Whittaker–Shannon_interpolation_formula

# Whittaker–Shannon interpolation formula The **Whittaker–Shannon interpolation formula**, also called sinc interpolation or the cardinal series, reconstructs a continuous-time, band-limited [[Signal|signal]] from an evenly spaced sequence of its samples by placing a scaled and shifted sinc function at every sample and adding the results: `x(t) = sum_n x[n]*sinc((t - nT)/T)`, where `sinc(u) = sin(pi*u)/(pi*u)` and T is the spacing between samples. The sum reconstructs the original signal exactly whenever the samples were taken at or above the rate the [[Nyquist–Shannon_sampling_theorem|sampling theorem]] of the same name requires. The microsim on this page draws the individual sinc kernels the formula sums, lets the reader move the sample spacing across that Nyquist boundary, and reports the reconstruction error a necessarily truncated sum still carries. The formula's two names trace a longer history than either alone suggests. [[Claude_Shannon|Claude Shannon]] presented it as the exact recovery step of his sampling theorem in a 1949 paper on communication in the presence of noise[^shannon49], but the same infinite sum had already appeared decades earlier under a different name: E. T. Whittaker called it the cardinal series in 1915, and J. M. Whittaker developed its convergence properties further in the 1930s[^citneeded1]. ## Definition Formally the reconstructed signal is `x(t) = sum_n x[n]*sinc((t - nT)/T)`, where the samples `x[n] = x(nT)` come from a signal already known to be band-limited — its [[Fourier_transform|Fourier transform]] vanishes above some highest frequency B — and T is chosen small enough that `1/T >= 2*B`, the condition the [[Nyquist–Shannon_sampling_theorem|sampling theorem]] states as a rate rather than a spacing[^tib116]. Two properties make the sum work. First, `sinc(u) = sin(pi*u)/(pi*u)` is exactly zero at every non-zero integer u and exactly 1 at u = 0, so at any sample instant t = mT every kernel in the sum vanishes except the one centred there, which contributes x[m] alone; the reconstructed curve is therefore guaranteed to pass through every sample exactly, by construction rather than by fitting. Second, the sinc pulse is itself the impulse response of an ideal, brick-wall [[Low-pass_filter|low-pass filter]] with a cutoff at the Nyquist frequency 1/(2T)[^tib116], so summing shifted copies of it is mathematically the same operation as passing the sample train through that filter — the definition and the filtering view are two descriptions of one calculation, made explicit in the next section. Because the sinc kernel extends without limit in both directions, the sum as defined needs every sample the signal ever had, past and future alike, a requirement no real-time system can meet, which is why the formula is a standard against which practical reconstruction methods such as the [[Zero-order_hold|zero-order hold]] inside a real [[Digital-to-analog_converter|digital-to-analog converter]] are measured, rather than a circuit built directly from its own definition. ## Equivalent formulation: convolution/lowpass filter The same reconstruction can be written as a single convolution rather than an infinite sum, and the two descriptions are identical rather than merely similar. Model the sampling process — the operation linking [[Discrete_time_and_continuous_time|discrete and continuous time]] representations of the same signal — as multiplying the original signal by an ideal train of unit impulses spaced T apart, one at every sampling instant, so that the sampled signal is `x_s(t) = sum_n x(nT)*delta(t - nT)`[^tib109]. In the frequency domain that multiplication becomes a convolution, and the spectrum of the sampled signal is the original spectrum repeated at every multiple of the sampling frequency, `X_s(f) = sum_k X(f - k*fs)`[^tib109]; when the signal is band-limited to less than half of fs, these repeated copies do not overlap, and the original spectrum can be recovered by discarding everything outside the baseband copy. Discarding everything outside a band is exactly what an ideal [[Low-pass_filter|low-pass filter]] does, unity for `abs(f) < fs/2` and zero elsewhere, and convolving the impulse train with that filter's impulse response — the very sinc pulse the definition uses — replaces each impulse with a copy of the sinc, scaled by the sample it carries and centred on the sample's instant, because convolution with a shifted impulse simply reproduces the other function shifted to that impulse's location. Summing those shifted, scaled sinc copies is precisely the Whittaker–Shannon sum: the definition is what the arithmetic looks like written out, and the filtering picture is why it recovers the right answer. ## Convergence The infinite sum converges to the original signal exactly when two conditions hold together: the signal must truly be band-limited, with no energy at or above half the sampling frequency, and the sampling rate must meet or exceed the Nyquist rate for that bandwidth. When either condition fails the sum still converges to something — every term is well defined and the series remains well behaved for a reasonable class of signals — but the something it converges to is not the original signal. A worked case makes the failure concrete: sampling a signal shaped like sinc-squared at only 1.5 Hz, below what its bandwidth requires, produces a reconstruction equal to a mixture of two different sinc terms rather than the original waveform[^tib120], the same [[Aliasing|aliasing]] error that shows up in the time domain as a wrong-looking but perfectly smooth curve. Practical reconstruction faces a second, unrelated limit on convergence: an implementation can only ever sum finitely many terms, and because a sinc kernel decays as slowly as 1/t, truncating the sum to the nearest handful of samples leaves a small ripple everywhere except exactly at the kept samples, shrinking toward zero only as more distant terms are added back in. The primary microsim's kernel-reach control makes this truncation error a number the reader watches directly rather than a claim taken on faith, converging visibly as more terms are included whenever the Nyquist condition is satisfied, and failing to converge to the right answer no matter how many terms are added when it is not. ## Stationary random processes The same interpolation formula extends from a single deterministic signal to an entire ensemble of random ones. A wide-sense-stationary random process has no single Fourier transform, since a random function is not one fixed signal, but it does have a well-defined [[Spectral_density|power spectral density]], the Fourier transform of its autocorrelation function; when that spectral density is zero above some frequency B, the process is band-limited in the statistical sense the theorem needs, and the Whittaker–Shannon sum built from the process's own samples converges to the process not pointwise but in mean square, meaning the expected squared difference between the sum and the true process value shrinks to zero at every instant even though no single realization is guaranteed to match term by term. This distinction matters wherever a signal is modelled as a random process riding on top of, or instead of, a known deterministic waveform: a communication channel's [[Noise_(electronics)|noise]] is one example, and estimating or predicting such a process from a finite run of samples is a standard problem in [[Detection_theory|detection]] and estimation theory built on this mean-square version of the reconstruction formula rather than the sample-by-sample guarantee the deterministic case offers. The practical consequence is a change in what a good reconstruction is judged against: instead of asking whether the reconstructed curve equals one particular signal, the question becomes whether its statistics, its mean and its correlation structure, match the process it was drawn from. ## Microsims The primary sketch sums real sinc kernels rather than only asserting that the formula works. A fixed two-cosine test signal, band-limited to 0.34 Hz, is sampled at a spacing the reader drags from 0.5 to 3 seconds, deliberately straddling the spacing the signal's own bandwidth allows; a second slider sets how many kernels on each side of a point are actually summed, from a bare ±1 up to ±24. The time panel plots the true signal, the sample stems, and the resulting sum, colouring the sum green when the sampling rate satisfies the Nyquist condition and red when it does not, so that a reconstruction which still threads every sample dot but no longer matches the curve between them is visually distinct from a correct one. A toggle overlays the individual scaled kernels so the reader can watch them add up to the green curve rather than take the sum on faith, and a second panel isolates one kernel and marks the integer zero crossings responsible for the interpolation property. A diagnostics readout reports the sample rate, the Nyquist rate the signal requires, the number of terms the truncated sum is using, and both the root-mean-square and peak error measured against the true signal. A three.js companion, a new framework sketch, sets the same reconstruction beside the zero-order hold it is compared against throughout this article. *Try:* With the sample spacing kept below the Nyquist limit, raise the kernel-reach slider from ±1 toward ±24 and watch the sketch's own reported RMS error fall toward zero; then push the spacing past the limit and watch the same error stop falling no matter how far the reach is raised. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Whittaker–Shannon_interpolation_formula) : [Wikitube](https://en.wikitube.io/wiki/Whittaker–Shannon_interpolation_formula) Skeleton mirrored at revision 1366710891. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Nyquist–Shannon_sampling_theorem]] - [[Zero-order_hold]] - [[Digital-to-analog_converter]] - [[Discrete_time_and_continuous_time]] - [[Aliasing]] - [[Low-pass_filter]] - [[Spectral_density]] ## References The primary sketch's reported RMS and peak errors are measured only across the interior of the displayed time window, away from its edges, to avoid penalizing the truncated sum for boundary effects that a longer window would remove; this is an illustrative simplification of the measurement, not a claim about the formula, which needs no truncation to converge when the Nyquist condition holds. Page numbers below are PDF pages of the open edition cited. [^shannon49]: Shannon, C. E. "Communication in the Presence of Noise." *Proceedings of the IRE*, vol. 37, no. 1, 1949, pp. 10–21. [^citneeded1]: Citation needed: primary bibliographic details for E. T. Whittaker's 1915 paper introducing the name "cardinal series" and for J. M. Whittaker's 1930s work extending its convergence theory would settle this precisely. [^tib116]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 116–119 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib109]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 108–110 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib120]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 119–120 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY.