The Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949. It is also commonly called Shannon's interpolation formula and Whittaker's interpolation formula. E. T. Whittaker, who published it in 1915, called it the Cardinal series.
Definition
Given a sequence of real numbers, x [ n ] = x ( n T ) {\displaystyle x[n]=x(nT)} , representing a sequence of samples at time intervals of T {\displaystyle T} seconds, consider the following continuous function:
x ( t ) = ∑ n = − ∞ ∞ x [ n ] s i n c ( t − n T T ) , {\displaystyle x(t)=\sum _{n=-\infty }^{\infty }x[n]\,{\rm {sinc}}\left({\frac {t-nT}{T}}\right),}
where s i n c ( t ) {\displaystyle {\rm {sinc(t)}}} denotes the normalized sinc function sin ( π t ) π t {\displaystyle {\frac {\sin(\pi t)}{\pi t}}} .
x ( t ) {\displaystyle x(t)} has a Fourier transform X ( f ) {\displaystyle X(f)} , whose non-zero values are confined to the region | f | ≤ 1 2 T {\displaystyle |f|\leq {\frac {1}{2T}}} , and a bandlimit of 1 / ( 2 T ) {\displaystyle 1/(2T)} cycles/sec (hertz). The quantity f s = 1 / T {\displaystyle f_{s}=1/T} is known as the sample rate, and f s / 2 {\displaystyle f_{s}/2} is the corresponding Nyquist frequency. When the sampled function has a bandlimit less than the Nyquist frequency, x ( t ) {\displaystyle x(t)} is a perfect reconstruction of the original function (see sampling theorem). Otherwise, the frequency components above the Nyquist frequency will "fold" into the sub-Nyquist region of X ( f ) {\displaystyle X(f)} , resulting in distortion.
Equivalent formulation: convolution/lowpass filter The interpolation formula is derived in the Nyquist–Shannon sampling theorem article, which points out that it can also be expressed as the convolution of an infinite impulse train with a sinc function:
x ( t ) = ( ∑ n = − ∞ ∞ T ⋅ x ( n T ) ⏟ x [ n ] ⋅ δ ( t − n T ) ) ∗ ( 1 T s i n c ( t T ) ) , {\displaystyle x(t)=\left(\sum _{n=-\infty }^{\infty }T\cdot \underbrace {x(nT)} _{x[n]}\cdot \delta \left(t-nT\right)\right)*\left({\frac {1}{T}}{\rm {sinc}}\left({\frac {t}{T}}\right)\right),}
where the convolution of two functions f {\displaystyle f} and g {\displaystyle g} is the following integral transform, denoted using "*":
( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .}
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