Matching pursuit with damped sinusoids

The matching pursuit algorithm derives an expansion of a signal in terms of the elements of a large dictionary of time-frequency atoms. This paper considers the use of matching pursuit for computing signal expansions in terms of damped sinusoids. First, expansion based on complex damped sinusoids is explored; it is shown that the expansion can be e ciently derived using the FFT and simple recursive lterbanks. Then, the approach is extended to provide decompositions in terms of real damped sinusoids. This extension relies on generalizing the matching pursuit algorithm to derive expansions with respect to dictionary subspaces; of speci c interest is the subspace spanned by a complex atom and its conjugate. Developing this particular case leads to a framework for deriving real-valued expansions of real signals using complex atoms. Applications of the damped sinusoidal decomposition include system identi cation, spectral estimation, and signal modeling for coding and analysis{modi cation{synthesis. 1. SIGNAL DECOMPOSITIONS In signal processing applications it is often useful to decompose a signal into elementary building blocks. In such a decomposition, a signal x[n] is represented as a linear combination of expansion functions gm[n]; in matrix notation, x = D ; D = [g1 g2 gm gM ] (1) where the signal x is a column vector (N 1), is a column vector of weights (M 1), and D is an N M matrix whose columns are the expansion functions gm[n] as indicated. The subscript m denotes an index set that describes the features of the building block gm[n], for instance time location, modulation, and scale. A wide variety of such decompositions, ranging from Fourier and sinusoidal models to wavelet and frame expansions, have been explored in the literature. These approaches nd use in coding applications, where compression is often achieved by discarding components with low-valued expansion coe cients; in such cases the expansion is intended to provide an accurate but not necessarily perfect reconstruction of the signal. For Fourier transforms, wavelets, and other expansions where the functions gm[n] constitute a basis (N = M), the matrix D is invertible and the expansion coe cients for a given signal are unique. These basis expansions exhibit a certain rigidity, however, in that a given basis is not well-suited for a wide variety of signals. Consider the Fourier case: for a time-localized signal, the frequency domain representation does not readily indicate the time localization; the Fourier analysis does not provide information about the relevant signal features. This shortcoming results from attempting to represent arbitrary signals in terms of a very limited set of functions. Better representations can be achieved by using a larger number of expansion functions, i.e. by choosing the gm[n] from a highly redundant dictionary that not only spans the signal space but also includes a wide range of functions beyond the spanning set; this enables appropriate representation of a wide range of time-frequency behaviors. When the functions gm[n] constitute a redundant set (M > N), the linear system in equation 1 is underdetermined. One solution is provided by the pseudo-inverse of D, which can be derived using the singular value decomposition (SVD); the weight vector ~ = Dx has the minimum twonorm of all solutions. This minimization of the two-norm in the SVD solution is inappropriate for compression, however, in that it tends to spread energy throughout all of the elements of ~ ; compression is only achievable by discarding elements below some threshold. In comparison to this SVD approach of computing a non-sparse solution and thresholding it, the goal of compression is better served by simply searching for a sparse approximate solution to the underdetermined inverse problem. One algorithm for computing such sparse approximate solutions is known as matching pursuit [1]. Investigation of matching pursuit and similar methods is readily motivated by the compaction improvement that can be achieved with respect to traditional linear methods such as the SVD. An example of this is given in gure 1, which shows a plot of the thresholded SVD expansion coe cients (solid) and a sparse matching pursuit solution (circles) for the same reconstruction error. To achieve the same error as the sparse representation, which has 16 nonzero elements, the SVD approach must use a very small threshold, which leads to poor compaction; the SVD solution retains 330 non-zero elements. In this simulation, the original signal is the sum of ve dictionary elements, which means there is an exact solution with only ve non-zero values. For reasons to be discussed, the matching pursuit does not nd this optimal sparse solution; however, it does reliably identify the fundamental signal structure.