Sinc Interpolation Error
Estimate the aliasing/truncation error floor when reconstructing a band-limited signal from finite samples.
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The engineering
Ideal reconstruction of a band-limited signal is a convolution with an infinitely long sinc kernel (Whittaker-Shannon). On real hardware you truncate that kernel to a finite number of taps N and you sample faster than the strict Nyquist rate to leave a guard band. Both choices set an error floor: the more oversampling and the longer the kernel, the lower the residual.
Rule of thumb: doubling the tap count buys you ~6 dB, and every extra bit of guard band (moving f_s away from 2·f_max) helps more than adding taps. If your interpolator sits right at Nyquist there is no guard band, the sinc tails never decay fast enough, and the error blows up — always oversample before you truncate.
Where this math comes from
The interpolation formula itself is Whittaker's (1915) cardinal series, popularized for communications by Claude Shannon in his 1949 sampling paper. But turning ideal sinc reconstruction into something a fixed-point DSP could actually run — windowed, truncated, and multirate — was the work of the digital filter community through the 1970s and 80s.
P. P. Vaidyanathan's 1993 text Multirate Systems and Filter Banks pinned down the truncation-versus-oversampling tradeoff that governs practical polyphase interpolators, giving DAQ and sample-rate-conversion designers the error bounds this card estimates.
- 1915E. T. WhittakerIntroduces the cardinal (sinc) interpolation series.
- 1949Claude ShannonTies sinc reconstruction to the sampling theorem for communications.
- 1973R. W. Schafer & L. R. RabinerDigital-signal-processing treatment of finite sinc interpolation.
- 1993P. P. VaidyanathanMultirate Systems and Filter Banks formalizes the truncation/oversampling error tradeoff.
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