基于欠采样的信号重建算法研究
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摘要
数字信号处理的一般过程是先把模拟信号转换成数字信号,然后对数字信号进行处理,最后把处理的结果再转换成模拟信号。上述过程包含了两个方面的内容:一是对信号的采样;二是原信号的重建。
传统Shannon采样定理解决了带限信号的采样与重建问题:若用不低于信号对应最高频率二倍(Nyquist率)的采样率对带限信号进行采样,则原信号可以用采样后的离散序列唯一的表示。根据Shannon采样定理,完全重建超宽带信号需要很高的采样率,而有时高采样率是难以实现的。对于实际中经常遇到的非带限信号,如紧支信号(时限信号),根据Shannon采样定理是无法进行采样和重建的。论文围绕着信号的采样和重建展开,主要研究如下两个问题:
1、对于带限信号,本文研究了在采样率低于Nyquist率——称之为欠采样的情况下进行采样与重建的两种方法:构造冲激采样脉冲序列法和导数采样法,仿真试验结果证明了两种方法所得结论的正确性。
2、对于非带限信号,本文利用小波采样理论对尺度函数取B样条函数时的紧支信号的采样和重建展开分析,得到了类似于Shannon采样定理中的信号重建形式。对分析过程中的一些不足之处,利用小波空间中的非均匀采样法和导数采样法进行了改进,通过仿真验证了小波采样理论对处理紧支信号的有效性和改进方法的可行性。
最后研究了几种和本文相关的信号采样和重建方法,从理论推导和实例说明两方面讨论了如何在欠采样情况下通过这些方法实现原信号的重建。
In digital signal processing applications, the first step is to convert analog signals to digital signals, then, the digital signals are processed, and finally, the processing results are converted back to analog signals. The above process involves two contents: the first one is sampling; and the other one is reconstruction of the original signals.
Shannon sampling theorem resolves sampling and reconstruction problem for the band-limited signals. A continuous time signal can be exactly reconstructed from its discrete samples if the sampling rate is higher than the Nyquist rate. However, for reconstruction of the UWB signal according to Shannon sampling theorem, the required sampling rate is very high. Such high sampling rate is sometimes very difficult to achieve. For nonband-limited signals, e.g., time-limited signals, which are usually used in practice, we can not sample and reconstruct them according to the Shannon sampling theorem. In this paper, we focus on the signal sampling and reconstruction to investigate the following two problems.
Firstly, we investigate two methods for sampling and reconstruction under the condition that the sampling rate is lower than the Nyquist rate, which is termed as undersampling case. The first method is constructing a impulse sampling sequence method and the second method is derivative sampling method. Theorem analysis and computation simulation results show the effectiveness of these two methods.
Secondly, we investigate the sampling and reconstruction for the compactly supported signal using wavelet sampling theorem. We obtain the signal reconstruction formulation which is similar to the form of Shannon sampling theorem. We also propose to use the nonuniform sampling and derivative sampling in wavelet space to improve the disadvantages in the process. Simulation results are presented to verify the efficacy of the improved method.
Finally, we investigate several methods for sampling and reconstruction of signal that related to this paper. We discuss the method to reconstruct original signal under the undersampling cases via theory analysis and examples.
传统Shannon采样定理解决了带限信号的采样与重建问题:若用不低于信号对应最高频率二倍(Nyquist率)的采样率对带限信号进行采样,则原信号可以用采样后的离散序列唯一的表示。根据Shannon采样定理,完全重建超宽带信号需要很高的采样率,而有时高采样率是难以实现的。对于实际中经常遇到的非带限信号,如紧支信号(时限信号),根据Shannon采样定理是无法进行采样和重建的。论文围绕着信号的采样和重建展开,主要研究如下两个问题:
1、对于带限信号,本文研究了在采样率低于Nyquist率——称之为欠采样的情况下进行采样与重建的两种方法:构造冲激采样脉冲序列法和导数采样法,仿真试验结果证明了两种方法所得结论的正确性。
2、对于非带限信号,本文利用小波采样理论对尺度函数取B样条函数时的紧支信号的采样和重建展开分析,得到了类似于Shannon采样定理中的信号重建形式。对分析过程中的一些不足之处,利用小波空间中的非均匀采样法和导数采样法进行了改进,通过仿真验证了小波采样理论对处理紧支信号的有效性和改进方法的可行性。
最后研究了几种和本文相关的信号采样和重建方法,从理论推导和实例说明两方面讨论了如何在欠采样情况下通过这些方法实现原信号的重建。
In digital signal processing applications, the first step is to convert analog signals to digital signals, then, the digital signals are processed, and finally, the processing results are converted back to analog signals. The above process involves two contents: the first one is sampling; and the other one is reconstruction of the original signals.
Shannon sampling theorem resolves sampling and reconstruction problem for the band-limited signals. A continuous time signal can be exactly reconstructed from its discrete samples if the sampling rate is higher than the Nyquist rate. However, for reconstruction of the UWB signal according to Shannon sampling theorem, the required sampling rate is very high. Such high sampling rate is sometimes very difficult to achieve. For nonband-limited signals, e.g., time-limited signals, which are usually used in practice, we can not sample and reconstruct them according to the Shannon sampling theorem. In this paper, we focus on the signal sampling and reconstruction to investigate the following two problems.
Firstly, we investigate two methods for sampling and reconstruction under the condition that the sampling rate is lower than the Nyquist rate, which is termed as undersampling case. The first method is constructing a impulse sampling sequence method and the second method is derivative sampling method. Theorem analysis and computation simulation results show the effectiveness of these two methods.
Secondly, we investigate the sampling and reconstruction for the compactly supported signal using wavelet sampling theorem. We obtain the signal reconstruction formulation which is similar to the form of Shannon sampling theorem. We also propose to use the nonuniform sampling and derivative sampling in wavelet space to improve the disadvantages in the process. Simulation results are presented to verify the efficacy of the improved method.
Finally, we investigate several methods for sampling and reconstruction of signal that related to this paper. We discuss the method to reconstruct original signal under the undersampling cases via theory analysis and examples.
引文
[1]C.E.Shannon,"Communication in the presence of noise," Proc.IRE,Vol.37,pp.10-21,Jan.1949
[2]P.L.Butzer,"A survey of the Whittaker-Shannon sampling theorem and some of its extension",J.Math.Res.Exposition,3(1983).185-212
[3]H.Nyquist,"Certain topics in telegraph transmission theory," AIEE Trans.,vol.47,pp.617-644,Jan.1928
[4]H.L.Landau,"Sampling,data transmission and the Nyquist rate,"Proc.IEEE,vol.55,pp.1701-1706,1967
[5]D.L.Jagerman and L.Fogel,"Some general aspects of the sampling theorem," IRE Trans.Inform.Theory,vol.IT-2,pp.139-146,Dec.1956.
[6]N.T.Gaarder,"A note on multidimensional sampling theorem," Proc.IEEE,Vol.60,pp.247-248,Feb.1972.
[7]E.T.Whittaker," On the functions which are represented by the expansions of the interpolation-theorey," Proc.Roy.Soc.,Edinburgy,Vol.35,1915
[8]V.A.Kotel'nikov,"On the transmission capacity of 'ether' and wire in electro-communications",First All-Union Conf.Questions of Commun.,Jan.1933
[9]A.J.Jerri,"The Shannon sampling theorem-Its various extensions and applications:A tutorial review," Proc.IEEE,Vol.65,No.11,pp.1565-1598,November,1977
[10]A.Papoulis,"Generalized sampling expansion," IEEE Trans.Circuits Syst.,Vol.24,pp.652-654,Nov.1977
[11]G.G.Walter,"A sampling theorem for wavelet subspaces," IEEE Trans.Inform.Theory,Vol.38,pp.881-884,Mar.1992.
[12]X.G.Xia and Z.Zhang,"On sampling theorem,wavelet,and wavelet transforms",IEEE Trans.On Signal Processing,41(12),pp.3524-3525,Dec.1993
[13]孙文昌,周性伟.向量小波子空间上的采样定理.科学通报.1999,44(3):262-265
[14]D.A.Linden,"A discussion of sampling theorems," Proc.IRE,Vol.47,pp.1219-1226,1959
[15]冉启文.小波变换与分数傅里叶变换理论及应用.哈尔滨:哈尔滨工业大学出版社,2001
[16]宗孔德.多抽样率信号处理.北京:清华大学出版社,1996
[17]吴大正,杨林耀,张永瑞.信号与线性系统分析.(第三版).北京:高等教育出版 社.1998
[18]和卫星,许波.信号与系统分析.西安:西安电子科技大学出版社.2007.3
[19]N.T.Gaarder,"A note on multidimensional sampling theorem," Proc.IEEE,Vol.60,pp.247-248,Feb.1972
[20]W.H.Nicholson,A.A.Sakla.The sampling and full reconstruction of band-limited signals at sub-Nyquist rate.U.S.patent No.CH2868-8/90/000,pp.1796-1800,Aug.1990
[21]J.L.Brown Jr.and S.D.Cabrera,"On well-posedness of the Papoulis generalized sampling expansion," IEEE Trans.Circuits Syst.,Vol.38,no.5,pp.554-556,1991
[22]J.L.Yen,"On the nonuniform sampling of bandwidth limited signals," IRE Trans.Circuit Theory,Vol.CT-3,pp.251-257,1956
[23]李衍达,常迥.信号重构理论及其应用.北京:清华大学出版社.1991
[24]成礼智,王红霞,罗永.小波的理论与应用.北京:科学出版社.2004
[25]A.Papoulis,"Error Analysis in sampling theory," Proc.IEEE,Vol.54,pp.947-955,July.1996
[26]Panlo Jorge,S.G.Ferreira,"Nonuniform sampling of nonbandlimited signals",IEEE Signal Processing Letters,Vol.2,No.5,pp.89-91,May.1995
[27]胡广书.正弦信号抽样中若干基本问题的讨论.清华大学学报(自然科学版),1997(1):74-77
[28]余越,柯有安.非均匀子波空间采样定理.电子学报,1997.7(7)1-6
[29]Unser.M.,Aldroubi.A.,and Eden.M.,"B-spline signal processing:Part Ⅰ-Theory",IEEE Trans.Signal Proc.,Vol.41,No.2,pp.821-833,Feb.1993
[30]Unser.M.,Aldroubi.A.,and Eden.M.,"B-spline signal processing:PartⅡ-Efficient design and applications," IEEE Trans.Signal Proc.,Vol.41,No.2,pp.834-848,Feb.1993
[31]余越,柯有安.具有紧支采样函数的子波空间采样定理.信号处理,1997.3(13)12-20
[32]S.Mallat,"A theory for multiresolution signal decomposition:the wavelet representation," IEEE Trans.Pattern Anal.Machine Intell.,pp.674-693,July 1989
[33]P.P.Vaidyanathan and S.M.Phoong,"Reconstruction of sequences from nonuniform samples," IEEE Proc.Int.Symp.Circuits and Systems,pp.601-604,Apr-May 1995
[34]P.P.Vaidyanathan,"Generalization of the sampling theorem:Seven Decades after Nyquist," IEEE Trans.,Vol.48,No.9,pp.1094-1109,Sep.2001
[35]Michael Unser,"Sampling-50 years after Shannon",Proceeding of the IEEE,Vol.88,No.4,pp.569-587,Apr.2000
[36]郑治真,沈萍,杨选辉.小波变换及其MATLAB工具的应用.北京:地震出版社.2001.10
[37]冯象初,甘小冰,宋国乡.数值泛函与小波理论.西安:西安电子科技大学出版社.2003.3
[38]林从容,贾平,谢求成.Shannon采样定理的由来和一些数学方面的发展.遥测遥控,1998.5(13):41-46
[39]张建康,保铮,焦李成.Shannon采样点到Daubechies小波采样点的转换.通信学报,2000.9(9)58-61
[40]刘立祥,谢剑英,张敬辕.非均匀采样信号的一种重建方法.信号处理,2000.12(6):543-546
[41]冯锦,任长学.可分解时域信号的采样分析.通信学报.2002(8):45-51
[42]李落清.多维非带限信号的抽样定理.湖北大学学报.1997(6):106-110
[43]Grigore Braileanu,"Exploding the Nyquist barrier misconception",IEEE Signal Proceeding Magzine,pp32-42,Sep.1996
[44]F.Marvasti,"Nonuniform sampling theorem for bandpass signals at or below the Nyquist density",IEEE Transactions on Signal Processing,Vol.44,No.3,pp.572-576,Mar.1996
[45]A.Nathan,"On sampling a function and its derivatives," Information and Control,Academic Press,Vol.22,pp.172-182,1973
[46]R.J.Marks,"Restoration of a continuously sampled band-limited signal from aliased data," IEEE ASSP,Vol.ASSP-30,No.5,pp.937-942,December 1982
[47]K.F.Cheung,and R.J.Marks,"Ill-posed sampling theorems," IEEE Trans.Ckts.Sys.,Vol.CAS-32,No.5,May 1985
[48]P.E.Pace,R.E.Leino,and D.Styer,"Use of the symmetrical number system in resolving single-frequency undersampling aliases," IEEE Trans.SP,Vol.45,No.5,pp.1153-1160,May 1997
[49]Y.C.Eldar and A.V.Oppenheim,Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples,IEEE Trans.Signal Processing,48(10):2864-2875,October 2000
[50]M.B.Matthews,"On the linear minimum-mean-squared-error estimation of an undersampled wide-sense stationary random process," IEEE Trans.SP,Vol.48,No.1,pp.272-275,January 2000
[2]P.L.Butzer,"A survey of the Whittaker-Shannon sampling theorem and some of its extension",J.Math.Res.Exposition,3(1983).185-212
[3]H.Nyquist,"Certain topics in telegraph transmission theory," AIEE Trans.,vol.47,pp.617-644,Jan.1928
[4]H.L.Landau,"Sampling,data transmission and the Nyquist rate,"Proc.IEEE,vol.55,pp.1701-1706,1967
[5]D.L.Jagerman and L.Fogel,"Some general aspects of the sampling theorem," IRE Trans.Inform.Theory,vol.IT-2,pp.139-146,Dec.1956.
[6]N.T.Gaarder,"A note on multidimensional sampling theorem," Proc.IEEE,Vol.60,pp.247-248,Feb.1972.
[7]E.T.Whittaker," On the functions which are represented by the expansions of the interpolation-theorey," Proc.Roy.Soc.,Edinburgy,Vol.35,1915
[8]V.A.Kotel'nikov,"On the transmission capacity of 'ether' and wire in electro-communications",First All-Union Conf.Questions of Commun.,Jan.1933
[9]A.J.Jerri,"The Shannon sampling theorem-Its various extensions and applications:A tutorial review," Proc.IEEE,Vol.65,No.11,pp.1565-1598,November,1977
[10]A.Papoulis,"Generalized sampling expansion," IEEE Trans.Circuits Syst.,Vol.24,pp.652-654,Nov.1977
[11]G.G.Walter,"A sampling theorem for wavelet subspaces," IEEE Trans.Inform.Theory,Vol.38,pp.881-884,Mar.1992.
[12]X.G.Xia and Z.Zhang,"On sampling theorem,wavelet,and wavelet transforms",IEEE Trans.On Signal Processing,41(12),pp.3524-3525,Dec.1993
[13]孙文昌,周性伟.向量小波子空间上的采样定理.科学通报.1999,44(3):262-265
[14]D.A.Linden,"A discussion of sampling theorems," Proc.IRE,Vol.47,pp.1219-1226,1959
[15]冉启文.小波变换与分数傅里叶变换理论及应用.哈尔滨:哈尔滨工业大学出版社,2001
[16]宗孔德.多抽样率信号处理.北京:清华大学出版社,1996
[17]吴大正,杨林耀,张永瑞.信号与线性系统分析.(第三版).北京:高等教育出版 社.1998
[18]和卫星,许波.信号与系统分析.西安:西安电子科技大学出版社.2007.3
[19]N.T.Gaarder,"A note on multidimensional sampling theorem," Proc.IEEE,Vol.60,pp.247-248,Feb.1972
[20]W.H.Nicholson,A.A.Sakla.The sampling and full reconstruction of band-limited signals at sub-Nyquist rate.U.S.patent No.CH2868-8/90/000,pp.1796-1800,Aug.1990
[21]J.L.Brown Jr.and S.D.Cabrera,"On well-posedness of the Papoulis generalized sampling expansion," IEEE Trans.Circuits Syst.,Vol.38,no.5,pp.554-556,1991
[22]J.L.Yen,"On the nonuniform sampling of bandwidth limited signals," IRE Trans.Circuit Theory,Vol.CT-3,pp.251-257,1956
[23]李衍达,常迥.信号重构理论及其应用.北京:清华大学出版社.1991
[24]成礼智,王红霞,罗永.小波的理论与应用.北京:科学出版社.2004
[25]A.Papoulis,"Error Analysis in sampling theory," Proc.IEEE,Vol.54,pp.947-955,July.1996
[26]Panlo Jorge,S.G.Ferreira,"Nonuniform sampling of nonbandlimited signals",IEEE Signal Processing Letters,Vol.2,No.5,pp.89-91,May.1995
[27]胡广书.正弦信号抽样中若干基本问题的讨论.清华大学学报(自然科学版),1997(1):74-77
[28]余越,柯有安.非均匀子波空间采样定理.电子学报,1997.7(7)1-6
[29]Unser.M.,Aldroubi.A.,and Eden.M.,"B-spline signal processing:Part Ⅰ-Theory",IEEE Trans.Signal Proc.,Vol.41,No.2,pp.821-833,Feb.1993
[30]Unser.M.,Aldroubi.A.,and Eden.M.,"B-spline signal processing:PartⅡ-Efficient design and applications," IEEE Trans.Signal Proc.,Vol.41,No.2,pp.834-848,Feb.1993
[31]余越,柯有安.具有紧支采样函数的子波空间采样定理.信号处理,1997.3(13)12-20
[32]S.Mallat,"A theory for multiresolution signal decomposition:the wavelet representation," IEEE Trans.Pattern Anal.Machine Intell.,pp.674-693,July 1989
[33]P.P.Vaidyanathan and S.M.Phoong,"Reconstruction of sequences from nonuniform samples," IEEE Proc.Int.Symp.Circuits and Systems,pp.601-604,Apr-May 1995
[34]P.P.Vaidyanathan,"Generalization of the sampling theorem:Seven Decades after Nyquist," IEEE Trans.,Vol.48,No.9,pp.1094-1109,Sep.2001
[35]Michael Unser,"Sampling-50 years after Shannon",Proceeding of the IEEE,Vol.88,No.4,pp.569-587,Apr.2000
[36]郑治真,沈萍,杨选辉.小波变换及其MATLAB工具的应用.北京:地震出版社.2001.10
[37]冯象初,甘小冰,宋国乡.数值泛函与小波理论.西安:西安电子科技大学出版社.2003.3
[38]林从容,贾平,谢求成.Shannon采样定理的由来和一些数学方面的发展.遥测遥控,1998.5(13):41-46
[39]张建康,保铮,焦李成.Shannon采样点到Daubechies小波采样点的转换.通信学报,2000.9(9)58-61
[40]刘立祥,谢剑英,张敬辕.非均匀采样信号的一种重建方法.信号处理,2000.12(6):543-546
[41]冯锦,任长学.可分解时域信号的采样分析.通信学报.2002(8):45-51
[42]李落清.多维非带限信号的抽样定理.湖北大学学报.1997(6):106-110
[43]Grigore Braileanu,"Exploding the Nyquist barrier misconception",IEEE Signal Proceeding Magzine,pp32-42,Sep.1996
[44]F.Marvasti,"Nonuniform sampling theorem for bandpass signals at or below the Nyquist density",IEEE Transactions on Signal Processing,Vol.44,No.3,pp.572-576,Mar.1996
[45]A.Nathan,"On sampling a function and its derivatives," Information and Control,Academic Press,Vol.22,pp.172-182,1973
[46]R.J.Marks,"Restoration of a continuously sampled band-limited signal from aliased data," IEEE ASSP,Vol.ASSP-30,No.5,pp.937-942,December 1982
[47]K.F.Cheung,and R.J.Marks,"Ill-posed sampling theorems," IEEE Trans.Ckts.Sys.,Vol.CAS-32,No.5,May 1985
[48]P.E.Pace,R.E.Leino,and D.Styer,"Use of the symmetrical number system in resolving single-frequency undersampling aliases," IEEE Trans.SP,Vol.45,No.5,pp.1153-1160,May 1997
[49]Y.C.Eldar and A.V.Oppenheim,Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples,IEEE Trans.Signal Processing,48(10):2864-2875,October 2000
[50]M.B.Matthews,"On the linear minimum-mean-squared-error estimation of an undersampled wide-sense stationary random process," IEEE Trans.SP,Vol.48,No.1,pp.272-275,January 2000