Sampling Theory and Spline Interpolation 2.3 2003/04/23 2003/09/24 16:54:55.075 GMT-5 Rebecca Willett willett@rice.edu Charlet Reedstrom charlet@rice.edu Adan Galvan jago@rice.edu Sujesh Sreedharan sujesh@rice.edu Alena Ixmocane Scott oetting@stat.rice.edu Rebecca Willett willett@rice.edu Shannon samples analog sinc B-splines cardinal splines In this module, Shannon's classical sampling theory is compared to digital to analog signal reconstruction using spline interpolation. In the spline method, the signal is reconstructed using sample-weighted cardinal splines as opposed to sample-weighted sinc functions.

Shannon's sampling theory Shannon's sampling theory tells us that if we have a bandlimited signal ( s x ) that has been sampled at the Nyquist rate, then the signal can be reconstructed from its samples ( s k ) with the following relation: s x k k s k sinc x k This relation is frequently used in digital to analog converter. There are several desirable properties of the sinc function that make this strategy effective. First of all, the sinc function vanishes at all integers except at the origin. Secondly, sinc 0 1 . As a result, if T is the sampling frequency, then s x T s k . The reconstruction of the sequence of samples 1 2 3 3 1.5 0 1 4 can be seen in .

Sinc Interpolation Bandlimited continuous signals can be reconstructed from their samples using a linear combination of sinc functions, where the sinc functions are weighted by the sample values.

The disadvantage of this approach is that it depends on the initial assumption that the signal is bandlimited, but frequently we rely on only a finite number of samples, which cannot completely describe a bandlimited signal. As a result, we can only find an approximate estimate of the signal s x .

Signal Reconstruction with Cardinal Splines As described above, having only a finite number of samples leads to inaccuracies in estimating s x . Using cardinal splines instead of sinc functions can lessen the magnitude of the errors. The n th cardinal spline, η n , gives piecewise polynomial interpolation with order n polynomials. Like the sinc function, each cardinal spline vanishes at all integers except the origin, and η 0 n 1 . Furthermore n η x n sinc x . This means that cardinal splines can be used for signal reconstruction from samples just as sinc functions are used. Specifically, s x k k s k η x k n The reconstruction of the sequence of samples 1 2 3 3 1.5 0 1 4 can be seen in .

Cardinal Spline Interpolation Bandlimited continuous signals can be reconstructed from their samples using a linear combination of cardinal splines, where the spline functions are weighted by the sample values.

From images and , it may appear that the spline interpolation is smoother than the sinc interpolation. This is because the support of the cardinal splines is more compact than that of the sinc function. In fact, to compute the value of s x (when s x is a polynomial signal) with an error of less than 1%, one would need O 100 sinc functions, but just n 1 B-splines for exact evaluation.