Wireless Pers Commun DOI 10.1007/s11277-014-2109-0

Novel Compressed Sensing Algorithm Based on Modulation Classification and Symbol Rate Recognition Yifan Zhang · Xuan Fu · Qixun Zhang · Xiaomin Liu

© Springer Science+Business Media New York 2014

Abstract In the complex and changing radio environment, how to achieve the fast and accurate spectrum sensing over an ultra-wide bandwidth is a big challenge. A novel compressed sensing algorithm based on modulation classification and symbol rate recognition is proposed by using the minimal sampling rate to detect spectrum holes. It is more efficient than the Nyquist sampling rate and traditional compressed sampling rate, which requires the reconstruction of the original signal. Simulation results show that it can further decrease the compressed sampling rate depending on the relation between compression ratio with modulation scheme and symbol rate. Furthermore, in order to improve the accuracy of modulation classification and symbol rate recognition, a compressed sensing and wavelet transform (CSWT) feature detector is proposed to perform wideband detection in low SNR condition. Simulation results show that CS-WT feature detector can effectively reduce the noise introduced by CS process. Given the false alarm of 0.05 and the detection probability of 0.9, the detection probability of proposed CS-WT feature detection algorithm can be improved 4 dB compared to traditional cyclostationary detection. Therefore, the overall sampling rate can be dramatically reduced without spectrum detection performance deterioration compared to the conventional static sampling algorithm. Keywords Cognitive radio · Compressed sensing · Cyclostationary detection · Wavelet transform · Modulation classification · Symbol rate recognition

Y. Zhang · X. Fu · Q. Zhang (B) · X. Liu Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China e-mail: [email protected]; [email protected] Y. Zhang e-mail: [email protected] X. Fu e-mail: [email protected] X. Liu e-mail: [email protected]

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1 Introduction In terms of the explosive data service requirements nowadays, the spectrum is a requisite natural resource for the wireless communication system, which faces the spectrum shortage with the traditional fragmented spectrum allocation. However, the phenomenon of low spectrum usage exists in the specific spectrum band in [1]. Recently, many researchers have paid their attentions on spectrum sensing technology in cognitive radio (CR), whicn can detect the unoccupied frequency bands for temporal utilization [2]. Most widely used spectrum sensing methods include the energy detection [3], matched filtering [4] and cyclostationary detection method [5]. Energy detection method is mostly used in practice for its simplicity, but it is criticized for its susceptibility to noise uncertainty and poor performance at low signal to noise ratio (SNR). Whereas matched filtering method is much more complex and requires the detailed signal information, such as modulation, timing and pulse shape, which hinders its practical use in CR scenario, as the priori signal information of primary user can not be obtained in advance. Therefore, much more attentions are paid to the cyclostationary feature detection method, for its high accuracy under low SNR and independence on priori information of primary signal. There are many researches on cyclostationary detection [5–7]. The cycle frequency domain profile (CDP) is used for signal detection and classification in [5]. However, the problem of traditional cyclostationary detection method is complexity, which comes from two aspects: (1) It requires high sampling rate, which may result in high cost of signal processing and storage. And this problem may be even more serious when it comes to wideband sensing. (2) The computation of spectral correlation function is very complex. In fact, complexity has become the main obstacle for the application of feature detector. Considering the scenario of a large number of temporary spectrum holes over an ultra-wide bandwidth, compressed sensing (CS) technique is brought forward for the efficient spectrum sensing to minimize the complexity proposed by Tian and Giannakis in [8]. In this paper, compressed sensing is proposed to reduce the complexity of feature detector. The basic idea behind CS is that the radio spectrum is sparse, thus the original signal can be reconstructed from samples obtained with sub-Nyquist rate. However, the CS process will inevitably bring in noise because of the compression of original data into smaller number of sample points. As it will cut down SNR, CS noise will cause degradation to sensing performance. Thus noise reduction is necessary after CS. Wavelet transform (WT) [9] has long been used for noise reduction in image processing. In this paper, the spectral correlation function (SCF) is a two dimensional matrix depending on both cyclic frequency and frequency. Therefore, SCF can be viewed as a grey map, and two-dimensional WT can be applied to reduce the noise introduced by channel and CS process. Therefore, a feature detector based on CS and WT (CS-WT feature detector) is proposed to perform wideband detection at low SNR. Meanwhile, considering the fact that the transmission parameters of specific spectrum band will keep stable within a certain period, we propose an adjustive algorithm of compression ratio for compressed sensing based modulation classification and symbol rate recognition to optimize the sampling rate during this period. The purpose of the algorithm is to achieve a good detection performance using the minimum number of samples. This paper is organized as follows. Section 2 presents the problem statement and system model. Section 3 proposes the realization of CS by matrix and cyclic feature analysis. In Sect. 4, we illustrate the procedure of two-dimensional wavelet approach to process noisy SCF data and the procedure to deduce the minimal compressed sampling rate based on modulation classification and symbol rate recognition. Simulation results are provided in Sect. 5, followed by a concluding summary in Sect. 6.

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Fig. 1 System model

2 System Model and Problem Statement Spectrum sensing is the basis of cognitive radio. Secondary user (SU) should be able to detect spectrum holes quickly and accurately in order to realize the efficient dynamic spectrum access. Therefore, the ultra-wide band information collection is one of the key technologies for spectrum sensing. Traditional signal processing technology is based on the Nyquist sampling theorem, which requires that the sampling rate should be at least twice of the system bandwidth. Thus, this becomes a barrier for an efficient ultra-wideband bandwidth detection. Although the CS scheme develops a new path of ultra-wide bandwidth detection, the over-sampling phenomena still exits. Thus, this paper proposes an adjustive algorithm of compression ratio for compressed sensing based on modulation classification and symbol rate recognition to further reduce the sampling rate of compressed sampling. This algorithm is shown in Fig. 1. First, for a given bandwidth of W Hz, the AIC samples the incoming time-domain signal x(t) and represents it as y(t). Then the compressed discrete samples are delivered to reconstruct the 2D spectral coherence function (SCF) directly. In fact, the process of compressing the signal’s information into a small number of samples inevitably decreases the SNR and will have a great impact on modulation classification and symbol rate recognition. So, we preprocess the SCF data for noise reduction to make an accurate modulation classification and symbol rate recognition. Then, modulation scheme and symbol rate can be identified depending on SCF, which has been preprocessed. Finally, modulation scheme and symbol rate are feed back to AIC. AIC determines compression rate for CS according to the relation between compression rate with modulation scheme and symbol rate. Then, AIC completes compressed sensing to detect the spectrum holes without reconstructing the signal by different compression ratios based on the modulation scheme and symbol rate. Thus this algorithm can reduce the sampling rate significantly and detect spectrum holes accurately and efficiently.

3 Compressed Sensing In this section, we implement the CS with matrix realization and then extract useful secondorder statistics of wideband random signals from sub-Nyquist rate samples. 3.1 Sub-Nyquist-Rate Sampling Realization The fundamental implementation of CS is the analog-to-information conversion (AIC) presented in Fig. 2. x(t) is a continuous time domain signal, and the sampling duration of AIC is [0,T]. x(t) can be expanded in a bandlimited Fourier series as x (t) =

N /2−1

x (n) e j

2π T

nt

, t ∈ [0, T ]

(1)

n=−N /2

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wherein, N = T W , it W is the Nyquist sampling rate. To obtain the sub-Nyquist sampling values, AIC first multiplies x(t) with the pseudo-random sequence pc (t), then delivers x(t) pc (t) through low-pass filter, and at last complete interval low speed sampling in the sampling duration [0,T]. Here, pc (t) is the continuous random process obeying Bernoulli distribution, if z n (n = 0, 1, . . . , N − 1) is Bernoulli random variable with equal probability of ±1, so pc (t) can be expressed as   n n+1 pc (t) = z n , t ∈ , , n = 0, 1, . . . , N − 1 (2) W W and its Fourier series expansion is pc (t) =

∞ 

p (n) e j

2π T

nt

, t ∈ [0, T ]

(3)

n=−∞

The low-pass filter in AIC is equivalent to an ideal integrator, and its impulse response is h (t) = r ect (2Mt/T − 1)  1, t ∈ [−1, 1] r ect = 0, t ∈ (−∞, −1) ∪(1, ∞)

(4) (5)

M(M < N ) is the sub-Nyquist rate, so AIC can effectively reduce the sampling rate of the continuous signal. To sum up, the AIC time domain sampling process is given in (6).   T 2M (6) y (m) = x (t) pc (t) ∗ h (t) = ∫ x (τ ) pc (τ ) r ect (t − τ ) − 1 dτ |t= (m+1)T M T 0 In order to facilitate the analysis of above AIC sampling process and express the process in the mathematics representation, the original signal x(t) can be regarded as the Nyquist  T sampling interval discretization vector form x = x0 , x1 , . . . , x N −1 , and pseudo-random sequence pc (t) can be written in the form of a diagonal matrix D with elements z n equaled to ±1 with half probability. Then the action of the accumulate-and-dump sampler can be treated as an R × W matrix H whose r th row has W/R consecutive unit entries starting in column r W/R + 1 for each r = 0, 1, . . . , R − 1. 3.2 Recovery of SCF From Sub-Nyquist Samples Given the available data xt (n), as we all know, if its time-varying covariance r x (n, v) = E {x (nTs ) x (nTs + vTs )} = E{xt (n)xt (n + v)} satisfies the equations: r x (n, v) = r x (n + kT0 , v)

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where T0  = 0, then we say that it is cyclostationary. Since r x (n, v) is periodic, and the length of received data xt (n) is N, the cyclic covariance can be expanded in a Fourier series:

N −1−v   2π π 1 r x (n, v) e− j N an e− j N av (8) r x (c) = N n=0

where α = N1Ts a, a ∈ [0, N − 1] is the cyclic frequencies. The cyclic spectrum is the Fourier Series of the cyclic covariance with respect to v and given by: sx(c) (a, b) =

N −1 

r x (c) (a, v) e

v=0

where the frequency f =



N −1 2



∈ − f2s ,



N −1 v − j 2π N b− 2

fs 2

(9)

, b ∈ [0, N − 1].  N −1  (c) from Based on above foundation of cyclic theory, the whole recovery sx (a, b) a,b=0  N −1  (c) compressed samples yt (n) can be partitioned into two parts. First, we relate sx (a, b) 1 N Ts

b−

a,b=0

to Nyquist samples xt (n). Then we relate xt (n) to compressed samples yt (n) by compressed matrix .   N −1   N −1 (c) As illustrated in [10], the 2D quantities r x (n, v) , r x (c) (a, v) a,v=0 and n,v=0  N −1  (c) x(c) , S(c) can be represented as matrices R, R sx (a, b) x respectively, and the timea,b=0

varying covariance R is symmetric for real-valued signals and can be arranged into the vector expressed in Eq. (10).  T r (0, 0) , r x (1, 0) , . . . , r x (N − 1, 0) , r x (0, 1) , r x (1, 1) , . . . (10) rx = x r x (N − 2, 1) , . . . , r x (0, N − 1) The correspondence between the two can be mapped through a relation rx = B P vec{R}. x(c) and the cyclic spectrum matrix Following Eqs. (8) and (9), the cyclic covariance matrix R (c) Sx are related to R through Fourier Series transformation in Eqs. (11) and (12). x(c) = R S(c) x =

N −1 

Gv RDv

v=0 x(c) F R

where F is the N point DFT matrix, Gv =



(11)



v 1 − j 2π N a(n+ 2 ) Ne (a,n)

(12) ∈ C N ×N , and Dv is an

N × N matrix with only its (v, v)th element being 1 and all other elements being 0. (c) The spectrum of interest can be fully represented in a vector form as sx =  cyclic  (c) vec Sx , then we reach N −1

   (c) s(c) DvT ⊗ Gv vec {R} = Trx x = vec Rx F = (I ⊗ F)

(13)

v=0

where I is the identity matrix, ⊗ denotes Kronecker product operation. (c) Next, we aim to relate the unknown sx with the data vector yt (n) by deriving the relationship between rx and the time-varying data covariance matrix R y = E{yt ytT } ∈ R M×M . Similar to rx in Eq. (10), R y can be organized into the vector expressed as Eq. (14).

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 ry =

r y (0, 0) , r y (1, 0) , . . . , r y (M − 1, 0) , r y (0, 1) , r y (1, 1) , . . . r y (M − 2, 1) , . . . , r y (0, M − 1)

T

Likewise, r y = Q M vec{Ry }. Meanwhile, it holds from Eq. (2) that R y = Rx T . Thus, it can be derived that   r y = Q M vec Rx T = Q M ( ⊗ ) vec {Rx } = rx

(14)

(15)

Finally, using Eqs. (13) and (15), we can express the measurement vector r y as a linear (c) function of the vector-form cyclic spectrum sx as r y = T+ s(c) x

(16)

For the 2D cyclic spectrum is highly sparse and the covariance matrix R is symmetric and positive semi-definite (psd), the problem of reconstructing rx subject to a psd constraint imposed on R is recognized as an unconstrained l1 norm regularized least squares (LR–LS) problem in Eq. (17).  2 min Trx 1 + λr y − rx 2 (17) rx

The optimization formulation in Eq. (17) is a convex problem, and can be feasibly solved. (c) Then, the vector-form cyclic spectrum can be estimated as sx = Trx .

4 A Novel Compressed Sensing Algorithm 4.1 Wavelet Approach for Noise Reduction In fact, the process of compressing the signal’s information into a small number of samples inevitably decreases the SNR and do have a great impact on detection. Here, we preprocess the SCF data for noise reduction to make accurate modulation classification and symbol rate recognition. This section we propose the novel method of wavelet approach to smooth noisy SCF data. First, the necessity of noise reduction is illustrated. Second, the wavelet approach for smoothing SCF data is suggested and proposed. 4.1.1 Necessity of Noise Reduction In fact, the process of CS and recovery SCF from sub-Nyquist samples generate additional noise, inevitably decreasing the SNR and having a great impact on modulation classification and symbol rate recognition. Besides, the feature detection is based on the theory that: for different cyclic frequency values α  = 0 , SCF is cross-spectral density between the signal and the signal shifted in frequency by α. So, if the signal being analyzed exhibits cyclostationarity, SCF will be non-zero for α  = 0. Otherwise, we will have non-zero values only for α = 0. Especially, Quaternary Phase Shift Keying (QPSK) has the fewest number of distinguishing peaks in its SCF and therefore will have the smallest amount of total distance between the ideal QPSK profile and random noise with no distinguishing. When α = 0, the SCF can also be interpreted as power spectral density of the signal. While the white noise signal in the case of low SNR exhibit energy dispersal, some small peaks also appear in non-zero frequencies, raising false alarm rate. So it is necessary to take measures to process the recovered noisy SCF data before making detection. Here we propose the two dimensional wavelet transform to realize noise reduction.

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A large number of methods based on wavelet edge detection and noise suppression technology have been brought in image processing and been more and more widely used. But the noise suppression has not been applied to the signal processing. The features of the wavelet transform are to maintain the characteristics of the local signal in the time domain and frequency domain, and the outline and detail part of the image can also be analyzed by its resolution characteristics. Therefore, wavelet transform is suitable for dealing with lowfrequency and high-frequency non-stationary image signal. Filtering through the wavelet transform method in image noise suppression can be divided into three types: modulus maxima reconstruction filtering, spatial filtering, and wavelet threshold filtering. These three methods can effectively suppress image noise, while the details of the image can be better reversed. Among them, wavelet threshold filtering method [11] takes the advantage of iterative process and intensive computation over the modulus maxima reconstruction filtering and spatial filtering methods. Therefore, we adopt the wavelet threshold filtering method in this paper due to its simplicity. 4.1.2 Wavelet Approach for Processing SCF Since SCF is a two-dimensional matrix function with sudden value change, we choose the − (x

2 +y 2 )

two-dimensional Gaussian function θ (x, y) = e 2s 2 as the wavelet transform filter. Then derivations on its two arguments are deduced to obtain the two-dimensional wavelet function in Eqs. (18) and (19). ∂θ (x, y) ∂x ∂θ y) (x, θ 2 (x, y) = ∂y θ 1 (x, y) =

The telescopic function can be induced in Eq. (20). 1 x y , θs (x, y) = 2 θ s s s And the matrix realization is shown in Eq. (21). ⎤  ⎡ 1  x1 y1  θ s , s · · · s12 θ xsN , ys1 s2 ⎢ ⎥ .. .. .. s (x, y) = ⎣ ⎦ . . .     x N yN x1 y N 1 1 · · · θ , θ , s s s s s2 s2

(18) (19)

(20)

(21)

The wavelet transform of matrix Sc ( f, α) of wavelet 1 (x, y) and 2 (x, y) on scale s are defined as shown in Eqs. (22) and (23). Ws1 Scx ( f, α) = Scx ( f, α) ∗ 1s (x, y)

(22)

Ws2 Scx ( f, α) = Scx ( f, α) ∗ 2s (x, y)

(23)

wherein s takes values from powers of 2(s = 2 j ) , representing the bandwidth of Gaussian filter. The bigger the value of s is, the wider the bandwidth of filter is. And the little change (c) of SCF may be eliminated by the convolution of Sx and Gaussian function, leaving alone the larger saltatory points. As a result, only the low frequency of signals can be detected by the wavelet transformation. However, the smaller s may cause the narrower bandwidth of Gaussian filter, thus the high frequency cannot be eliminated. We can achieve a reasonable adjustment between too fuzzy features and excessive mutations caused by noise and fine

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texture, and get the acutely changing points at different scales, so as to realize multi-scale edge detection by adjusting scale parameter values. According to the processing result, we choose j = 2. The process of convolution can be viewed as two components of the edge detection vector of received samples Scx ( f, α) smoothed by function 2 j (x, y). The modulus of edge detection vector is shown in Eq. (24).  2  2     c M2 j Sx ( f, α) = W21j Scx ( f, α) + W22j Scx ( f, α) (24) The inflexion of smoothing received samples Scx ( f, α) ∗ 2 j (x, y) corresponding to the local modulus maximum of edge detection. Then we can get edge through the comparison of the threshold γ and M2 j Scx ( f, α) to distinguish between useful information and noise. We consider it feature point if the modulus is larger than the given threshold value, otherwise, we consider it noise and the modulus equals to 0. Finally, the processed SCF S ( f, α) is obtained. 4.2 Spectrum Detection Based on Modulation Classification and Symbol Rate Recognition A random demodulator is used to deduce the minimum measurement rate required for accurate signal reconstruction in [12]. As we know, the sampling rate R is relative to the sparse level K and the band-limited width W. The measurement of the sampling rate required to recover random sparse signals accurately is given in Eq. (25). R∼ = 1.7 × K × log(W/K + 1)

(25)

The aim of sampling rate in Eq. (25) is to reconstruct the signal perfectly, however, we just need to use the samples to detect the spectrum holes and unveil the fact that spectrum holes exist, regardless of the location or number of signals. So further study is made on the Eq. (25) and a theorem is derived below. Theorem 1 R is an increasing function of the sparse level K . Proof The derivative of R on K is dR 1.7 × K × log e = 1.7 × log (W/K + 1) − dK W+K W = 1.7 × log e × [ln (W/K + 1) − ] W+K



As we all know, the authorized bandlimited number is much smaller than the whole sensing W W bandlimited number, so W K 2 and W +K  1. That is ln (W/K + 1) − W +K > 0 hang established. In spectrum sensing, K large or small is equal to K = 1, while in recovering signals, we require more samples if K is larger. Hence, regardless of recovering the signals accurately, deciding exist spectrum holes requires less samples. And sampling rate is related to K. As shown in Eq. (25), signal sparsity and the detection bandwidth determine the desired compression ratio for signal reconstruction. Similarly, signal sparsity and the detection bandwidth determine the desired compression ratio for detecting exist spectrum holes. Therefore factors of affecting the signal sparsity will affect the compression ratio for compressed sensing. Modulation scheme and symbol rate have a relatively large impact on the signal sparsity. So modulation scheme and symbol rate will affect the compression ratio. SCF is relatively different in terms of signals with different modulation and symbol rate. So, modulation scheme can be classified and symbol rate can be recognized by SCF. Meanwhile, the relationship between compression rate with modulation scheme and symbol rate

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can be gotten by simulation. Therefore, compression ratio can be determined based on modulation classification and symbol rate recognition, which can be determined by processed SCF data. Then, spectrum detection can be made. Cyclostationary detection will be made based on scanning all cycle frequencies. First the two dimension function is projected onto the cycle frequency domain profile (CDP), the vector representation is S (α) = max fi S ( f, α). Then the crest factor (CF) is applied to find the highest peak in CDP. The CF is a dimensionless quantity and its expression of CF has been derived in [5]. !$ % " N "  # 2 S (α) C T H = max (S (α)) N (26) α=0

For signal detection, the threshold C T H is first calculated when no signal is present, i.e., when x (t) = n(t) , and then the detection statistic C R compares with the threshold C T H . In summary, the following binary hypothesis testing of signal existence in additive white Gaussian noise (AWGN) is shown in Eq. (27) and it can be performed in Eq. (28). H0 : x (t) = n (t) H1 : x (t) = s (t) + n (t)

(27)

C R ≤ C T H : Declar eH0 C R > C T H : Declar eH1

(28)

According to the above analyses, this paper presents an adjustive algorithm of the compression ratio for compressed sensing to further reduce the sampling rate. This algorithm is based on modulation classification and symbol rate recognition. The steps are shown below. Step 1: Realize Sub-Nyquist-rate sampling and recover SCF from Sub-Nyquist samples by cyclostationary detection; Step 2: Processing SCF by wavelet approach to reduce noise; Step 3: Detect the spectrum holes; Step 4: Achieve modulation classification and symbol rate recognition depending on processed SCF data; Step 5: Determine the compression ratio based on modulation scheme and symbol rate, and return to step 1.

5 Results and Analysis First, the SCF of the QPSK and 2ASK before noise deduction is shown in Figs. 3a and 4a. It can be seen that apart from the inherent original distinct peaks, the noise peaks also exhibit along the axis. However, noise peaks after wavelet transform are well suppressed out, as shown in Figs. 3b and 4b, which prove the effectiveness of the wavelet transform noise reduction, so that the SNR of processed SCF data is greatly increased to enhance the detection accuracy. Then, the cyclostationary feature of AWGN is depicted in Fig. 5, showing the difference between noisy SCF and the SCF after noise reduction. It depicts that the wavelet transform filters out some irregular peaks and more clearly reflects the characteristics of the AWGN. In other words, the wavelet transform allows us to distinguish signal from noise, making the detection more accurate.

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Fig. 3 Wavelet transformation noise reduction for QPSK. a Before the noise reduction. b After the noise reduction

Fig. 4 Wavelet transformation noise reduction for 2ASK. a Before the noise reduction. b After the noise reduction

Fig. 5 Wavelet transformation noise reduction for AWGN. a Before the noise reduction. b After the noise reduction

To show the performance of the CS-WT feature detector, simulations of Binary Phase-Shift Keying (BPSK) modulation and Minimum Shift Keying (MSK) signals are also performed in AWGN channel. The performance metrics include the probability of detection Pd and

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Detection Probability

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

Energy detection Traditional cyclostationary detection CS−WT feature detection

0.1 0 −30

−25

−20

−15

−10

−5

0

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SNR(dB) Fig. 6 S N RV s Pd curves for several detection methods

probability of false-alarm P f a . The two probabilities indicate the performance of the sensing quality. Sensing algorithms based on energy detection, traditional cyclostationary feature detection and CS-WT feature detection are also compared by simulation results below. The S N RV s Pd graphs at fixed false alarm probabilities (P f a = 0.05) are presented in Fig. 6. Compared to the energy detection and traditional cyclostationary feature detection, CS-WT feature detection has a higher Pd in the case of the same SNR. Because under the same condition achieving a certain P f a , CS-WT feature detection requires less samples than the other two methods, especially in the case of below −25d B. It proves that the noise reduction of the wavelet transform and CS-WT feature detection can achieve a better probability of detection in a low SNR condition. As demonstrated in Fig. 6 with different SNRs, the detection probability of proposed CS-WT feature detection performs exceptionally better than the energy detection and traditional cyclostationary detection, even at a low SNR condition. Given P f a = 0.05, Pd = 0.9, the detection probability of CS-WT feature detection can be improved 4 dB compared to traditional cyclostationary detection. This is significant for detection theory, and suggests that the proposed detector can detect spread spectrum signals such as CDMA or Bluetooth. The normalized mean square error can be applied as the performance metric of the accuracy of signal reconstruction, depicted by MSE = & x − x 2 / x 2 , where & x is the reconstructed signal. This metric can affect the sensing performance. To check the influence of modulation scheme on compression ratio, four typical modulation schemes are chosen, including 2PSK, 4PSK, MSK and 2ASK with symbol rate f d = 5H z in Fig. 7. It can be observed that MSE increases with the decrease of compression ratio, which is reasonable and is line with practice. The compression ratio of these modulation signals is different. And it is evident that the compression ratio of 2ASK and MSK modulation signals is smaller, because 2ASK and MSK modulation signals are more sparse. Meanwhile, to verify the influence of symbol rate on compression ratio, MSK modulation signals with three different symbol rates f d = 5, 10 and 20H z are chosen in Fig. 8. Similar to modulation scheme, the compression ratio of these signals is different. And it is evident that the compression ratio increases with symbol rate when the compression ratio is big, because signals with low symbol rate are more sparse. When the compression ratio is small, compressed sensing algorithm fails to reconstruct the

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Fig. 7 Effect of modulation scheme on compression ratio

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Fig. 8 Effect of symbol rate on compression ratio

signals with symbol rate f d = 10, 20H z. So the MSE is big and is also unrelated to the symbol rate (Figs. 9, 10). To depict the advantage of the adjustive algorithm of compression ratio algorithm using the modulation classification, we set the symbol rate f d = 5H z. Modulation scheme includes BPSK, QPSK, BFSK, MSK and 2ASK. As denoted in Fig. 9, the modulation classification can reduce the average compression ratio. Furthermore, to depict the advantage of the adjustive algorithm of compression ratio algorithm using the symbol rate recognition, the modulation scheme of MSK modulation is chosen with the symbol rate of f d = 5, 10 and 20H z. As denoted in Fig. 10, the recognition of symbol rate can also reduce the average compression ratio. In a word, the adjustive algorithm of compression ratio based on modulation classification and recognition of symbol rate can reduce the compression ratio for reconstructing signals. Similarly, it can reduce the compression ratio for sensing the spectrum holes.

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Fig. 9 Benefit of modulation classification on reduction of compression ratio 1 Unrecognize Symbol Rate Recognize Symbol Rate

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Fig. 10 Benefit of symbol rate recognition on reduction of compression ratio

6 Conclusion Based on the proposed adjustive algorithm of compression ratio for compressed sensing using modulation classification and symbol rate recognition, this paper deduces the compressed sampling rate required to detect spectrum holes over relatively stable PUs. Unlike the conventional CS, we improve the efficiency of compressed sensing by reducing the redundancy of detailed information in the signal. To use the modulation classification and symbol rate recognition to guide the sampling rate is the key procedure. Simulation results show that the proposed algorithm can indeed improve the sampling efficiency. In order to improve the accuracy of modulation classification and symbol rate recognition, a CS-WT feature detector is proposed to perform wideband detection in the low SNR. The WT is applied to reduce the noise brought in by the CS process and improve the detection accuracy in the low SNR condition. Simulation results show that we can restrain the noise brought in by CS and extract the

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feature of modulated signals by a two-dimensional wavelet analysis. The innovation includes two points: (1) We find the relation between compression ratio and modulation scheme and symbol rate in order to determine the sampling rate of CS process. (2) We regard the noisy SCF as a two dimensional image with frequency range in horizontal direction and cyclic frequency range in vertical direction and then apply the WT for noise reduction. In a word, by using the adjustive algorithm of compression ratio for compressed sensing based on modulation classification and symbol rate recognition, we can perform more accurate, rapid and effective spectrum sensing, and then secondary users can dynamic access to the unlicensed bands much more accurately and feasibly. Given P f a = 0.05, Pd = 0.9, the detection probability of CS-WT feature detection can be improved 4 dB. More kinds of modulated signals in different environments will be involved in the follow-up study.

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Novel Compressed Sensing Algorithm Based on Modulation Classification Yifan Zhang is a researcher in the School of Information and Communication Engineering of Beijing University of Posts and Telecommunications (BUPT). He received his B.S. degree, Master degree and Ph.D. degree in BUPT. His current research interests include compressed sensing, cognitive radio network implementation, optimization algorithm in wireless networks.

Xuan Fu is a graduate student in the School of Information and Communication Engineering of Beijing University of Posts and Telecommunications (BUPT). He is currently working hard towards his Master degree in BUPT. His current research interests include compressed sensing, optimization algorithm in wireless networks.

Qixun Zhang is a researcher in the School of Information and Communication Engineering of Beijing University of Posts and Telecommunications (BUPT). He received his B.S. degree and Ph.D. degree both in BUPT. His current research interests include heterogeneous wireless networks convergence, cognitive pilot channel, cognitive radio networks, compressed sensing, cooperative relay, self-organizing network (SON) technology, femtocell and small cells.

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Y. Zhang et al. Xiaomin Liu is a graduate student in the School of Information and Communication Engineering of Beijing University of Posts and Telecommunications (BUPT). She is currently working hard towards her Master degree in BUPT. Her current research interests include compressed sensing, optimization algorithm in wireless networks.

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