Question

What is the ideal reconstruction formula or ideal interpolation formula for x(t) = _________

(a) \(\sum_{-\infty}^\infty x(nT) \frac{sin⁡(π/T) (t-nT)}{(π/T)(t-nT)}\)

(b) \(\sum_{-\infty}^\infty x(nT) \frac{sin⁡(π/T) (t+nT)}{π/T)(t+nT}\)

(c) \(\sum_{-\infty}^\infty x(nT) \frac{sin⁡(2π/T) (t-nT)}{2π/T)(t-nT}\)

(d) \(\sum_{-\infty}^\infty x(nT) \frac{sin⁡(4π/T) (t-nT)}{(4π/T)(t-nT)}\)

I got this question in homework.

Question is taken from Digital to Analog Conversion Sample and Hold in portion Sampling and Reconstruction of Signals of Digital Signal Processing

Answer 1

The correct answer is (a) \(\sum_{-\infty}^\infty x(nT) \frac{sin⁡(π/T) (t-nT)}{(π/T)(t-nT)}\)

To explain: x(t) = \(\sum_{-\infty}^\infty x(nT) \frac{sin⁡(π/T) (t-nT)}{(π/T)(t-nT)}\) where the sampling interval T = 1/Fs=1/2B, Fs is the sampling frequency and B is the bandwidth of the analog signal.