Ideal Reconstruction

Ideal reconstruction converts a sequence of discrete samples back into a perfectly smooth continuous-time signal, with zero distortion. This operation underpins digital-to-analog conversion in audio playback, where stored samples must become a continuous waveform reaching your speakers.

Core Concept

When a band-limited signal is sampled above the Nyquist rate, all the original information is preserved in the samples. Ideal reconstruction is the mathematical process of recovering that original continuous signal exactly, without any approximation.

The reconstructed signal is formed by placing a sinc function at each sample location. Each sinc is scaled by the sample value. Summing all these shifted sinc functions gives back the original signal perfectly. This sum is called the Whittaker-Shannon interpolation formula.

In the frequency domain, ideal reconstruction is a perfect low-pass filter with cutoff at the Nyquist frequency fs/2. It passes the baseband copy of the spectrum and removes all spectral replicas created during sampling. The filter gain is T (the sampling period) to restore correct amplitude.

Key Formula

The Whittaker-Shannon interpolation formula reconstructs x(t) from its samples x[n] = x(nT):

x(t) = sum over n of x[n] * sinc((t - nT) / T)

Here T is the sampling period, n ranges over all integers, and sinc(u) = sin(pi*u)/(pi*u). The ideal low-pass filter in frequency domain has transfer function H(f) = T for |f| <= fs/2 and 0 otherwise, where fs = 1/T.

Given: x[n] = {1, 2, 1} sampled at T = 1s, reconstruct x(t) using sinc interpolation

Formula: x(t) = sum_n x[n] * sinc(t - n)

Step 1: Identify non-zero samples
  x[0] = 1, x[1] = 2, x[2] = 1

Step 2: Write each sinc term
  Term n=0: 1 * sinc(t - 0) = sinc(t)
  Term n=1: 2 * sinc(t - 1)
  Term n=2: 1 * sinc(t - 2)

Step 3: Combine
  x(t) = sinc(t) + 2*sinc(t-1) + sinc(t-2)

Step 4: Verify at sample points
  t=0: sinc(0) + 2*sinc(-1) + sinc(-2) = 1 + 0 + 0 = 1  (correct)
  t=1: sinc(1) + 2*sinc(0)  + sinc(-1) = 0 + 2 + 0 = 2  (correct)
  t=2: sinc(2) + 2*sinc(1)  + sinc(0)  = 0 + 0 + 1 = 1  (correct)

Final Answer: x(t) = sinc(t) + 2*sinc(t-1) + sinc(t-2)

Exam Tip: Ideal reconstruction requires two conditions to work perfectly: the signal must be strictly band-limited, and sampling must occur at or above the Nyquist rate. If either condition fails, reconstruction produces aliased output. In GATE problems, the phrase 'ideal reconstruction' always implies a perfect brick-wall low-pass filter with gain T and cutoff fs/2. Do not confuse the sinc function sinc(u) = sin(pi*u)/(pi*u) with the normalized sinc used in some textbooks.

Properties Summary

• Perfect recovery: if x(t) is band-limited to W Hz and fs >= 2W, then x(t) is recovered exactly from its samples.
• Sinc interpolation kernel: each sample contributes a sinc pulse x[n]*sinc((t-nT)/T) centered at t = nT.
• Frequency domain filter: ideal reconstruction filter H(f) = T * rect(f/fs), a brick-wall LPF of gain T.
• Orthogonality: sinc functions at different sample times are orthogonal, sinc((t-nT)/T) evaluated at t=mT equals delta[m-n].
• Energy preservation: by Parseval, energy in the reconstructed signal equals T times the sum of squared samples.
• Causality: the ideal sinc filter is non-causal and infinite in duration, so true ideal reconstruction is physically unrealisable.

Quick Revision

• Ideal reconstruction = sinc interpolation in time = ideal LPF in frequency.
• LPF cutoff frequency is fs/2 = 1/(2T).
• Filter gain must be T to restore the correct amplitude after sampling.
• Sinc(0) = 1, sinc(n) = 0 for all nonzero integers n. This ensures each sample is reproduced exactly.
• The formula works only if no aliasing occurred during sampling.
• Practical DACs approximate the sinc filter using zero-order hold plus equalisation filters.
• Exam trap: forgetting the gain T in the ideal LPF means the reconstructed amplitude is off by a factor of T.