I want to know what is the difference between sample method and sample and hold method ,and why are these two methods important to study analog to digital conversion and its reconstruction back to analog signal, and what is the effect of different sampling frequencies on the reconstructed signal.
The basis for sampling of signals is the concept of the "unit impulse" function, which is defined as a pulse at t=0 of infinitesimal width and infinite height that has an area of exactly 1. The integral of the unit impulse is the "unit step" function, which is 0 for t<0 and 1 for t>0. These two functions have some useful mathematical properties that can be taken advantage of in general signal analysis.
When we talk about "sampling a signal", what we mean is multiplying the signal by a train of unit impulses that repeat at some sample rate. This produces a train of infinitesimal pulses whose height (and area) now varies based on the original signal's value at those moments in time. Much of the work of people like Shannon and Nyquist is based on the characteristics of a signal sampled in this way.
However, in practice, it's difficult to build an electronc circuit that can process such an impulse signal. In particular, we often want to digitize each sample, and this generally takes some non-infinitesimal amount of time. Therefore, we build what are called sample-and-hold circuits, which convert the modulated unit impulses into a series of rectangular pulses that have the same width as the sample period and the same area as the impulses, which means that the heights of these pulses still represent the values of the original (unsampled) signal, but now in a range that can be easily measured by an ADC.
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