M2.18 MAXIMUM LIKELIHOOD ESTIMATION OF POLES, AMPLITUDES

AND PHASES FROM 2-D Nh4R TIME DOMAIN SIGNALS

R. de Beer, D. van Ormondt, and W WF. Pijnappel,

Applied Physics Department, Delft University of Technology P.O. Box 5046, 2600 GA Delft, The Netherlands

ABSTRACT

2-D Magnetic Resonance (MR) is a powerful spec- troscopic tool. MR signals are usually measured in the t i m e domain. For analytical purposes, i t is often required to quantify a signal in t e r m s of physically relevant model parameters. To this end we f i t a sum of exponentially damped 2-D sinusoids to the data. Our approach can accommodate a full 2-D amplitude matrix a s well as unequal numbers of signal poles in the t w o dimensions.

1. INTRODUCTION

The method of choice for studying molecular structure in solution is 2-D high resolution Nu- clear Magnetic Resonance (NMR). This technique enables one to infer, among other things, the struc- ture of proteins and nucleic acids. See e.g. A. Bax and L. Lerner [l].

The N M R phenomenon is induced by applying a sequence of pulsed radio frequency waves to the molecules under study, and is subsequently obser- ved and recorded in the time domain. A typical 2-D NMR signal comprises 512 rows of 1024 complex- valued data points. Frequently, t h e data acquisi- tion t ime is of the order of ten hours.

Traditionally, 2-D NMR signals are transformed to the frequency domain using FFT in both dimen- sions. Subsequently, the spectra thus obtained are studied and quantified. As many a s several thou- sand spectral features (peaks) may be distinguish- able. In m o s t 2-D experiments one obtains a spec- t rum which is symmetric with respect to the bisector of the two frequency axes. However, 2-D experiments yielding asymmetric spectra exist also. See [ll for details.

The aforementioned determination of molecular structure hinges on the ability to detect and para- metrize weak peaks, which lie off the above men- tioned bisector and 'connect' strong peaks on the bisector in the sense that the respective projec- tions on the frequency axes coincide. These con- necting peaks are called cross-peaks in the 2-D NMR jargon. Unfortunately, various experimental conditions pose practical limits to the measuring t ime. In practice, one may therefore be forced to reduce the number of rows, which entails distor- tion of the spectrum as obtained by FFT. This in turn hampers the search for, and analysis of, weak peaks in a contour plot of a 2-D spectrum. An alternative to usinR FFT and processing in the

frequency domain is to introduce a model func- tion for the 2-D NMR time domain signal, and fi t this function directly to the data. In the present work several such methods are proposed and tested on simulated signals. Our methods are based on previous work by Kung et al. [lo], Kuma- resan et al. C7,81, Bresler and Macovski [41, and Park and Cordaro [121. New features are that the numbers of signal poles in the two dimensions, K and K , need not be equal, and that a l l K x K ele- ments of the amplitude matrix are allowed to be non-zero. The latter is required to quantify the above-mentioned cross-peaks.

The remainder of this paper is organized a s fol- lows. In Sec2 w e introduce the model function, and then w e treat two fundamentally different approaches to retrieve the wanted model parame- ters from the data. In Sec.3 results of quantifica- tion of noise corrupted simulated data are given and discussed.

2. THEORY

2.1 The model function

The model function that best approximates actual 2-D NMR t ime domain signals obtained from liquid or liquidlike samples consist of a linear combinati- on of exponentially damped 2-D sinusoids, i.e.,

in which the symbols have the following meaning: n: 1,2, ... ,N, N being the number of samplesper column, n'= 1,2, ... ,N', N being the number of samples per row, C k p , k=l, ... ,K, and k = l , .._ , K , are the, generally complex-valued, amplitudes of the sinusoids, i.e.

Zk' exp[(ak+iwk)Atl, k=l , ... ,K, are the signal poles in the unprimed (column) space, ak being damping factors, wk angular frequencies (wk=2mk). and A t the sampling interval, L'F= expl(a'p+iwk)At'], k'=l, ... , K , are the signal poles in the primed (row) space, the primed sym- bols having the same meaning as their unprimed counterparts. K need not be equal to K .

A possible inadequacy of the above model func- tion is that the actual damping of the signal can be more complicated than expht) , e.g. exp(at+gt2).

ckk:lckk.lexp(ipkkf), vkk being the phases,

1504 CH2673-2/89/0OWl504 $1.00 0 1989 IEEE

2.2 Decomposition of the data matrlx

One way to f i t Eq.( l ) t o the data, xnn, , n = l , ... , N , n '=l , .._ , N , is to seek a decomposition of the N x N data matrix X , formed f rom the 2-D data, into a product of three matrices, namely an NxK Vandermonde matrix CN,K, a K x K amplitude matrix C, and the transpose of another Vandermonde matrix & * , K * , according to

x= CN,K c tL ' ,K' =

H,=

1 1 . . . 1

Zl ' . . . Z A

-

xnM+l

-XnL XnL+l. . . 7 xnN'

, n=l , .._ ,N, (3)

Xrll %z ' ' '

Xnz X n 3 . . . . . . . . .

. . . . 1 Z;c,. . .

, (2)

where - indicates transposition. In absence of noise, and provided Eq.(l) ade-

quately describes the signal, Eq.(2) is exact. In the presence of noise, Eq.(2) can be satisfied in the least squares sense. Once the decomposition has been accomplished, the model parameters are immediately available. In the present contribution we distinguish two approaches, namely non-itera- tive ones based on SVD [lo], and iterative ones based on Variable Projection plus Linear Predic- tion [7,8,4,121. A common advantage of all methods mentioned is that t h e model function of Eq.il) is fitted to the data without the need to provide starting values. A common disadvantage is that spectroscopic prior knowledge other than the number of sinusoids can not be imposed.

2.3 Decompositlon by SVD (Method I )

W e s tar t by considering the rank of t h e data matrix, R(X), and this in absence of noise so that the decomposition of Eq.(2) is exact. Perusing Eq.(2). one can see that R(X)= min[K,Kl, provided the rank of the amplitude matrix, R(C), is full. Using the s ta te space formalism, Kung e t al. have shown [lo] that the wanted decomposition can be brought about by SVD if K:K and C is diagonal.

We have found that restriction of the amplitude matrix in Eq.(2) to diagonal form is not necessary. In fact , various simulated 2-D signals with full, nonsingular amplitude matrices were successfully processed using the above method. The latter result is not sufficient for the purpose of NMR. however. Firstly, signals with K + K exist, and secondly, even if K = K and R(C)=K, Monte Carlo experiments show that the variances of the re- sulting model parameters are unnecessarily far above the Cram&-Rao lower bounds. These two points will now be remedied.

W e propose to rearrange the data in the fol- lowing manner. First, we form Hankel matrices H, (n=l. ... ,N) f r o m the N entries of each of t h e N rows of X, i.e.

E2 is to be minimized as a function of the com- plex-valued amplitudes, frequencies, and damping factors, using an iterative nonlinear least squares procedure. The special feature of VARPRO is that the amplitudes entering in 2,,. are eliminated

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from Eq.6) so that, a t least initially, minimization is to be achieved only with respect to the fre- quencies and damping factors. Recently, i t has been pointed o u t [7,8.4,121 that further simplifica- tion of Eq.(7) is possible by invoking Linear Pre- diction (LP). E2 can then be minimized with re- spect to the LP coefficients rather than the fre- quencies and damping factors. A very important advantage of this is that in spite of t h e iterative character of the minimization procedure, starting values of the LP coefficients need not be supplied. Subsequent calculation of the wanted signal poles f rom the LP coefficients is a routine task.

Refs.[7,5,4,121 are primarily aimed a t processing 1-D signals with the LP/VARPRO method. In this work, an extension to 2-D is made. Our strategy is described in Ref.[3], and is only very briefly indi- cated here for reasons of space.

First, X is decomposed. in the least squares sense, a s (see e.g. C71)

(6)

where cN,K is defined in Eq.(2) and A is a K,N' matrix that carries the information about t h e signal poles in the primed dimension and the amplitudes. Subsequently, A is decomposed, using the same decomposition method once more which leads to

the right-hand side of which is defined in Eq.(2). A t this point all model parameters in Eq.(l) have been quantified. Note that the number of 1-D com- ponents in each dimension had to be known befo- rehand. Estimates of the latter can be obtained either from previous studies or f rom perusing the spectrum obtained by FFT. Since an N M R signal is usually contaminated by noise, choosing K and K somewhat too large results in a limited number of insignificant amplitudes which can be discarded.

An important aspect of t h e LPIVARPRO method, not yet mentioned, is the potential to reduce the computational load relative to SVD-based me- thods, for large data sets. Our 2-D implementation of the LP-approach of Refs 17,8,41 requires N'N2 complex-valued multiplications in the initial stage and subsequently in each iteration cycle approx- imately K(K+4)(N-K)2 complex-valued multiplica- tions for executing Eq.(6), and K(Nr2+3N'K) for Eq.(7). (Method Ila.) Note that for large data sets, these expressions depend on the squares of the number of rows and columns o f X. The appealing feature of the aooroach of Ref.rl21 is that for a 1-D signal the computational load is proportional to the first power of N . Our 2-D implementation requires approximately 3 K " ' complex-valued multiplications for Eq.(7), 3KKN for Eq.(8), and N N for the initial stage.(Method Ilb.)

Finally, w e point o u t that Kumaresan and Shaw have devised [8,131 two alternative 2-D implemen- tations of the LPIVARPRO method. However, these authors restricted themselves to K = K and ckk.=O for k+k'.

3 RESULTS and DISCUSSION

In t h i s section we present results of the SVD- and LP/VARPRO-based methods treated in Secs. 2.3 and 2.4. Our aims were: 1) to ascertain that retrieval of all model parameters entering Eq.(l) is feasible, 2) to investigate to what extent the standard deviations of the quantified parameters approach the theoretical (Cramer-Rao) lower bounds, 3) to investigate whether or not there is evidence for a significant bias in the values of the quantified parameters. To this end we have exe- cuted a Monte Carlo study using 50 different complex-valued normally distributed white noise realizations with standard deviation equal to one, for the the real and imaginary parts each. The phases of all sinusoids were set to zero; the va- lues of the frequencies, damping factors, and amplitudes are given in Fig.i. Note that two sinu- soids had amplitudes smaller than the noise. For economy of space we present here only the re- sul ts for one of the two smallest 2-D sinusoids. See Table 1. (See [31 for all results of method lla.). The results for t h e complete signal in t h i s example can be summarized a s follows:

1) All three methods were indeed able to re- trieve all model parameters of the signal without posing problems. The SVD-based method, I, easily found the number of sinuoids involved in each dimension. In methods IIa and Ilb these numbers were assumed to be known a priori.

2) Methods Ila and IIb approached the Cramk- Rao bounds to within several tens of percents. This result is to be expected, since LP/VARPRO is a maximum likelihood method. Method I (SVD) also performed well, but occasionally the stan- dard deviation of some parameter (most ly a phase or an amplitude) was of t h e order of twice the standard deviation. Since SVD-based methods are not maximum likelihood, this result is satisfying.

3) There is no evidence for significant bias in the quantified parameters, a t least a t t h e given SNR.

4) Quantification of one noise-contaminated version of the signal presently investigated re- quired about 4.5 , 6.7, and 4.5 s. for method 1, Ila, and Ilb, respectively, using an IBM 3083-3x1 Mainframe computer.

From the above results we conclude that all three methods are candidates for quantifying 2-D signals. However, further tests are required. The following aspects should ye t be investigated. 1) The robustness under possible inadequacy of the model function of Eq.(l). 2) The resolving power. 3) The feasibility of analysing large real-world data sets comprising many sinusoids. 4) The SNR a t which each of the methods ceases to function properly.

Acknowledgment This work was carried o u t in the program of

t h e Foundation for Fundamental Research on Mat- ter (FOM) and was supported (in part) by the Ne- therlands Technology Foundation (STW). The authors thank Th.J.L. Bosman, C. W. Hilbers, and F.J.M. van der Ven for useful discussions.

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V '

0 0.10 0.20 0.30 0.40 0 I I I I I I I I

-0.03; j -0.01 j -0.02 -0.035 : I , I ,

1 0.10

Fig.1. Top view of simulated signal comprising twelve 2-D sinusoids, displayed in the Frequency domain. The sinusoids are indicated by small cir- cles. The frequencies can be read from the projec- tions of the centres of the circles on the axes, in units of VN,, uist. The damping factors are given in italic num%ers, and the amplitudes are under- lined. Note that in real-world N M R signals the damping factors of all 'cross-peaks' with the same frequency in either dimension need not have the same damping factor, a s was assumed here. All phases were put equal to zero. The standard de- viation of the added complex-valued noise is l , for both the real and imaginary parts.

Table 1. Results of Monte Carlo study for one small 2-D sinuoid present in the simulated signal ('cross-peak', a t u=O.15, d.0.17, see Fig.1). Three me- thods were used: Method I is based on SVD and the State Space formalism, methods Ila and Ilb on a combination of Linear Prediction and Variable Projection. See text for details.

v 2 a2 P a r a m .

True Val. 0.150000 0.170000 -0.01 000 Mean ( 1 ) 0.150025 0.170006 -0.00995 Mean (Ila) 0.149972 0.169993 -0.01016 Mean (Ilb) 0.149983 0.170042 -0.01019

C.R. Bound 0.000076 0.0001 1 2 0.00042 St. Dev.(l) 0.000105 0.0001 78 0.00062 St. Dev.(lIa) 0.000078 0.00011 5 0.00042 St. Dev.(lIb) 0.000066 0.000105 0.00047

-0.01000 -0.01015 -0.00995 -0.01014

0.00068 0.00 1 os 0.00065 0.00064

P a r a m . I C 1 2 I q12 (degrees)

True Val. 0.50 0.0 Mean ( 1 ) 0.51 - 2.5 Mean (IIa) 0.48 0.6 Mean (IlbJ 0.49 - 2.5

C.R. Bound 0.05 5.7 St. Dev. ( 1 ) 0.08 7.7 St. Dev.(lla) 0.06 6.3 St. Dev.(lIb) 0.05 5.6

REFERENCES

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Retrievai'Problem, J Opt SOC A m , Vol 73, pp 1799-1811, 1983 [111 Liu, Q.-C.. and L. -H. Zou, A New Separable Eigenstructure Algorithm for Parameter Estimation of 2-D Sinusoids in White Noise, in "Signal Process- ing IV: Theory and Applications", J.L. Lacoume et al. Eds., Elsevier, pp. 443-446, 1988. [121 Park, S.-W., and J.T. Cordaro, Maximum Likeli- hood Estimation of Poles from Impulse Response Data in Noise, Proc. ICASSP, pp. 1501-1504, 1987. I131 Shaw. A.K.. and R. Kumaresan. Some Structured Matrix Approximation Problems, 'hoc . ICASSP, pp. 2324-2327, 1988.

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