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Sampling Theory and Spline Interpolation

Module by: Rebecca Willett

Summary: 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.

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Shannon's sampling theory

Shannon's sampling theory tells us that if we have a bandlimited signal ( sx s x ) that has been sampled at the Nyquist rate, then the signal can be reconstructed from its samples ( sk s k ) with the following relation:

sx=∑k∈ℤsksincx−k s x k k s k sinc x k (1)
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, sinc0=1 sinc 0 1 . As a result, if TT is the sampling frequency, then sxT=sk s x T s k . The reconstruction of the sequence of samples 12331.5014 1 2 3 3 1.5 0 1 4 can be seen in Figure 1.
Figure 1: 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.
Sinc Interpolation
Sinc Interpolation (SincInterp.png)

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 sx s x .

Signal Reconstruction with Cardinal Splines

As described above, having only a finite number of samples leads to inaccuracies in estimating sx s x . Using cardinal splines instead of sinc functions can lessen the magnitude of the errors. The n th n th cardinal spline, ηn η n , gives piecewise polynomial interpolation with order nn polynomials. Like the sinc function, each cardinal spline vanishes at all integers except the origin, and ηn0=1 η 0 n 1 . Furthermore limn→∞ηnx=sincx 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,

sx=∑k∈ℤskηnx−k s x k k s k η x k n (2)

The reconstruction of the sequence of samples 12331.5014 1 2 3 3 1.5 0 1 4 can be seen in Figure 2.

Figure 2: 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.
Cardinal Spline Interpolation
Cardinal Spline Interpolation (SplineInterp.png)

From images Figure 1 and Figure 2, 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 sx s x (when sx s x is a polynomial signal) with an error of less than 11%, one would need O100 O 100 sinc functions, but just n+1 n 1 B-splines for exact evaluation.

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