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Proc Appl Math Mech. Author manuscript; available in PMC 2010 August 16.

Published in final edited form as:

Proc Appl Math Mech. 2009 February 26; 8(1): 10761–10762.

PMCID: PMC2922022

NIHMSID: NIHMS175361

See other articles in PMC that cite the published article.

This paper is devoted to estimation of parameters for a noisy sum of two real exponential functions. Singular Spectrum Analysis is used to extract the signal subspace and then the ESPRIT method exploiting signal subspace features is applied to obtain estimates of the desired exponential rates. Dependence of estimation quality on signal eigenvalues is investigated. The special design to test this relation is elaborated.

In this paper we address the task of estimation of signal parameters having noisy measurements when the signal is the sum of two real exponential functions. For this purpose we apply the method ESPRIT [1] originally aimed at estimation of parameters of complex-valued exponentials (cisoids).

The ESPRIT method exploits the subspace properties of the signal and this is what connects it with Singular Spectrum Analysis (SSA) [2]. SSA is the method of time series analysis aimed at decomposition of the time series into the sum of components, e.g. signal and noise. As an essential step, SSA includes the construction of a subspace which approximates the subspace generated in some way by the signal. ESPRIT uses a basis of this subspace for estimating the rates of the exponential components. After that, the coefficients before the exponentials can be evaluated using the conventional least squares method.

We introduce the signal subspace as follows. For the given series *S* = (*s*_{0}, …, *s*_{N−1}), we select the *window length L*, where 2 ≤ *L* ≤ (*N* + 1)/2. Then *S* is embedded into ^{L} constructing *S*_{j} = (*s*_{j−1},…,*s*_{j+L−2})^{T} ^{L}, *j* = 1,…, *K*, *K* = *N* − *L* + 1. The linear span of {*S*_{j}} is called *L-trajectory space* of the series *S*. The *L-rank* (or simply *rank*) of the series is defined to be the dimension *r* of the trajectory space, naturally *r* ≤ *L*. The rank of the series *S* is equal to (i) the rank of the *trajectory matrix* **S** with the column vectors {*S _{j}*}, and (ii) the number of non-zero eigenvalues of the matrix

If the signal *S* has rank *r* < *L*, then the trajectory subspace of *S* is called the *signal subspace* and determines the shape of the signal up to coefficients before summands. When the series contains both the signal and noise, the signal subspace can be found only approximately (e.g. by means of SSA) that leads to approximate estimation of parameters of the signal.

Algorithms of SSA and ESPRIT are briefly described in Section 2. The design of numerical tests aimed to reveal the relation between properties of ESPRIT estimates and the signal eigenvalues λ_{j}(*S*) is constructed in Section 3. The results on mean square deviations of estimates are presented and discussed in Section 4.

Let us consider a noisy signal = *S* + = (*f*_{0}, …., *f*_{N−1}) and the window length selected is *L*. The first step of both SSA and ESPRIT is the construction of the trajectory matrix **F**, having the columns *F _{j}* = (

Under conditions of approximate separability, the space spanned by {*U _{j}*}

Let us describe ESPRIT for the signal *S* = + consisting of the sum of two exponential components and with elements ${a}_{n}={c}_{a}\phantom{\rule{thinmathspace}{0ex}}\text{exp}(\alpha n)={c}_{a}{r}_{a}^{n}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{b}_{n}={c}_{b}\phantom{\rule{thinmathspace}{0ex}}\text{exp}(\beta n)={c}_{b}{r}_{b}^{n}$. The aim is to estimate the parameters *r _{a}*,

Let us consider the signal *S* = + , define signal eigenvalues λ_{min} = λ_{1}(*S*), λ_{max} = λ_{2}(*S*) and denote their divided by *K* versions as μ_{min} and μ_{max}. Suppose that we observe = *S* + , where is the white gaussian noise with variance σ^{2}. Let **S** and **N** be the trajectory matrices of *S* and , correspondingly, and ‖ · ‖ is the Frobenius norm which is defined to be the square root of the sum of squares of the entries of the matrix. Hence ‖**S**‖^{2} = λ_{min} + λ_{max}, ${\Vert \mathbf{N}\Vert}^{2}={\displaystyle {\sum}_{j=1}^{L}{\lambda}_{j}(\mathcal{N})}$ ≈ *LK* σ^{2} and the signal-to-noise ratio ‖**S**‖^{2}/‖**N**‖^{2} for the given *L* is specified by (μ_{min} + μ_{max})/(*L*σ^{2}), where the numerator is actually the weighted sum of squares of the elements of the signal. Below we refer to the Frobenius norm of the trajectory matrix as *L-norm* of the series. Note that the average of noise eigenvalues λ_{j}(), *j* = 1, …, *L*, is approximately equal to *K*σ^{2}. To obtain the proper approximation of the two-dimensional signal subspace, we need λ_{min} to be larger than the maximal of noise eigenvalues. Hence, the relation between μ_{min} and σ^{2} is the key point here.

Obviously, the eigenvalues depend on *L*. We choose *L* = *N*/2 in order to increase μ_{min} and to improve also the separability between signal and noise following the recommendations of SSA theory [2]. Let us fix *N* = 99 and *L* = 50. We consider eight test sets of parameters producing exponential components with equal *L*-norms and the fixed signal *L*-norms: μ_{1}() = μ_{1}(), μ_{min} + μ_{max} = 2. These test sets differ in the value of μmin, see Table 1 with the test sets ordered by increasing μ_{min}. Note that we consider as the first exponential the one with the largest absolute value of rate: |α| ≥ |β|, i.e. the first exponential component changes not slower than the second one.

The statistical simulation was performed to estimate the parameters of the model. Two levels of noise were considered, σ^{2} = 1.e–2 and σ^{2} = 1.e–4. The signal-to-noise ratio is equal to 4 and 400, correspondingly. The two obtained ESPRIT estimates * _{a}* and

In our experiments, the noise variance σ^{2} is relatively low for the signal to be reconstructed with good accuracy. However, based on the previous section, the accuracy of the parameter estimates is assumed to depend not only on σ^{2} but on μ_{min} also.

The simulation involving 1000 replicates confirmed this guess. Though the signal estimates are precise (not shown in Table 1), the parameters estimates are close to the original values only when μ_{min} is several times larger than σ^{2}, see error values in Table 1. Moreover, it turns out that the estimates of *r _{a}* and

This work was supported by NSF/NIGMS BioMath 1-R01-GM072022 grant and GAP award RUB1-1643-ST-06 from the CRDF.

1. Roy R, Kailath T. IEEE Trans. Acoust. 1989;37:984–995.

2. Golyandina N, Nekrutkin V, Zhigljavsky A. Analysis of Time Series Structure: SSA and Related Techniques. Chapman&Hall/CRC; 2001.