Introduction to Interpolation

August 22, 2019

It can be useful to recover an original waveform from a sample sequence. If sampling is the mathematical operator 𝐒 that maps a waveform 𝗑(𝑑) to a sequence 𝗒(𝑛) such that 𝗒(𝑛) = 𝐒𝗑(𝑑), then, ideally, a waveform can be recovered by using the inverse operator π’βˆ’1 such that 𝗑(𝑑) = π’βˆ’1𝗒(𝑛).

Mapping a sequence back to a waveform may be referred to as synthesis, reconstruction, or interpolation. There is an infinite number of interpolation methods. This course presents two general methods: waveform approximation using Lagrange polynomial interpolation and waveform perfect reconstruction using sinc interpolation.

Waveform Interpolation

Lagrange polynomial and sinc interpolation are commonly used together in a single system.

DACs (digital to analog converters) commonly use zero-order hold circuitry (β€œsample and hold”) at its outputs equivalent to order-0 Lagrange polynomial interpolation. This circuitry is easy to implement in silicon and tends to translate to a lower component cost.

Sinc interpolation can be approximated using an analog low-pass filter called a reconstruction filter. The closer the filter is to an ideal β€œbrick-wall” filter, the closer it performs sinc interpolation. Low-pass filters can be realized on a circuit board using simple components such as resistors, capacitors, and op-amps.

Distortion

Both methods introduce distortion in the frequency domain of the system.

The zero-hold circuitry acts as a low-pass filter that has a magnitude in the shape of a sinc function. If the original waveform has been sufficiently over-sampled (well above the Nyquist rate), the distortion can be canceled out by the reconstruction filter.

By definition, the low-pass filter introduces distortion by attenuating spectral energy above the cut-off frequency. However, if the waveform is band-limited such that it has no energy above the cut-off frequency, then the distortion is effectively eliminated. A waveform can be (and typically is) forced to be band-limited by the use of an analog anti-aliasing low-pass filter.