Practical Reconstruction

Practical reconstruction approximates the ideal sinc interpolation using realisable filters that can be built in hardware. It is the core operation inside every digital-to-analog converter used in smartphones, audio equipment, and communication transmitters.

Core Concept

The zero-order hold (ZOH) is the simplest practical reconstruction method. Between each pair of consecutive samples, the DAC output holds the previous sample value constant. This produces a staircase waveform. The staircase is a rough approximation of the original signal.

The ZOH has a frequency response shaped like a sinc function. It attenuates high frequencies within the baseband, so the reconstructed signal has a slightly droop near the Nyquist frequency. This is called ZOH distortion. An equalisation filter with a 1/sinc response corrects this droop.

After the ZOH, a smoothing analog low-pass filter removes the spectral replicas at multiples of the sampling frequency. This filter does not need to be a perfect brick wall. A Butterworth or elliptic filter of sufficient order is adequate in practice.

Key Formula

The ZOH frequency response for sampling period T is:

H_ZOH(f) = T * sinc(fT) * exp(-j*pi*f*T)

Here sinc(fT) = sin(pi*f*T)/(pi*f*T) is the magnitude distortion, and exp(-j*pi*f*T) is a linear phase shift corresponding to a half-sample delay. The equaliser needed to correct ZOH droop has response 1/sinc(fT) within the passband.

Problem: Find the ZOH output for samples x[0]=1, x[1]=3, x[2]=2 with T=1s

Step 1: ZOH holds each sample for one period T
  Interval 0 <= t < 1: output = x[0] = 1
  Interval 1 <= t < 2: output = x[1] = 3
  Interval 2 <= t < 3: output = x[2] = 2

Step 2: Staircase output x_ZOH(t) is a piecewise constant function

Step 3: ZOH distortion at f = 0.4*fs (near Nyquist)
  |H_ZOH(0.4*fs)| = T * |sinc(0.4)|
  sinc(0.4) = sin(0.4*pi)/(0.4*pi) = 0.935/(1.257) = 0.744
  So amplitude at 0.4*fs is attenuated to 74.4% of ideal

Step 4: Equaliser gain needed at 0.4*fs
  G_eq = 1/0.744 = 1.344

Final Answer: ZOH introduces ~2.6 dB droop at 0.4*fs, corrected by equaliser gain 1.344

Exam Tip: In GATE, the ZOH transfer function is often written as H(s) = (1 - e^(-sT))/s in the Laplace domain. Taking the magnitude on the j-omega axis gives the sinc shape. The half-sample delay exp(-j*pi*f*T) does not cause distortion in magnitude, only a linear phase shift. Do not confuse ZOH distortion with aliasing: aliasing happens before sampling, ZOH distortion happens after.

Properties Summary

• ZOH response: H_ZOH(f) = T*sinc(fT)*exp(-j*pi*f*T), magnitude is a sinc envelope.
• ZOH distortion: amplitude at frequency f is attenuated by sinc(fT) relative to ideal reconstruction.
• Equalisation: post-filter with gain 1/sinc(fT) restores flat response across the passband.
• Smoothing filter: analog LPF after ZOH removes images at fs, 2fs, 3fs, etc.
• Oversampling: running the DAC at 4x or 8x the Nyquist rate relaxes the analog filter order requirement.
• First-order hold (FOH): linearly interpolates between samples, providing better approximation than ZOH at the cost of complexity.

Quick Revision

• Practical reconstruction = ZOH staircase + analog smoothing LPF.
• ZOH Laplace transfer function: H(s) = (1 - e^(-sT))/s.
• ZOH magnitude distortion at f is sinc(fT), worst near Nyquist.
• Equaliser compensates for ZOH droop with a 1/sinc response.
• Smoothing filter removes spectral replicas above fs/2.
• Oversampling reduces the sharpness required of the analog smoothing filter.
• Exam trap: applying ZOH distortion correction to aliasing problems; the two effects are separate and occur at different stages.