A Compression Sampling System based on ... - Semantic Scholar
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A Compression Sampling System based on Sparse AR Model Dandan Cheng, Qingwei Ye, Yu Zhou, and Xiaodong Wang Information Science and Engineering College, Ningbo University, Ningbo 315211, China Email: [email protected]
Abstract—A compression sampling system based on sparse AR (auto regression) model is designed in this paper. The sparse samples are non-uniform sampled using uniform random sampling (URS) method by mono-chip computer. To guarantee the reconstruction effectiveness, a basis matrix is constructed with prior signal information to represent the received signal sparsely. Then the received signal is recovered from the samples using optimization algorithms in PC. The URS method can scale down the sampling frequency effectively. The basis matrix is constructed based on the known AR model and named sparse AR (SAR) basis in this paper. Since the SAR basis is a self-adaptive basis, better reconstruction quality at a low sampling rate can be obtained. The performance of the proposed compression sampling system is illustrated using normal vibration signal. From both simulation and experiment, the transmitted signal can be reconstructed effectively and accurately. Index Terms—Compression Sampling; Sparse AR Model; Uniform Random Sampling; Signal Reconstruction
I.
INTRODUCTION
The Shannon/Nyquist sampling theorem specifies that to avoid losing information when capturing a signal, one must sample at least two times faster than the signal bandwidth. Nowadays, with the popularization of the real-time monitoring technology in many practical applications, such as global positioning system, communication system and structural health monitoring system, it brings great pressure to data storage and data transmission especially in the wireless sensor network. Like in structural health monitoring system, vibration signal provides useful information for diagnosis of structure failure. In order to identify the features of the fault, a significant amount of vibration data needs to be collected by sensors. It may face bottlenecks, due to limited transmission bandwidth and long-term transmission time. Thus, to minimize the storage and transmission load, it is vital to consider signal compression issue. Besides the data compression, it must ensure that the received signal can be recovered back without losing any information. Therefore, the main issue of a compression approach is able to achieve a high compression ratio while keeping a reasonable The work has been financially supported by National Nature Science Funds of China (No. 61071198), Nature Science Funds of Zhejiang Province (No. LY13F010015) and Innovation Project of Zhejiang Province Science and Technology Department (2012-R10009-08). © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.9.2536-2542
reconstruction performance. Besides the aforementioned performance criteria, the level of complexity is also a considerable factor. Compression sensing (CS) theory [1, 2] presents a new method to capture and represent compressible signal at a rate significantly below the Nyquist rate and then reconstructs back the original signal effectively using an optimization algorithm. Theory CS has been extensively treated in the literature and widely used in the field like GPS system [3], acoustic signals [4], electrocardiograph signals [5, 6]. The existing method for signal compression mainly focus on transformation methods (discrete Fourier transform [7], discrete cosine transform [8], and wavelet transform [9-11]), which transform the signal in the time domain to another domain and then compress the transform coefficients just transmitting a small portion of them. Among the transformation methods, wavelet transform has shown promising performance for speech signal due to its good localization properties [12]. But the work dedicated to vibration signal compression is rather limited [13, 14]. The major problem is the fixed basis because the selection process of the best basis is dominated by the vibration signal components [15]. The earlier work for signal compression mainly aimed at digital signal. To obtain compressed samples directly from an analog signal, analog-to-information converter (AIC) structure has been proposed [16]. It consists of several parallel branches of mixers and integrators, where each integrator multiplies the signal with a sampling waveform and the result is integrated. The effectiveness of AIC has been proved being used as a receiver of an ultra wideband signal system [17, 18]. However it is complex to implement for the complicated hardware system. In this paper, we propose a compression sampling system based on sparse AR model. The system includes two steps: signal compression sampling and signal reconstruction. First to simplify the requirement of hardware, the sparse samples are non-form sampled using uniform random sampling (URS) method by mono-chip computer. URS refers to each sample is taken random in a fixed time period adjacent. The URS method is with good advantages of low sampling rate and easy realization. Second a sparse AR (SAR) basis is designed to represent the received signal sparsely. The SAR basis is constructed based on the known AR (auto regression) model, which is composed with the prior signal
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components. For the SAR basis is adaptive and captures the property of vibration signal, thus the effectiveness of signal reconstruction can be guaranteed. At last with the interior-point method of L1 norm algorithm [19], the received signal can be successfully recovered from the sparse samples. As the compression sampling system using URS method and adaptive SAR basis, good reconstruction performance can be obtained preserving system simplicity and high compression ratio. In this paper, the performance of the proposed compression sampling system is demonstrated with normal continuous vibration signal from both theoretical and practical aspects respectively compared with exiting approaches. The remainder of the paper is organized as follows. Section II first gives an overview of theory CS and then provides an introduction to the architecture of hardware and proposes the URS method. Section III first proposes the approach to construct the SAR basis in details and then depicts the process of signal reconstruction. Section IV runs numerical simulations to illustrate the performance of the proposed compression system using URS sampling method and SAR basis compared with the popular DFT basis. Section V applies the proposed compression sampling system to a real vibration signal collected from a cable-stayed bridge model. Finally, conclusions are drawn in Section VI. II.
COMPRESSION S AMPLING
A. Background of Compression Sensing Compression sensing (CS) theory is one of hot issue in signal and information processing field. It indicates that if a signal is sparse, it can be reconstructed with overwhelming probability with optimization algorithms from the samples sampled at a rate significantly below the Nyquist rate. Any signal in R N can be represented in terms of a basis of N 1 vectors { i }iN1 . Using the N N basis matrix (1 , 2 , , N ) , a discrete-time signal x ( x1 , x2 ,
This is a dimension reduction process. The problem is to design a stable measurement matrix. According to CS, any matrix who obeys “restricted isometry principle” (RIP) can be used as a measurement matrix [20]. It is known that most of the random matrices satisfy the property “RIP”. It can be noted that the recovery of signal x is an NPhard problem for (2) is underdetermined. Some algorithms have been developed to solve the problem. One kind is the basis pursuit (BP) approach, which relaxes the L0 norm condition by the L1 norm and solves the problem through linear programming. The other is known as the greedy algorithm which approximates the signal through a sequence of incremental approximations by selecting atoms suitably. The greedy algorithms usually require the knowledge of sparsity in advance while BP algorithms can produce more accurate solutions [21]. Hence in this paper we use the L1 norm interiorpoint algorithm [22]. B. Compression Sampling 1) Architecture of Sampling System Theory CS provides a novel framework to process sparse signals. In this section, we develop the architecture of hardware for vibration signal compression sampling. At present, structure AIC is usually used to collect analog signal whose components mainly include mixers, integrators and ADCS. This part proposes a structural diagram of the circuit for the realization of the compression sampling, which only consists of accelerometer ADXL335 and mono-chip computer ADuC7020 as shown in Fig. 1 simplifying the hardware requirement. The ADuC7020 is fully integrated, 1 MSPS, 12 bit data acquisition system incorporation a high performance multichannel ADC on a signal chip. It contains 16 ADC channels and supports in circuit serial download via the UART, widely applied in industrial control, automation systems and smart sensors.
, xN )T can be expressed as N
x i i
(1)
i 1
where (1 ,2 , , N )T is the coefficients vector. Obviously, x and are the equivalent representations of the signal, with x in the time domain and in the domain. If vector has only K ( K N ) nonzero values, the signal x is called K-sparse signal with respecting to the matrix . In some ways, signal x can be considered compressible if vector has just a few large coefficients and many small coefficients. Most signals are sparse with respecting to special basis in practice. In CS, the signal x is not acquired directly, but measured by a measurement matrix RM N M N . The measurements y can be expressed as followed
y x
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(2)
Figure 1. Accelerometer and micro-controller
In this scheme, an input signal from accelerometer is projected onto sparse signal by a set of pulse signals which are generated by the Timer of ADuC7020. Next the ADC is used to sample and quantize the projection signals, producing the sparse vibration signal samples. With the proposed sampling diagram, there is no need to store the measurement matrix in hardware system beforehand, which simplifies the mixers of AIC structure.
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The compressive samples are selected by fixed time control, easy to implement. 2) Uniform Random Sampling According to CS theory, the signal received should be sampled at a low rate and then recovered successfully by optimization algorithms. The core of the compression sampling is to realize the projection of multidimensional input signal onto a lower dimensional subspace [23]. In the case of uniform sampling, all received signal data are collected at a specified time interval. Simple random sampling refers to every atom is randomly acquired. In this paper, the uniform random sampling (URS) method is taken as a form of non-uniform sampling. For example, if there needs to acquire M samples in period T , each one is acquired in fixed time interval t , t T / M . Thus, the sampling instant sequences can be formulated as
ti (i 1)* t (i 1, 2,
(3)
,M)
where satisfies uniform distribution between 0 and t . After down-conversion, the received signal y can be modeled as
yi x(ti )(i 1, 2,
III.
SIGNAL RECONSTRUCTION BASED O N SPARSE AR MODEL
A. The Construction of Sparse AR Basis The essential idea of sparse representation is to find a basis which can linearly approximate a given signal. Therefore, the selection of a basis that can sparsely represent the desired signal is crucial. However it relies on the degree of the fit between the signal and the basis. According to the recent research, there are two main approaches: the analytical approach and the learningbased approach. In the first approach, fixed-bases are commonly used, like discrete wavelet transform, the discrete cosine transform, and discrete Fourier transformation. The second method is to train the basis from a set of data so that its atoms can represent the features of the signal [24]. Based on the property of the transmitted signal, an adaptive basis is constructed to represent the received signal instead of using a predefined transform such as DFT [25]. We focus on a situation where the transmitted signal matches the AR (Auto Regressive) model: p
xn ak xn k
(4)
M)
The process can be expressed as y x , where is the {ij }(i 1, 2, , M ; j 1, 2, , N ) measurement matrix. In this case the atoms of measurement matrix are either “1” or “0”: only 1,t1 , 2,t2 , , M ,tM 1 while the others are all zero.
where xn is the output response, p is the order of the model. From (5), we can know that by using AR model, the future signals can be forecasted, expressed like
Therefore the compression sampling method proposed makes the hardware implementation in an easier way. An example of uniform random sampling signal samples in time domain is shown in Fig. 2.
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Figure 2. Uniform random sampling of a sine signal
The original sine signal is generated by generator GFG-8355A. There are 10 sparse samples uniform random sampled in one second by mono-chip computer. It only takes from a continuous signal to complete analog to digital conversion at sampling instants controlled by the Timer of ADuC7020. And with the same random seed, the system can generate the same sequence repeatedly. That is to say the measurement matrix will be the same which can improve efficiency during signal rebuilding.
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(6)
a p x p N 1
Suppose we have got a set of signal ( x1 , x2 , , xn )(n 2 N 1) which are normally sampled at Nyquist rate, then the extended segments ( xn1 , xn 2 , , xn N )( N p) can be expressed as
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k 1
cN xN cN xN 1
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cN x2 N 1
Equations (7) above can be written as xn 1 x1 xn 2 x2 xn N xN
x2 x3 xN 1
xN c1 xN 1 c2 x2 N 1 cN
(8)
We define
x in which x ( xn 1 , xn 2 , signal, (c1 , c2 ,
(9)
, xn N )T denotes the predicted
, cN )T is the projection coefficients
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min 1 s.t. y
vector, the matrix x1 x 2 xN
x2 x3 xN 1
xN xN 1 x2 N 1
(10)
From (5) we can know there are only p nonzero values in vector . So the projection coefficients vector is p-sparse. That is to say signal x can be sparsely represented by matrix . Here, we call the matrix ( N N ) sparse AR (SAR) basis. So the sparsity of the received signal can be well guaranteed with projecting to the SAR basis which is composed by prior signal components. Therefore it is feasible to recover the original signal with SAR basis constructed as (10) shown. B. Signal Reconstruction As depicted, the signal is non-uniform sampled with URS method and then the SAR basis is constructed to represent the received signal sparsely. Finally, the signal can be recovered successfully with optimization algorithms. Based on all the analysis, a compression sampling system based sparse AR model is developed in the framework of CS, shown in Fig. 3. The system includes two parts: the mono-chip module and the PC module. The hardware realizes function of uniform random sampling. The main work of software in PC is to achieve the reconstruction of the received signal.
(11)
where . is euclidean norm. For noisy signal, it can be expressed as the following form
arg min{ y 1} 2
(12)
where is the regularization parameter. The reconstruction signal is approximately given by x . IV.
SIMULATION
The compression sampling system based on sparse AR model has been described in previous parts. In this section we run numerical simulations to demonstrate the performance of the proposed sampling system in this paper using SAR basis compared with traditional predefined DFT basis. The sparse samples are collected using URS method. The reconstruction is obtained via interior-point method of L1 norm algorithm then. Here we set 0.01 . For objective quality assessment, we define criteria Signal Noise Ratio (SNR), Compression Ratio (CR) as
xT * x SNR 10*log10 xx 2 2 CR
M N
(13)
(14)
where x is the original signal, xr is the reconstruction signal, M is the number of samples, N is the length of signal x . We take a continuous vibration signal for example
x(t ) 5e 0.5t cos(200 t ) 7
(15)
4et cos(150 t ) 3e2t cos(100 t ) 5 3 Figure 3. The proposed compression sampling system
In this system, the measurement matrix is defined by sampling instant sequences ti . Then based on the matrix sparse samples y are uniform random sampled from the input signal x(t ) at a low rate, which has been explained in Section II. Then the coefficients vector of the received signal can be recovered by optimization algorithms. At last, the original signal is approximately reconstructed. It is important to note that the SAR basis has been constructed in advance as (10) shown which is composed by prior vibration signal components. Hence it will not burden the compression sampling system implementation. As we know the coefficients vector can be recovered using optimization algorithms. Here the L1 minimization algorithm is used to solve the underdetermined equation (2). The recovery of the coefficients vector can be formulated as © 2014 ACADEMY PUBLISHER
At first the original signal x(t ) is sampled at Nyquist rate f s 300Hz in 0-2s and 600 normal samples are collected to construct the SAR basis (300 300) as directed by (10). In the next third second 20 sparse samples are taken by URS. Here, M=20, N=300, the CR drops to about 7% (20/300). Fig. 4 displays the original signal and the uniform random samples. In Fig. 5 we show the reconstruction signals using SAR and DFT respectively recovered from the sparse samples. For better observation of the reconstruction performance, we just focus on the signal in 2-2.2s. For the SAR case we obtain a reconstruction SNR of about 60dB, meanwhile, reconstruction error can reach to the level of 104 . As a comparison, the result of reconstruction signal using DFT is presented in Fig. 5, too. The reconstruction SNR for the DFT case is about 10dB and the reconstruction error is much larger. So the SAR basis proposed can greatly raise the reconstruction accuracy.
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Finally, compression results obtained by averaging SNR values over the whole dataset are shown in Fig. 7. It can be noted the reconstruction SNR rises as the CR improves for both cases. But the reconstruction performance using SAR is much better than DFT. For CR = 10%, the reconstruction gain provided by SAR, in terms of SNR, is approximately 64dB, while the reconstruction SNR with DFT is just about 15dB. And it can also be noted that, using SAR, we are able to obtain the reconstruction quality about 50 dB over DFT on average. The results prove that the reconstruction performance of the SAR basis proposed degrades obviously compared with DFT especially in the situation of low sampling rate. Since the above simulation results demonstrate the proposed compression sampling system based on sparse AR model can obtain a good reconstruction quality using uniform random sampling method and sparse AR basis. 3 original signal samples
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Figure 7. Reconstruction SNR
V.
EXPERIMENT
For further demonstration the performance, the compression sampling system proposed is now illustrated with real vibration signal collected from the cable-stayed bridge model as Fig. 8 shown. The test model is made of metal material, which includes the main beam assembled together the main towers, cable-stayed, stand, pedestal and standoffs. The accelerometer ADXL335 is mounted on the main beam of the model. Use a hammer to stimulate the beam producing vibration signal and then the signal is acquired using the sampling circuits in Fig. 1 shown.
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Figure 4. Original signal and uniform random samples 3 original signal
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Figure 8. The cable-stayed bridge model 2
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Figure 6. Reconstruction error
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Figure 10. Reconstruction signals 1400 original signal SAR DFT
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Figure 11. FFT spectra 20
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the reconstruction signal using SAR basis proposed can still used for fault detection. Compression results obtained by averaging SNR values over the whole dataset of real vibration signal of bridge model are shown in Fig. 12. When CR is below 15%, with the increase of CR, the reconstruction SNR grows fast simultaneously. It is easy to see that the reconstruction SNR using SAR is much larger than DFT when CR is under 10%. Thus the proposed SAR basis has a stable and effective performance in the solution of low sampling rate which agrees with the simulation results. The proposed compression sampling system based on sparse AR model provides a simple but useful compression scheme. The experiment results demonstrate the proposed system is also applicable to the case of real situation. VI.
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CONCLUSION
In this paper, we introduced a compression sampling system based on sparse AR model. The sparse samples are non-uniform acquired using uniform random sampling method by mono-chip computer. SAR basis is constructed to represent the received signal sparsely and then signal reconstruction is obtained by optimization algorithm. Applying the URS method, it simplifies the design of hardware and reduces production costs. The sparse AR basis is applied to represent signal sparsely which can lead to efficient signal reconstruction. The emulational and experimental results showed the effectiveness of the approach in the terms of reconstruction accuracy (SNR) and compression ratio (CR). It also showed that sparse AR basis, due to capturing the property of vibration signal, obtained accurate reconstruction performance compared to DFT while keeping a high compression ratio. Hence the compression sampling system based on sparse AR model proposed can be useful to scale down the cost and energy consumption. REFERENCES
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Figure 12. Reconstruction SNR
Here we set N=300, M=15. The CR is 5% (15/300). First we collected 600 samples at Nyquist rate to construct the SAR basis. Then 15 sparse samples are acquired using URS in three seconds. Fig. 9 displays the original signal and uniform random samples. The reconstruction signals are shown in Fig. 10. We can see the SAR basis proposed can still recover the signal in good quality giving a reconstruction SNR of 16.6dB, while the DFT is only 6dB. Due to the noisy components and quantization error in real-world applications, the reconstruction performance of experiment will not be as accurate as the simulation. From the Fourier analysis (Fig. 11) it is clear that the real vibration signal is more complex than the simulation signal. But the spectrum of reconstruction signal using SAR is almost coincident with the original signal. Hence
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[22] [23]
[24]
[25]
Dandan Cheng received her B.E. degree in communication engineering from Anhui Normal University, China in June 2012. She is currently working towards her M.E. degree in electronics and communication engineering at Ningbo University, China. Her current research interests include vibration signal sampling and compression sensing. Qingwei Ye received the B.S. degree in applied physics from Ningbo University, China in 1992. He received his M.E. degree in computer application from Zhejiang University, China in 1999 and Ph.D. degree in instrument science and technology from Chongqing University, China in 2009. He is currently an associate professor in the Information Science and Engineering College, Ningbo University, China. His research interests include signal Processing, optimization algorithm and computer network. Xiaodong Wang received his B.S. degree in applied physics from Ningbo University, China in 1992. He received his M.E. degree in software engineering from Zhejiang University, China in 2007. He is currently an associate professor in the Information Science and Engineering College, Ningbo University, China. His research interests include computer network, multimedia information processing, multimedia communication and information security. Yu Zhou received his B.S. degree in physics from Shandong University, China in 1978. He received his M.S. degree from University of Electronic Science and Technology, China in 1986. He is currently a professor in the Information Science and Engineering College, Ningbo University, China. His research interests include Multimedia communication network, computer network and information security.
JOURNAL OF NETWORKS, VOL. 9, NO. 9, SEPTEMBER 2014
A Compression Sampling System based on Sparse AR Model Dandan Cheng, Qingwei Ye, Yu Zhou, and Xiaodong Wang Information Science and Engineering College, Ningbo University, Ningbo 315211, China Email: [email protected]
Abstract—A compression sampling system based on sparse AR (auto regression) model is designed in this paper. The sparse samples are non-uniform sampled using uniform random sampling (URS) method by mono-chip computer. To guarantee the reconstruction effectiveness, a basis matrix is constructed with prior signal information to represent the received signal sparsely. Then the received signal is recovered from the samples using optimization algorithms in PC. The URS method can scale down the sampling frequency effectively. The basis matrix is constructed based on the known AR model and named sparse AR (SAR) basis in this paper. Since the SAR basis is a self-adaptive basis, better reconstruction quality at a low sampling rate can be obtained. The performance of the proposed compression sampling system is illustrated using normal vibration signal. From both simulation and experiment, the transmitted signal can be reconstructed effectively and accurately. Index Terms—Compression Sampling; Sparse AR Model; Uniform Random Sampling; Signal Reconstruction
I.
INTRODUCTION
The Shannon/Nyquist sampling theorem specifies that to avoid losing information when capturing a signal, one must sample at least two times faster than the signal bandwidth. Nowadays, with the popularization of the real-time monitoring technology in many practical applications, such as global positioning system, communication system and structural health monitoring system, it brings great pressure to data storage and data transmission especially in the wireless sensor network. Like in structural health monitoring system, vibration signal provides useful information for diagnosis of structure failure. In order to identify the features of the fault, a significant amount of vibration data needs to be collected by sensors. It may face bottlenecks, due to limited transmission bandwidth and long-term transmission time. Thus, to minimize the storage and transmission load, it is vital to consider signal compression issue. Besides the data compression, it must ensure that the received signal can be recovered back without losing any information. Therefore, the main issue of a compression approach is able to achieve a high compression ratio while keeping a reasonable The work has been financially supported by National Nature Science Funds of China (No. 61071198), Nature Science Funds of Zhejiang Province (No. LY13F010015) and Innovation Project of Zhejiang Province Science and Technology Department (2012-R10009-08). © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.9.2536-2542
reconstruction performance. Besides the aforementioned performance criteria, the level of complexity is also a considerable factor. Compression sensing (CS) theory [1, 2] presents a new method to capture and represent compressible signal at a rate significantly below the Nyquist rate and then reconstructs back the original signal effectively using an optimization algorithm. Theory CS has been extensively treated in the literature and widely used in the field like GPS system [3], acoustic signals [4], electrocardiograph signals [5, 6]. The existing method for signal compression mainly focus on transformation methods (discrete Fourier transform [7], discrete cosine transform [8], and wavelet transform [9-11]), which transform the signal in the time domain to another domain and then compress the transform coefficients just transmitting a small portion of them. Among the transformation methods, wavelet transform has shown promising performance for speech signal due to its good localization properties [12]. But the work dedicated to vibration signal compression is rather limited [13, 14]. The major problem is the fixed basis because the selection process of the best basis is dominated by the vibration signal components [15]. The earlier work for signal compression mainly aimed at digital signal. To obtain compressed samples directly from an analog signal, analog-to-information converter (AIC) structure has been proposed [16]. It consists of several parallel branches of mixers and integrators, where each integrator multiplies the signal with a sampling waveform and the result is integrated. The effectiveness of AIC has been proved being used as a receiver of an ultra wideband signal system [17, 18]. However it is complex to implement for the complicated hardware system. In this paper, we propose a compression sampling system based on sparse AR model. The system includes two steps: signal compression sampling and signal reconstruction. First to simplify the requirement of hardware, the sparse samples are non-form sampled using uniform random sampling (URS) method by mono-chip computer. URS refers to each sample is taken random in a fixed time period adjacent. The URS method is with good advantages of low sampling rate and easy realization. Second a sparse AR (SAR) basis is designed to represent the received signal sparsely. The SAR basis is constructed based on the known AR (auto regression) model, which is composed with the prior signal
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components. For the SAR basis is adaptive and captures the property of vibration signal, thus the effectiveness of signal reconstruction can be guaranteed. At last with the interior-point method of L1 norm algorithm [19], the received signal can be successfully recovered from the sparse samples. As the compression sampling system using URS method and adaptive SAR basis, good reconstruction performance can be obtained preserving system simplicity and high compression ratio. In this paper, the performance of the proposed compression sampling system is demonstrated with normal continuous vibration signal from both theoretical and practical aspects respectively compared with exiting approaches. The remainder of the paper is organized as follows. Section II first gives an overview of theory CS and then provides an introduction to the architecture of hardware and proposes the URS method. Section III first proposes the approach to construct the SAR basis in details and then depicts the process of signal reconstruction. Section IV runs numerical simulations to illustrate the performance of the proposed compression system using URS sampling method and SAR basis compared with the popular DFT basis. Section V applies the proposed compression sampling system to a real vibration signal collected from a cable-stayed bridge model. Finally, conclusions are drawn in Section VI. II.
COMPRESSION S AMPLING
A. Background of Compression Sensing Compression sensing (CS) theory is one of hot issue in signal and information processing field. It indicates that if a signal is sparse, it can be reconstructed with overwhelming probability with optimization algorithms from the samples sampled at a rate significantly below the Nyquist rate. Any signal in R N can be represented in terms of a basis of N 1 vectors { i }iN1 . Using the N N basis matrix (1 , 2 , , N ) , a discrete-time signal x ( x1 , x2 ,
This is a dimension reduction process. The problem is to design a stable measurement matrix. According to CS, any matrix who obeys “restricted isometry principle” (RIP) can be used as a measurement matrix [20]. It is known that most of the random matrices satisfy the property “RIP”. It can be noted that the recovery of signal x is an NPhard problem for (2) is underdetermined. Some algorithms have been developed to solve the problem. One kind is the basis pursuit (BP) approach, which relaxes the L0 norm condition by the L1 norm and solves the problem through linear programming. The other is known as the greedy algorithm which approximates the signal through a sequence of incremental approximations by selecting atoms suitably. The greedy algorithms usually require the knowledge of sparsity in advance while BP algorithms can produce more accurate solutions [21]. Hence in this paper we use the L1 norm interiorpoint algorithm [22]. B. Compression Sampling 1) Architecture of Sampling System Theory CS provides a novel framework to process sparse signals. In this section, we develop the architecture of hardware for vibration signal compression sampling. At present, structure AIC is usually used to collect analog signal whose components mainly include mixers, integrators and ADCS. This part proposes a structural diagram of the circuit for the realization of the compression sampling, which only consists of accelerometer ADXL335 and mono-chip computer ADuC7020 as shown in Fig. 1 simplifying the hardware requirement. The ADuC7020 is fully integrated, 1 MSPS, 12 bit data acquisition system incorporation a high performance multichannel ADC on a signal chip. It contains 16 ADC channels and supports in circuit serial download via the UART, widely applied in industrial control, automation systems and smart sensors.
, xN )T can be expressed as N
x i i
(1)
i 1
where (1 ,2 , , N )T is the coefficients vector. Obviously, x and are the equivalent representations of the signal, with x in the time domain and in the domain. If vector has only K ( K N ) nonzero values, the signal x is called K-sparse signal with respecting to the matrix . In some ways, signal x can be considered compressible if vector has just a few large coefficients and many small coefficients. Most signals are sparse with respecting to special basis in practice. In CS, the signal x is not acquired directly, but measured by a measurement matrix RM N M N . The measurements y can be expressed as followed
y x
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(2)
Figure 1. Accelerometer and micro-controller
In this scheme, an input signal from accelerometer is projected onto sparse signal by a set of pulse signals which are generated by the Timer of ADuC7020. Next the ADC is used to sample and quantize the projection signals, producing the sparse vibration signal samples. With the proposed sampling diagram, there is no need to store the measurement matrix in hardware system beforehand, which simplifies the mixers of AIC structure.
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The compressive samples are selected by fixed time control, easy to implement. 2) Uniform Random Sampling According to CS theory, the signal received should be sampled at a low rate and then recovered successfully by optimization algorithms. The core of the compression sampling is to realize the projection of multidimensional input signal onto a lower dimensional subspace [23]. In the case of uniform sampling, all received signal data are collected at a specified time interval. Simple random sampling refers to every atom is randomly acquired. In this paper, the uniform random sampling (URS) method is taken as a form of non-uniform sampling. For example, if there needs to acquire M samples in period T , each one is acquired in fixed time interval t , t T / M . Thus, the sampling instant sequences can be formulated as
ti (i 1)* t (i 1, 2,
(3)
,M)
where satisfies uniform distribution between 0 and t . After down-conversion, the received signal y can be modeled as
yi x(ti )(i 1, 2,
III.
SIGNAL RECONSTRUCTION BASED O N SPARSE AR MODEL
A. The Construction of Sparse AR Basis The essential idea of sparse representation is to find a basis which can linearly approximate a given signal. Therefore, the selection of a basis that can sparsely represent the desired signal is crucial. However it relies on the degree of the fit between the signal and the basis. According to the recent research, there are two main approaches: the analytical approach and the learningbased approach. In the first approach, fixed-bases are commonly used, like discrete wavelet transform, the discrete cosine transform, and discrete Fourier transformation. The second method is to train the basis from a set of data so that its atoms can represent the features of the signal [24]. Based on the property of the transmitted signal, an adaptive basis is constructed to represent the received signal instead of using a predefined transform such as DFT [25]. We focus on a situation where the transmitted signal matches the AR (Auto Regressive) model: p
xn ak xn k
(4)
M)
The process can be expressed as y x , where is the {ij }(i 1, 2, , M ; j 1, 2, , N ) measurement matrix. In this case the atoms of measurement matrix are either “1” or “0”: only 1,t1 , 2,t2 , , M ,tM 1 while the others are all zero.
where xn is the output response, p is the order of the model. From (5), we can know that by using AR model, the future signals can be forecasted, expressed like
Therefore the compression sampling method proposed makes the hardware implementation in an easier way. An example of uniform random sampling signal samples in time domain is shown in Fig. 2.
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x p 1 a1 x1 a2 x2
ap xp
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Figure 2. Uniform random sampling of a sine signal
The original sine signal is generated by generator GFG-8355A. There are 10 sparse samples uniform random sampled in one second by mono-chip computer. It only takes from a continuous signal to complete analog to digital conversion at sampling instants controlled by the Timer of ADuC7020. And with the same random seed, the system can generate the same sequence repeatedly. That is to say the measurement matrix will be the same which can improve efficiency during signal rebuilding.
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(6)
a p x p N 1
Suppose we have got a set of signal ( x1 , x2 , , xn )(n 2 N 1) which are normally sampled at Nyquist rate, then the extended segments ( xn1 , xn 2 , , xn N )( N p) can be expressed as
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cN xN cN xN 1
(7)
cN x2 N 1
Equations (7) above can be written as xn 1 x1 xn 2 x2 xn N xN
x2 x3 xN 1
xN c1 xN 1 c2 x2 N 1 cN
(8)
We define
x in which x ( xn 1 , xn 2 , signal, (c1 , c2 ,
(9)
, xn N )T denotes the predicted
, cN )T is the projection coefficients
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min 1 s.t. y
vector, the matrix x1 x 2 xN
x2 x3 xN 1
xN xN 1 x2 N 1
(10)
From (5) we can know there are only p nonzero values in vector . So the projection coefficients vector is p-sparse. That is to say signal x can be sparsely represented by matrix . Here, we call the matrix ( N N ) sparse AR (SAR) basis. So the sparsity of the received signal can be well guaranteed with projecting to the SAR basis which is composed by prior signal components. Therefore it is feasible to recover the original signal with SAR basis constructed as (10) shown. B. Signal Reconstruction As depicted, the signal is non-uniform sampled with URS method and then the SAR basis is constructed to represent the received signal sparsely. Finally, the signal can be recovered successfully with optimization algorithms. Based on all the analysis, a compression sampling system based sparse AR model is developed in the framework of CS, shown in Fig. 3. The system includes two parts: the mono-chip module and the PC module. The hardware realizes function of uniform random sampling. The main work of software in PC is to achieve the reconstruction of the received signal.
(11)
where . is euclidean norm. For noisy signal, it can be expressed as the following form
arg min{ y 1} 2
(12)
where is the regularization parameter. The reconstruction signal is approximately given by x . IV.
SIMULATION
The compression sampling system based on sparse AR model has been described in previous parts. In this section we run numerical simulations to demonstrate the performance of the proposed sampling system in this paper using SAR basis compared with traditional predefined DFT basis. The sparse samples are collected using URS method. The reconstruction is obtained via interior-point method of L1 norm algorithm then. Here we set 0.01 . For objective quality assessment, we define criteria Signal Noise Ratio (SNR), Compression Ratio (CR) as
xT * x SNR 10*log10 xx 2 2 CR
M N
(13)
(14)
where x is the original signal, xr is the reconstruction signal, M is the number of samples, N is the length of signal x . We take a continuous vibration signal for example
x(t ) 5e 0.5t cos(200 t ) 7
(15)
4et cos(150 t ) 3e2t cos(100 t ) 5 3 Figure 3. The proposed compression sampling system
In this system, the measurement matrix is defined by sampling instant sequences ti . Then based on the matrix sparse samples y are uniform random sampled from the input signal x(t ) at a low rate, which has been explained in Section II. Then the coefficients vector of the received signal can be recovered by optimization algorithms. At last, the original signal is approximately reconstructed. It is important to note that the SAR basis has been constructed in advance as (10) shown which is composed by prior vibration signal components. Hence it will not burden the compression sampling system implementation. As we know the coefficients vector can be recovered using optimization algorithms. Here the L1 minimization algorithm is used to solve the underdetermined equation (2). The recovery of the coefficients vector can be formulated as © 2014 ACADEMY PUBLISHER
At first the original signal x(t ) is sampled at Nyquist rate f s 300Hz in 0-2s and 600 normal samples are collected to construct the SAR basis (300 300) as directed by (10). In the next third second 20 sparse samples are taken by URS. Here, M=20, N=300, the CR drops to about 7% (20/300). Fig. 4 displays the original signal and the uniform random samples. In Fig. 5 we show the reconstruction signals using SAR and DFT respectively recovered from the sparse samples. For better observation of the reconstruction performance, we just focus on the signal in 2-2.2s. For the SAR case we obtain a reconstruction SNR of about 60dB, meanwhile, reconstruction error can reach to the level of 104 . As a comparison, the result of reconstruction signal using DFT is presented in Fig. 5, too. The reconstruction SNR for the DFT case is about 10dB and the reconstruction error is much larger. So the SAR basis proposed can greatly raise the reconstruction accuracy.
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Finally, compression results obtained by averaging SNR values over the whole dataset are shown in Fig. 7. It can be noted the reconstruction SNR rises as the CR improves for both cases. But the reconstruction performance using SAR is much better than DFT. For CR = 10%, the reconstruction gain provided by SAR, in terms of SNR, is approximately 64dB, while the reconstruction SNR with DFT is just about 15dB. And it can also be noted that, using SAR, we are able to obtain the reconstruction quality about 50 dB over DFT on average. The results prove that the reconstruction performance of the SAR basis proposed degrades obviously compared with DFT especially in the situation of low sampling rate. Since the above simulation results demonstrate the proposed compression sampling system based on sparse AR model can obtain a good reconstruction quality using uniform random sampling method and sparse AR basis. 3 original signal samples
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Figure 7. Reconstruction SNR
V.
EXPERIMENT
For further demonstration the performance, the compression sampling system proposed is now illustrated with real vibration signal collected from the cable-stayed bridge model as Fig. 8 shown. The test model is made of metal material, which includes the main beam assembled together the main towers, cable-stayed, stand, pedestal and standoffs. The accelerometer ADXL335 is mounted on the main beam of the model. Use a hammer to stimulate the beam producing vibration signal and then the signal is acquired using the sampling circuits in Fig. 1 shown.
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Figure 4. Original signal and uniform random samples 3 original signal
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Figure 8. The cable-stayed bridge model 2
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Figure 6. Reconstruction error
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Figure 10. Reconstruction signals 1400 original signal SAR DFT
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the reconstruction signal using SAR basis proposed can still used for fault detection. Compression results obtained by averaging SNR values over the whole dataset of real vibration signal of bridge model are shown in Fig. 12. When CR is below 15%, with the increase of CR, the reconstruction SNR grows fast simultaneously. It is easy to see that the reconstruction SNR using SAR is much larger than DFT when CR is under 10%. Thus the proposed SAR basis has a stable and effective performance in the solution of low sampling rate which agrees with the simulation results. The proposed compression sampling system based on sparse AR model provides a simple but useful compression scheme. The experiment results demonstrate the proposed system is also applicable to the case of real situation. VI.
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CONCLUSION
In this paper, we introduced a compression sampling system based on sparse AR model. The sparse samples are non-uniform acquired using uniform random sampling method by mono-chip computer. SAR basis is constructed to represent the received signal sparsely and then signal reconstruction is obtained by optimization algorithm. Applying the URS method, it simplifies the design of hardware and reduces production costs. The sparse AR basis is applied to represent signal sparsely which can lead to efficient signal reconstruction. The emulational and experimental results showed the effectiveness of the approach in the terms of reconstruction accuracy (SNR) and compression ratio (CR). It also showed that sparse AR basis, due to capturing the property of vibration signal, obtained accurate reconstruction performance compared to DFT while keeping a high compression ratio. Hence the compression sampling system based on sparse AR model proposed can be useful to scale down the cost and energy consumption. REFERENCES
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5
10 CR
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Figure 12. Reconstruction SNR
Here we set N=300, M=15. The CR is 5% (15/300). First we collected 600 samples at Nyquist rate to construct the SAR basis. Then 15 sparse samples are acquired using URS in three seconds. Fig. 9 displays the original signal and uniform random samples. The reconstruction signals are shown in Fig. 10. We can see the SAR basis proposed can still recover the signal in good quality giving a reconstruction SNR of 16.6dB, while the DFT is only 6dB. Due to the noisy components and quantization error in real-world applications, the reconstruction performance of experiment will not be as accurate as the simulation. From the Fourier analysis (Fig. 11) it is clear that the real vibration signal is more complex than the simulation signal. But the spectrum of reconstruction signal using SAR is almost coincident with the original signal. Hence
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[22] [23]
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[25]
Dandan Cheng received her B.E. degree in communication engineering from Anhui Normal University, China in June 2012. She is currently working towards her M.E. degree in electronics and communication engineering at Ningbo University, China. Her current research interests include vibration signal sampling and compression sensing. Qingwei Ye received the B.S. degree in applied physics from Ningbo University, China in 1992. He received his M.E. degree in computer application from Zhejiang University, China in 1999 and Ph.D. degree in instrument science and technology from Chongqing University, China in 2009. He is currently an associate professor in the Information Science and Engineering College, Ningbo University, China. His research interests include signal Processing, optimization algorithm and computer network. Xiaodong Wang received his B.S. degree in applied physics from Ningbo University, China in 1992. He received his M.E. degree in software engineering from Zhejiang University, China in 2007. He is currently an associate professor in the Information Science and Engineering College, Ningbo University, China. His research interests include computer network, multimedia information processing, multimedia communication and information security. Yu Zhou received his B.S. degree in physics from Shandong University, China in 1978. He received his M.S. degree from University of Electronic Science and Technology, China in 1986. He is currently a professor in the Information Science and Engineering College, Ningbo University, China. His research interests include Multimedia communication network, computer network and information security.
