Explain the reconstruction formula for a bandlimited signal from its samples using sinc interpolation.

• A. x(t) = ∑ x[n] sinc((t − nT)/T), with T = 1/Fs
• B. x(t) = ∑ x[n] sinc((t − nT)/2T)
• C. x(t) = ∑ x[n] sinc((t + nT)/T)
• D. x(t) = ∑ x[n] sinc((t − nT)²/T)

Answer: (A)

The idea is that a bandlimited signal can be perfectly reconstructed from its samples by using sinc interpolation, which is the ideal way to reverse sampling when the signal’s bandwidth is below Fs/2. Each sample x[n] contributes a shifted sinc function centered at t = nT, and summing all these shifted kernels rebuilds the original waveform. The spacing between samples is the sampling period T, and the reconstruction uses the kernel sinc((t − nT)/T) with T = 1/Fs. The sinc kernel is the impulse response of an ideal low-pass filter, so this combination exactly recovers the original frequencies without aliasing, provided the signal is truly bandlimited. The other forms would misplace the centers or distort the interpolation kernel, so they wouldn’t reproduce the signal correctly.