an -band filter bank in DTFD by combining one -band filter bank with one -band filter bank.
3.4. The Cross-Correlation Analysis
In this part, we will analyze the cross-correlation between the outputs of the two filter banks and 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 banks and in Figure 3 have a common chirp-stationary random sequence as the input, with sample spacing ; let ; as known in Definition 2 in Section 3.6, is the traditional WSS sequence with autocorrelation , . The two filter banks , , and , , have , outputs, respectively. Now take the output of the filter bank and the output of the filter bank . The cross-correlation between and is
Letting , we have , where is the autocorrelation of , . Since ,
Parseval’s relation yieldswhere is the power spectrum of . Since and are coprime, the product filter has a single passband, with width , and center at , for some integer in . For large , the product is a very narrow band filter, and soand is a constant which depends on the exact passband shapes, and . Assuming the WSS process of is ergodic, we use time domain averages to estimate correlations:where refers to the number of consumed snapshots. As can be seen from Section 2.2, for the low-pass filter in DTSFrFD, there is , where is the DTFT of . Thus, , the low-pass filter in DTSFrFD, is equivalent to , which is the low-pass filter in DTFD. Accordingly, is LTI in DTFD. And filter banks and are LTI in DTFD. Therefore, and are outputs of LTI systems with a common WSS input ; they are jointly WSS, sofor any integer . That is, we can simply multiply decimated versions of and and perform the averaging.
3.5. The Polyphase Representation of DSFrFT Filters
Suppose is the -type polyphase component of which has been defined in Section 2.2,i.e.,where .
Equation (25) is the equivalence polyphase representation of 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.
So, we can represent the filters in appropriate polyphase forms:
Figure 7 shows the polyphase forms of the filters in DTFD. Similarly, the polyphase forms of the filters areand its polyphase forms in DTFD are shown in Figure 8. And we have the noble identities of decimator  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 , an efficient polyphase implementation coprime DSFrFT filter banks under coprime sampling is shown in Figure 4. Besides, in Figure 4, and can be, respectively, acquired by employing two ADCs with the sampling rates , to discretize a continuous signal .
3.6. The Fractional Power Spectrum of Nonstationary Signals in DTSFrFD
In this section, we will prove that the output of the architecture in Figure 4 is the fractional power spectrum of nonstationary signals . To make it more clear, we first define the chirp stationarity random sequence.
Definition 2 (the chirp-stationary random sequence). For nonstationary random sequence , if is a stationary random sequence, we call the chirp-stationary random sequence.
And a useful conclusion associated with the fractional power spectrum of is shown here for convenience.
Theorem 4. The relationship between the fractional power spectrum of the chirp-stationary random sequence and the conventional power spectrum of the stationary random sequence isin which is the conventional power spectrum of in DTFD and is the fractional power spectrum of in DTSFrFD. See Appendix C for the certification process.
Therefore, as can be seen from equation (30), except for a constant coefficient, has the same spectral shape with . So, a traditional spectral estimate of the chirp-stationary sequence is transformed into the simplified fractional spectral estimation of the nonstationary sequence . So, equation (23) can be rewritten aswhere is a constant which depends on the exact passband shapes, and . That is, in DTSFrFD, the output of the system as shown in Figure 4 is the fractional power spectrum of multiplied by a constant.
3.7. Choice of the Transform Orders and
As shown in Figure 4, there are two transform orders and to be determined. The transform order is used to convert nonstationary signals into stationary signals, while decides the resolution of the proposed system which partially influences the sensing accuracy.
The received chirp-stationary signal is composed of a modulated signal and noise. Its model is given bywhere and are received signal and modulated signal, respectively. is assumed to be additive white Gaussian noise.
The modulated signal is given by , where is the amplitude, and and are the carrier frequency and the initial uniformly distributed random phase, respectively. is the Nyquist sampling rate. is the phase function, which determines the modulation type of the signal. For simplicity and without loss of generality, we assume that is an invariant constant.
In Definition 1, we know that for the chirp-stationary signal , its chirp modulated form is stationary, i.e., ; therefore, the transform order .
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 is equal to the optimal transform which can maximize the absolute amplitude, i.e., .
Furthermore, since the energy of Gaussian noise signals cannot effectively concentrate in FrFD, the Gaussian noise can be suppressed effectively.
The optimal transformation order corresponding to maximum magnitude obtained from the FrFT is given by ,
Where is the FrFT of the stationary signal in the -order FrFD. And 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  to directly obtain the optimal transformation order in DTSFrFD. Compared with the search-based algorithms, the NSOCM approach has higher computational efficiency because of its nonergodic search mechanism.
According to , the optimal order is normally given bywhere is the time-bandwidth product for , and the NSOCM of is defined bywhere is the normalized first-order origin moment of and is the normalized second-order origin moment of . The NSOCM represent the time width and frequency width of , respectively. Hence, equation (33) becomes
The NSOCM product is given bywhere is the mixed second-order moment. Setting the first derivative of with respect to the order equal to zero, we obtain
For this case where is equal to the extreme point , the product reaches the extremum values. This result demonstrates that when satisfies equation (38) as follows, the product reaches its minimum.
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 to obtain , (2)Calculate the normalized first-order origin moments and , the normalized second-order origin moments , and , the mixed second-order moment , and NSOCM and in accordance with the definition(3)Obtain the optimal order of by using (38) in the range of .
And the flowchart is shown in Figure 11.
4.1. Special Cases for Deterministic Signals
The above system as shown in Figure 4 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.
When is a deterministic signal, suppose that is the output of filter bank , is the output of filter bank , , and the two filter banks have a common input . Then, the product of the multiband filters and is
According to the fractional convolution theorem in Section 2.4, the spectrum of in FrFD is
According to the convolution theorem in FrFD, the spectrum of in FrFD is
So, the final output of the system iswhere denotes the traditional convolution operator.
The spectrum of in FrFD is
According to (2), there isand since , we havethat is, in DTSFrFD, the system output which is shown in Figure 12 is the square of the fractional spectrum of deterministic signal . And this conclusion can be used to estimate the fractional spectrum .
We demonstrate the simulations for three different cases: resolution performance, the sensing accuracy, and the influence of the order .
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  and the conventional polyphase filter banks  are given for comparison. The original multiband signal is denoted by . And the noisy signal is , where is white Gaussian noise. And is given by the following:
The choices of the parameters are listed in Table 1.
The optimal orders  of and are , ; and have the best energy concentration at orders and , respectively. For the system shown in Figure 4, we set and . And the sampling frequency is . The individual filter banks could only have resolved separations and , respectively. However, the coprime DFrFT filter banks could have resolved separations , which is more effective than the individual filter banks. The output of the original system  is shown in Figure 13(a); the fractional spectrum estimation in the 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 .
Figure 14 shows the resolution for different numbers of channels. In Figure 14, the number of channels (i.e., ) varies from 10 to 50 with a step of 10. And appropriate and 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 and to get the maximum resolution. The maximum resolution of the proposed system is while the coprime DFT filter banks is . It is observed that in Figure 14, the proposed system has the best performance because the parameter 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.
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 , spectrum reconstruction technique , and non-undersampling method .
To evaluate the sensing accuracy of the proposed system as shown in Figure 4, a chirp-stationary random signal with the initial uniformly distributed random phase is considered. And is given by the following:where is the amplitude of the signal which could be fixed. is the time scale factor which determines the signal duration. is the signal modulation rate. is the frequency carrier. is a rectangular time-window denoted by , if ; else, . The choices of the parameters for are listed in Table 2.