10.6 The Sample-and-Hold Element

Because most of the systems in which we are interested have both digital and analog parts, we need a circuit element which takes continuous-time signals and "converts" them to discrete-time signals. The sample-and-hold circuit fills this need. An ideal sample-and-hold element that samples at the rate T samples per second has as its input a signal r( t) and as its output c( t) a signal that satisfies:

(See Figure 10.1 for an example of a sine wave that has been passed through a sample-and-hold circuit.) A signal that has been processed by a sample-and-hold circuit only changes values at discrete times.

Figure 10.1: A Sampled-and-Held Sinewave (Dashed Lines) and the Unsampled Sinewave (Solid Lines). The Frequency of the Sinewave is 20 Hz, and 200 Samples Per Second Are Taken.

Taking the Laplace transform of both sides of (10.1), we find that:

Inspecting C( s) carefully, we find that it can be written:

Defining the "star-transform" of the sequence { r(0), r( T), r(2 T), } by:

we find that the Laplace transform of the output of the sample-and-hold circuit is:

In the next sections we will see that R*( S) corresponds to samples of r( t) made by an "ideal sampler" and that is the transfer function of a "zero-order hold."