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Hold operation

Module by: Anders Gjendemsjø

Summary: Short description of the hold operation for reconstruction

Any practical reconstruction system must input finite length pulses into the reconstruction filter. The reason is that we need nonzero energy in the nonzero pulses.

Introduction

The operation performed to produce these pulses is called hold. Using the hold-operation we get pulses with a predefined length and height proportional to the input to the digital-to-analog converter. By means of the hold operation we get nonzero pulses with energy.
hold.jpg
Figure 1: Output signal from the hold device
As we have made changes relative to the ideal reconstruction, we need to look at the output signal the reconstruction filter will give us. Quite obviously the output will not be the original signal. So, is it still useful?

Analysis

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 .
Hold_freq.jpg
Figure 2: Frequency response of the analog square filter as a function of digital frequency f.
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∈-0.50.5 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.

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