3.9 Systems view of sampling and reconstruction

Ideal reconstruction system

shows the ideal reconstruction system based on the results of the Sampling theorem proof .

consists of a sampling device which produces a time-discrete sequence $x_s(n)$ . The reconstruction filter, $h(t)$ , is an ideal analog sinc filter, with $h(t)=\mathrm{sinc}(\frac{t}{T_s})$ . We can't apply the time-discrete sequence $x_s(n)$ directly to the analog filter $h(t)$ . To solve this problem we turn the sequence into an analog signal using delta functions . Thus we write $x_s(t)=\sum_{n=()}$∞ ∞ x s n δ t n T .

Ideal system including anti-aliasing

To be sure that the reconstructed signal is free of aliasing it is customary to apply a lowpass filter, an anti-aliasing filter , before sampling as shown in .

But if the anti-aliasing filter removes the "higher" frequencies, (which in fact is the job of the anti-aliasing filter), we will never be able to exactly reconstruct the original signal, $s(t)$ . If we sample fast enough we can reconstruct $x(t)$ , which in most cases is satisfying.

The reconstructed signal, $\widehat{x}(t)$ , will not have aliased frequencies. This is essential for further use of the signal.

Reconstruction with hold operation

To make our reconstruction system realizable there are many things to look into. Among them are the fact that any practical reconstruction system must input finite length pulses into thereconstruction filter. This can be accomplished by the hold operation . To alleviate the distortion caused by the hold opeator we apply the output from the hold deviceto a compensator. The compensation can be as accurate as we wish, this is cost and application consideration.

• Introduction
• Proof
• Illustrations
• Matlab example
• Hold operation
• Aliasing applet
• Exercises