4.5 Hold operation

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 lengthand height proportional to the input to the digital-to-analog converter. By means of the hold operation we get nonzero pulses with energy .

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 $$ . The duration $$ is the holding time , i.e. how long we hold the incoming value. For the pulses not to overlap we must choose $< T_s$ . The convolution can be seen as a filtering operation, using the square pulse as theimpulse response. If we fourier transform the square pulse we obtain the frequency response of the filter, which is a sinc function.

shows the frequency response of the analog square pulse filter. We have plotted the frequency response for $=T_s$ and $=\frac{T_s}{2}$ .

• The signal will be attenuated more and more towards the band edge, $f=0.5$
• For $=T_s$ the maximum attenuation is 3 dB at $f=0.5$ .
• For $=\frac{T_s}{2}$ the maximum attenuation is 0.82 dB at $f=0.5$ .

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• Introduction
• Proof
• Illustrations
• Aliasing applet
• System view
• Exercises