A hold operation can be modeled using a step function over one sampling period i.e.

$R(t) = 1/T * (h(t) - h(t-T))$, $h(t)$ the step function

In frequency domain this is equivalent to

$R(jw) = e^{-jwT/2}*sinc(wT/2)$

We see there exists a phase distortion of $T/2$ and a magnitude distortion from the $ sinc(wT/2)$ term.

This means that all the reconstructed signal will be distorted unless operated at very low frequency.

Can someone explain what induces this distortion and what is done to minimize both the phase and magnitude distortion


What you are describing is the distortion introduced by an ideal digital-to-analog converter (DAC) in the analog domain. Two things are typically done to reduce this distortion:

  1. Analog filtering
  2. Oversampling

As you note in your question, the distortion is modeled, in the frequency domain, by a sinc rolloff. Increasing the sample rate before converting the digital signal to analog reduces $T$/2, which make the rolloff slower, or "flatter".

Analog filtering can range from simple low-pass filters to compensation filters that counteract the sinc rolloff. These filters do not, I believe, fix the phase issues, but they can greatly improve the frequency amplitude response.

CD players are a good example of these techniques. Audio CD's are sampled at 44.1 kHz, but before the DAC the CD players will typically increase the sample rate to a few hundred kHz. I'm not sure how sophisticated their analog filtering is. My guess would be that the cheap stuff does simple low-pass filtering and the expensive ones do sinc rolloff compensation.


I agree with Jim Clay's answer, but I think it is important to point out two things. First of all, there are no phase distortions due to the hold operation, just a simple delay of half a sampling interval. So nothing needs (and can) be done about the phase. Second, it is important to realize that the gain roll-off due to the sinc shape is relatively mild. Note that the gain at Nyquist ($\omega=\pi/T$) is


which corresponds to $-3.9224\,\text{dB}$ (without oversampling). This slight gain distortion can be easily compensated by the low pass filter following the hold operation.


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