As before, and as will be the situation later, using the frequency domain simplifies the analysis.
To model the hold operation we use convolution with a
delta function
and a square pulse. The square pulse has unit height and duration ττ.
The duration ττ is the *holding time*, i.e. how
long we *hold* the incoming value.
For the pulses not to overlap we must choose
τ<Ts
τ
Ts
.
The convolution can be seen as a filtering operation, using the square pulse as the
impulse response. If we fourier transform the square pulse
we obtain the frequency response of the filter, which is a
sinc function.

Figure 2 shows the frequency response of the analog square pulse filter.
We have plotted the frequency response for
τ=Ts
τ
Ts
and
τ=Ts2
τ
Ts
2
.

From the figure we can make the following observations

- The signal will be attenuated more and more towards the band edge,
f=0.5f0.5
- For
τ=Ts
τ
Ts
the maximum attenuation is 3 dB at
f=0.5f0.5.
- For
τ=Ts2
τ
Ts
2
the maximum attenuation is 0.82 dB at
f=0.5f0.5.

The distortion is a result of linear operations and can thus be compensated for
by using a filter with opposite frequency response in the passband,

f∈
−0.5
0.5
f
0.5
0.5
.
The compensation will not be exact, but we can make the approximation as accurate
as we wish. The compensation can be made in the reconstruction filter or after the reconstruction
by using a separate analog filter. One can also predistort the signal in a digital filter
before reconstruction. Where to put the compensator and it's quality are cost considerations.