Single-channel Sub-Nyquist sampling Structure

The aliasing effect of the modulated signal spectrum due to the period extension is the basis for the implementation of sub-Nyquist sampling in MWC systems. The spectrum of the modulated signal ({{tilde{X}}_i}left( f right)) appears as a weighted linear combination of ({f_p})-shift copies of (Xleft( f right)). Assuming that the spectrum range is divided into several intervals by ({f_p}) length, every ({f_p})-length interval contains the weighted spectrum information from each sub-band, which is considered as the drivers for single-channel sub-Nyquist sampling structure.

Using a single-channel equivalent to a conventional MWC system, the core idea is to combine the frequency-shifting properties of the Fourier transform. Taking full advantage of the spectral panning feature of single-channel signals, the mixed signals in each region of ({f_p}) length are considered as the signals acquired by one sampling channel. The period-weighted expansion of the spectrum of one sampling channel is extracted, and the signal in each region is equivalent to the signal obtained from multiple other sampling channels. Therefore, the single channel needs to add a frequency shift module between the mixer and the low-pass filter, and the modulated signal will be frequency shifted and passed through the low-pass filter in turn. Several linearly weighted signals in the ({f_p})-length region are reserved according to the actual requirements to equate to the road sampling channels of a conventional MWC system.

The processed signal ({hat{x}}left( t right) {mathrm{= }}{tilde{x}}left( t right) cdot {e^{ – j2pi left( {a cdot {f_p}} right) t}}) produces a frequency shift effect in the frequency domain.

$$begin{aligned} {hat{X}}left( f right)= & {} int _{ – infty }^infty {{hat{x}}left( t right) cdot {e^{ – j2pi ft}}{mathrm{d}}t} nonumber \= & {} int _{ – infty }^infty {{tilde{x}}left( t right) cdot {e^{ – j2pi left( {a cdot {f_p}} right) t}} cdot {e^{ – j2pi ft}}{mathrm{d}}t} nonumber \= & {} {tilde{X}}left( {f{mathrm{+ }}a cdot {f_p}} right) end{aligned}$$

(10)

Where ({tilde{X}}left( f right)) is the spectrum of the modulated signal and (a in mathbf{N } +). Different channels are equated by selecting different values of a.

The spectrum of the modulated signal ({tilde{x}}left( t right)), after different frequency shift operations, is used as the input signal ({hat{x}}left( t right)) for the low-pass filter.

$$begin{aligned} {hat{x}}left( t right) = {tilde{x}}left( t right) cdot {e^{ – j2pi left( {a cdot {f_p}} right) t}} = xleft( t right) pleft( t right) cdot {e^{ – j2pi left( {a cdot {f_p}} right) t}} end{aligned}$$

(11)

Its Fourier expansion takes the form

$$begin{aligned} {hat{X}}left( f right)= & {} int _{ – infty }^infty {{hat{x}}left( t right) {e^{ – j2pi ft}}{mathrm{d}}t} nonumber \= & {} int _{ – infty }^infty {xleft( t right) left( {sum limits _{l = – infty }^infty {{c_l}{e^{jfrac{{2pi }}{{{T_p}}}lt}}} } right) {e^{ – j2pi left( {a cdot {f_p}} right) t}} cdot {e^{ – j2pi ft}}{mathrm{d}}t} nonumber \= & {} sum limits _{l = – infty }^infty {{c_l}Xleft( {f – left( {l – a} right) {f_p}} right) } end{aligned}$$

(12)

The filtered signal (Yleft( f right) = {hat{X}}left( f right) cdot {mathbf{H }_{LPF}}left( f right)) retains only the spectrum located in the baseband portion, which is a linear combination of each sub-band with different weighting factors. By setting different values of a to obtain sub-Nyquist sampling values sufficient, it can satisfy the reconstruction of signal support information. These sample values can be equivalently considered as coming from different sample channels. In this paper, the relationship between sampling channel number i and parameter a is specified as follows:

$$begin{aligned} i = a + 1 end{aligned}$$

(13)

Where i and a take the values of (i = left[ {1,{} 2, cdots ,m} right]) and (a = left[ {0,{} 1,{} cdots ,m – 1} right]), respectively. m is the equivalent number of sampling channels, Sampling channel (i = 1) is the sub-Nyquist sampled value obtained when no frequency shift ((a = 0)) occurs, while sampling channel (i ge 2) is the modulated signal spectrum obtained by sequentially shifting ({f_p}) distances to the left.

Sensing matrix design

In the MWC system, the observation matrix (mathbf{A }) is constructed based on the Fourier series ({c_{il}}) of the different mixing functions ({p_i}left( t right)). Matrix (mathbf{A }) is (m times L), with the elements ({mathbf{A }_{il}} = {c_{i, – l}} = c_{il}^*), (- {L_0} le l le {L_0}), where m denotes the number of sampling channels and L denotes the number of regions of ({f_p}) length divided in the spectrum sensing range. The observation matrix (mathbf{A }) of MWC is composed of m mixing functions ({p_i}left( t right)) and the form is as follows:

$$begin{aligned} mathbf{A } = left[ {begin{array}{*{20}{c}} {{c_{1, – {L_0}}}}&{}{{c_{1, – {L_0} + 1}}}&{} cdots &{}{{c_{1,0}}}&{} cdots &{}{{c_{1,{L_0} – 1}}}&{}{{c_{1,{L_0}}}}\ {{c_{2, – {L_0}}}}&{}{{c_{2, – {L_0} + 1}}}&{} cdots &{}{{c_{2,0}}}&{} cdots &{}{{c_{2,{L_0} – 1}}}&{}{{c_{2,{L_0}}}}\ vdots &{} vdots &{} ddots &{} vdots &{} ddots &{} vdots &{} vdots \ {{c_{m, – {L_0}}}}&{}{{c_{m, – {L_0} + 1}}}&{} cdots &{}{{c_{m,0}}}&{} cdots &{}{{c_{m,{L_0} – 1}}}&{}{{c_{m,{L_0}}}} end{array}} right] end{aligned}$$

(14)

Each column of matrix (mathbf{A }) corresponds to a region of ({f_p}) length in the sensed spectrum range, and the weighting factor of each sub-band located in the baseband part is related to the location of the sub-band. The weighting factor of each sub-band period extension is known by Eq. 4.

Matrix (mathbf{A }) has conjugate symmetry. From the amplitude point of view, the leftward and rightward periodic extension spectrum in the baseband part has the same amplitude variation.

In the single-channel structure, the frequency-shifted modulated signal spectrum is as in Eq. 12. Let (l’ = l – a) and substitute (l = l’ + a) into Eq. 12,

$$begin{aligned} {hat{X}}left( f right) = sum limits _{l’ = – infty }^infty {{c_{l’ + a}}Xleft( {f – l'{f_p}} right) } = sum limits _{l = – infty }^infty {{c_{l + a}}Xleft( {f – l{f_p}} right) } end{aligned}$$

(15)

Comparing with Eq. 12, the weighting factor of the spectrum is ({c_{l + a}}).

In the single-channel structure, there is only one mixing function, and the observation matrix (mathbf{A }) is constructed using the Fourier series of one mixing function corresponding to the frequency shift operation. The weighting factor for each sampling channel corresponds to one row of matrix (mathbf{A }). Therefore, (mathbf{A }) can be obtained by a row-by-row translation in a single channel structure.

$$begin{aligned} {c_{i,l}} = {c_{l + a}} = {c_{l + left( {i – 1} right) }},quad 1 le l le {L_0} – left( {i – 1} right) end{aligned}$$

(16)

where (i = a + 1). At this point, the elements in matrix (mathbf{A }) no longer satisfy the conjugate symmetry relationship.

The samples (mathbf{Y }left( f right)) in MWC system are the weighted linear combination of ({f_p})-shift copies of (Xleft( f right)), and the difference between the sampling channels is the weighted coefficients, due to different mixing functions. The weighted coefficients are several specific values from the Fourier coefficients of different mixing functions. In the single-channel structure, the equivalent samples are still the weighted linear combination of ({f_p})-shift copies of (Xleft( f right)) and the weighted coefficients are acquired by shifting the only Fourier coefficients. As long as the constructed matrix (mathbf{A }) is consistent with the sampled values after the translation operation, it will not affect the subsequent reconstruction of the sub-band support information.

In the single-channel structure, the Fourier series ({c_l}) of the mixing function is used as the first row of matrix (mathbf{A }), and the remaining rows are generated by shifting the elements of the first row according to Eq. 16. After the translation, there are free positions in (mathbf{A }) that need to be filled to complete the construction of (mathbf{A }). In this paper, two methods are presented to build matrix (mathbf{A }) as follows,

Sampling structure improvement

The frequency shift module is the most central device of the sub-Nyquist sampling system based on the single-channel structure, which is not only the key to ensure the proper operation of the system, but also can increase the flexibility of sampling value acquisition. Two basic structures are given in this section, i.e., parallel structure and series structure, as shown in Figs. 3 and 4.

Fig. 3
figure 3

Frequency shifting module with parallel structure

Fig. 4
figure 4

Frequency shifting module with serial structure

In both structures, a low-pass filter is used to obtain multiple sub-Nyquist sampled values through the control of the timing. The difference between the two structures is that the parallel structure can acquire the baseband spectrum without frequency shifting as the sampled signal, while the sampled signal acquired in the series structure is the frequency shifted signal spectrum. The parallel structure uses different frequency shifters, while the series structure can use the same frequency shifters.

Wideband spectrum sensing has high requirements for timeliness. When the above-mentioned single channel advanced sampling structure is adopted, the efficiency of the frequency shift module will directly affect the working time and efficiency of the whole system and it becomes the key to system performance improvement. Considering that the low-pass filter bandwidth in the system matches the sub-band bandwidth of the multiband signal, the sampling rate can be appropriately increased in order to shorten the signal processing time delay caused by frequency shift. In addition, the sampling process is flexibly controlled according to actual needs, making it more adaptable to application scenarios where the number of signals is unknown and constantly changing.

The mixing function used in the mixer can be a pseudo-random sequence, which values switch between 1 and (- 1). For better hardware implementation, the mixing function can be improved to ensure that the total number of code pieces remains the same during the period.

First, the value of each code slice of the modulation function is selected as 0 or 1. The mixer can be implemented through the control of the high-frequency switch. The opening of the high-frequency switch corresponds to the code piece whose mixing function takes the value of 1 to ensure the normal passage of the signal, while the closing of the switch corresponds to the code piece whose mixing function takes the value of 0 to achieve the modulation of the input multiband signal. Secondly, in order to reduce the opening and closing frequency of the high frequency switch, the mixing function structure can be changed so that a number of successive adjacent elements take the same value, thus reducing the opening and closing frequency of the switch exponentially and reducing the difficulty of hardware implementation.

In the improved mixing function, it is required that the number of code pieces remains the same M for a cycle ({T_p}). As the number of code slices with the same value increases, it makes the mixing function less random and will affect the reconstruction results of the signal support set to some extent.

The proposed ARED algorithm

In this section, the proposed adaptive residual energy detection algorithm (ARED) is similar to other algorithms for support set solution based on the greedy compressed sensing algorithm. The processing of the proposed ARED algorithm includes matching the residuals with the observation matrix, and then finding the most relevant columns to obtain the corresponding support set information, updating the residuals again, and performing circular matching until the complete support set information is obtained. In this algorithm, the iteration termination condition is no longer the sparsity of the signal, but the difference of two mean square errors is chosen. It can effectively solve the difficult problem that the signal sparsity cannot be known in advance, and solve the reconstruction problem of signal support set with arbitrary sparsity more flexibly. The pseudo-code of the proposed algorithm is shown in Algorithm 1.

In step 1, at the beginning of each iteration, the correlation between the residuals and each column of the observation matrix is solved to find the best matching column, whose corresponding column number ({Z^kappa }) is a support set information. Considering that the spectrum of the real signal has conjugate symmetry, the signal support set ({{tilde{S}}^kappa }) is thus updated as ({Z^kappa }) and (L + 1 – {Z^kappa }).

In step 2, the multiband signal after this iteration is obtained through the pseudo-inverse matrix (mathbf{A }_{{{{tilde{S}}}^kappa }}^dag), and the residuals are updated.

In step 3, the observation matrix ({mathbf{A }^kappa }) is updated with the diagonal correction matrix so that the energy of each row of the observation matrix is normalized to 1.

The details of the algorithm are shown in Algorithm 1.

figure a

Compared with the traditional MWC system signal support set recovery algorithm, the proposed ARED algorithm no longer requires the a priori information of signal sparsity and uses the residual energy detection result as the iterative termination condition to achieve blind detection of the signal.

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