Interpolation in the Frequency Domain

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Lagrange and sinc interpolation create issues in the frequency domain. The functions multiply in the frequency domain, as illustrated below:

For best results, the system should be designed as follows:

• The zero-order hold circuitry in a DAC (linear or order-2 in more sophisticated DACs) introduces low-pass filtering into the signal path. As such, the sample rate 𝖥s should be set several times higher than the signal’s highest frequency 𝖿h component (256 times if practical). The frequency response of the hold circuitry is approximately flat in the resulting narrow-band occupied by the signal.
• The sinc function interpolation is introduced by the analog reconstruction low-pass filter. It should have a corner frequency greater than 𝖿h.