Systems view of sampling and reconstruction

Ideal reconstruction system

Figure 1 shows the ideal reconstruction system based on the results of the Sampling theorem proof.

Figure 1 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)=sinc(

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)=

∞

∑

n=−∞

(x_s(n)δ(t−nT)).

**Figure 1:** Ideal reconstruction system

But when will the system produce an output x(t)=x(t)? According to the sampling theorem we have x(t)=x(t) when the sampling frequency, F_s, is at least twice the highest frequency component of x(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 Figure 2.

**Figure 2:** Ideal reconstruction system with anti-aliasing filter

Again we ask the question of when the system will produce an output x(t)=s(t)? If the signal is entirely confined within the passband of the lowpass filter we will get perfect reconstruction if F_s is high enough.

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, 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 the reconstruction 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 device to a compensator. The compensation can be as accurate as we wish, this is cost and application consideration.