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# CONTINUOUS SIGNALS WITH DISCRETE SPECTRA

Let us suppose we have a one dimensional, continuous signal s(t) that has a finite time duration T. Within the time interval T, this signal can be represented by a Fourier series expansion as follows:
 (2)
where Sk is the discrete spectrum of the signal at the discrete frequency

Let be a sequence of samples of this signal. If one can reconstruct the original continuous signal from this finite number of samples, then one can interpolate or resample this signal arbitrarily. This reconstruction is possible if the spectrum of the signal is band-limited, in other words, if the Fourier series expansion in equation (2) has a finite number of terms. With this assumption, equation (2) tells us that the continuous signal can be fully described by a finite number of samples of its spectrum. Thus, the key step of the reconstruction is to estimate the spectrum.

Substituting the known samples into equation (2) gives
 (3)
If we define

and

where K is the number of frequencies required to model the signal, we can rewrite equation (3) into the corresponding matrix form
 (4)
The least squares solution of this system equation is defined by the following normal equation:
 (5)
where is a matrix with complex elements and is the conjugate transpose of .

There exist many numerical algorithms for solving a general normal equation. But the computation required is generally in an order of K3. Fortunately, as we will see later, the matrix has a special structure that makes it possible to solve equation with an order of K2 operations.

Next: TOEPLITZ STRUCTURE Up: Zhang: Interpolation Previous: Introduction
Stanford Exploration Project
12/18/1997