Sunzi Theorem and Signal Processing Xiang-Gen Xia Dept. of Electrical & Computer Engineering University of Delaware Newark, DE

Outline Sunzi (Sun Tzu) also known as Sun Wu Sunzi Theorem (also known as Chinese Remainder Theorem (CRT)) Sunzi Theorem and Signal Processing (Phase Unwrapping) A Generalized CRT Robust Phase Unwrapping and A Robust CRT Applications in SAR Imaging of Moving Targets –Multi-wavelength antenna array Synthetic Aperture Radar (SAR) –Non-uniformly spaced antenna array SAR –Non-uniform speed antenna array SAR Conclusion

Sunzi ( 孙子, Sun Tzu, 孙武, Sun Wu) Sunzi (544 BC-498 BC, about 2500 years ago), born in Guang Dong, China, served the state of Wu. Sunzi authored the internationally famous book The Art of War (Ancient Chinese book on military strategy). 孙子兵法 Portraits

Sunzi Theorem (Chinese Remainder Theorem) The following problem was posed by Sunzi in the book Sunzi Suanjing: There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? The answer is hidden in a Sunzi song named Sunzi Theorem later (also universally known as Chinese Remainder Theorem (CRT)) that gives the conditions necessary for multiple equations to have a simultaneous integer solution: He first determined the 'use numbers' 70, 21 and 15 which are multiples of 5*7, 3*7 and 3*5, respectively. Next, he noted that the sum (2*70)+(3*21)+(2*15) equals to 233. Thus 233 is one answer. He then casted out a multiple of 3*5*7 as many times as possible. With this, the least answer, 23, is obtained. The complete theorem was first given in 1247 by Qin Jiushao. (Sunzi Song in Chinese) ? Do not remember how many measures 5 3 7

; Error correction coding etc.

CRT is not robust: If k 1 =1, then N=4 and the error is 10.  k 1 =1)

By “optimally”, we mean with the highest probability of correct detection !

r r -

10 Hz Fourier Transform 1 Hz A 10 Hz harmonic signal Nyquist samplingUndersampling

Signal Processing = Fourier Transform? Many people think that signal processing is in fact Fourier transform. Not too bad claim but it probably should be “Fourier transform is the most important signal processing tool.” What happens when m<N called undersampling? –N can not be determined. –What is detected from the m-point DFT is the remainder of N modulo m called phase wrapping. –What can we do in this case? Use multiple samplings!

suspect low power/tiny spy sensors

This precisely follows the CRT problem!

Hz True frequency components Detected frequency components from multiple undersampled signals

total number of samplings number of frequencies new equivalent number of samplings to one frequency My Basic Idea for a Solution

(Xia’97)

 We have also obtained algorithms to determine multiple integers from their residue sets ---- Generalized CRT, IEEE Trans. on Signal Process. Feb  We have studied the noisy remainders case, IEEE Signal Process. Letters, Nov  We have obtained improved upper bounds for the uniqueness. H. Liao and X.-G. Xia, “A sharpened dynamic range of a generalized Chinese remainder theorem for multiple integers,” IEEE Trans. on Information Theory, Jan  This has been applied to a mechanical engineering problem: P. Beauseroy and R. Lengelle, “Nonintrusinve turbomachine blade vibration measurement system,” Mechanical Systems and Signal Processing, vol. 21, pp , 2007.

Robust Phase Unwrapping and A Robust CRT Phase unwrapping has tremendous applications in synthetic aperture radar (SAR) imaging: for example, moving target location and interferometric SAR (InSAR) for three dimensional mapping, where the information is included in variable x that needs to be solved. –When x >M, the M-point DFT can not uniquely determine x and a solution for x is called unwrapping. –Due to the additive noise, a solution of x may not be accurate (or robust). Chinese remainder theorem (CRT) is well-known non- robust: any small errors from remainders may cause a large error in the reconstruction from the remainders.

Problem Formulation Signal model: where are known and x (may not be integer) is unknown to be determined, after some traditional radar signal processing. For each i, taking the M-point FFT of s i (m) in terms of m, m=1,2,…,M, we obtain (only) integer remainders k i with : where n i is a nonnegative unknown integer and  i is an unknown real number with –The unknown  i is precision error that only causes small error for x. –The unknown n i is folding error that may cause large error for x. It has to be determined in order to robustly determine x. –Due to the additive noise in the signal, the integer remainders k i from the M-point DFT may have errors.

Problem Formulation Problem of interest: determine the folding integers n i from erroneous integer remainders from the M-point DFTs of the noisy signals s i (m) in terms of m, m=1,2,…,M, with for i=1,2,…,L.

A Robust Phase Unwrapping: Algorithm Let  be the smallest positive number such that  i =  i, i=1,2,…,L, are all integers. Without loss of generality, assume Let Find the following sets Let S i,1 denote the set of all the first components of the pairs in set S i, i.e., Let

A Robust Phase Unwrapping: Results Theorem (Xia-Wang’07): Assume and are co-prime for. If and, then set S contains only the element n 1, i.e.,, and implies, where n i are the true solutions. The estimate of the unknown x is, then, The estimation error is upper bounded by A remark: the condition that and are co-prime is easy to satisfy in practice since the parameters i are usually designed by ourselves.

A Robust Chinese Remainder Theorem For integers, the precision errors   =0. In this case, after the M-point FFT, we obtain Let Then, which is the CRT problem of determining n from its remainders r i and moduli M i, i=1,2,…,L. When our robust phase unwrapping results apply to the above integer problem, we obtain a robust CRT, where the known remainders (in this case ) may have errors.

A Robust Chinese Remainder Theorem Corollary (Xia-Wang’07): Let for some nonnegative integer n 0. If n 0 <M and, then n i, i=1,2,…,L, can be uniquely determined by the above algorithm. An estimate of n is and its error is upper bounded by Notice that the error from the CRT is at least in the order of

A Fast Implementation of Finding Set S i Theorem: Let (Li-Xu-Peng-Xia’07) then, ii  i 1=23…L1=23…L 11  1 ii

Some Simulations Consider L=3, 1 =0.4, 2 =0.5, 3 =0.7,  =1 In this case, we take  =10, then  1 =4,  2 =5,  3 =7 The maximal determinable range for x is 14 The unknown x is taken uniformly in the real interval [0,14) M=(1+2  )(  1 +  L ) The remainder error levels  =0,1,2,3,4,5 For the robust CRT, and the maximum determinable n 0 is M-1 and we take M=2  (  1 +  L )+1

Application in Antenna Array SAR In an antenna array SAR, there are M receive antennas on the radar platform. The received signal model after some signal processing (motion compensation and range compression) becomes where x represents the position (Doppler shifted) of the target due to its motion that is proportionally determined by the target velocity in the range direction v y. Radar platform

Multiwavelength Antenna Array SAR Imaging of Moving Targets (Wang-Xia-Chen-Fiedler’04) When the radar transmits waveforms with L different wavelengths 1, 2,…, L, the signals are which follows the signal model we studied in the beginning, where  =d/v. We then can apply our developed robust phase unwrapping algorithm to detect the Doppler shifted location x of the target. After the shifted location x is determined, its true location can be determined.

Non-uniformly Spaced Antenna Array SAR Imaging of Moving Targets (Li-Xu-Peng-Xia’07) When the receive antennas are spaced non-uniformly with L different groups, the signals are which also follows the signal model we studied in the beginning, where  =1/( v).

Non-uniform Speed Antenna Array SAR Imaging of Moving Targets (Li-Xu-Peng-Xia’07) When the radar moves with L different speeds v 1,v 2,…,v L, the signals are which follows the signal model we studied in the beginning, where  =d/.

Simulation Results for Non-uniform Spaced Antenna Array SAR An example of non-uniform spaced antenna array SAR Main parameters: wavelength 3cm slant range 10km spacings of two uniform arrays 2.0m and 1.5m the overall array is non-uniform the radial speeds of the targets -1.30m/s and 4.20m/s While VSAR mis-locates the fast target, the new system can correctly locate both fast and slow targets. Conventional VSAR New VSAR

Conclusion Sunzi Theorem (CRT) is about 2500 years old and I feel lucky that I was able to generalize it from determining single integer from its remainders to determining multiple integers from their remainder sets. We obtained a robust CRT with a special form that is the first and the only format of the CRT of robustness in the literature so far to my knowledge. I believe that these fundamental results will have many other applications beyond radar and sensor network signal processing.

Acknowledgement Thank: Genyuan Wang Huiyong Liao Gang Li Jia Xu Yingning Peng

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