Sinc interpolation

Sinc interpolation (also called Whittaker–Shannon interpolation) is the exact reconstruction of a sampled bandlimited signal from its samples. Given samples $x(n{T}_s)$ taken at rate $f_s=1/T_s$ above the Nyquist rate, the original continuous signal is

$x(t)={\sum }_{n=-\infty }^{\infty }x(n{T}_s)\mathrm{sinc}\left(\frac{t-n{T}_s}{T_s}\right).$

Each sample contributes a sinc-shaped pulse centered at $t=n{T}_s$, with height equal to the sample value. Summing all the sincs gives back the original signal exactly.

Derivation

The spectrum of the impulse-sampled signal $x_{\delta }(t)$ is $X_{\delta }(f)=f_s{\sum }_{n}X(f-n{f}_s)$ — copies of $X(f)$ spaced $f_s$ apart. When sampling above Nyquist ($f_s>2f_m$), the copies don’t overlap.

To recover $X(f)$, multiply by a frequency-domain rectangle $(1/f_s)\mathrm{rect}(f/f_s)$: this gates through the $n=0$ copy, scales it by $1/f_s$ to cancel the $f_s$ factor that sampling introduced, and zeros out all other copies. The result is $X(f)$.

In time, multiplying by a frequency rect becomes convolving with the sinc inverse transform. Using the normalized-sinc convention $\mathrm{sinc}(x)=\sin (\pi x)/(\pi x)$ and the pair $\mathrm{rect}(t/w)\leftrightarrow w\mathrm{sinc}(wf)$ read in reverse (with $w=1/f_s$, so the frequency-domain rectangle has width $f_s$):

${\mathcal{F}}^{-1}\{(1/f_s)\mathrm{rect}(f/f_s)\}=(1/f_s)\cdot f_s\mathrm{sinc}(f_{s}t)=\mathrm{sinc}(f_{s}t)=\mathrm{sinc}(t/T_s).$

So $x(t)=x_{\delta }(t)\ast \mathrm{sinc}(t/T_s)$. Since $x_{\delta }(t)={\sum }_{n}x(n{T}_s)\delta (t-n{T}_s)$, the convolution becomes a sum of shifted sincs — the formula above.

A beautiful property

The sinc function $\mathrm{sinc}((t-n{T}_s)/T_s)$ equals $1$ at $t=n{T}_s$ and equals $0$ at every other sample instant $t=m{T}_s$ (since the sinc has zeros at every nonzero integer argument). So at the sample instants, only one term in the sum contributes, and it contributes exactly $x(n{T}_s)$.

Between sample instants, the sinc tails from all the samples combine to give the original signal’s value at that intermediate time. This is interpolation in the most literal sense: the samples pin down the function at the integer sample times, and the sincs smoothly interpolate between them.

Why it’s not practical

The sinc function has infinite support — its tails extend to $\pm \infty$. To compute $x(t)$ at a single intermediate time, you’d need every sample from every time. Real systems can’t do this; they have finite memory.

Practical reconstruction uses approximations:

• Zero-order hold: each sample is held constant until the next. Square-wave reconstruction, easy to implement but high-frequency content distorted.
• First-order hold: linear interpolation between adjacent samples. Smoother than ZOH.
• FIR filters approximating sinc: a finite-length impulse response that approximates the ideal sinc with windowed truncation. Used in audio DACs, image upscaling, and most modern reconstruction systems.

The cost of approximation: small aliasing artifacts in the reconstructed signal. The fewer taps, the more artifacts; the more taps, the better the reconstruction (and the more computation).

The signal chain

Sinc interpolation (or its practical approximation) is the last step in the digital-to-analog conversion chain:

$\text{bits}\stackrel{\text{decode}}{\to }q[n]\stackrel{\text{dequantize}}{\to }x[n]\stackrel{\text{reconstruct}}{\to }x(t).$

A real DAC implements an approximation of sinc interpolation, followed by an analog reconstruction filter (also called a smoothing filter) that suppresses any remaining out-of-band content.