The shock function is one of the most basic functions of signal processing theory. Consider the following function:
This is a rectangular function characterized by the fact that the area of the rectangle always remains 1, regardless of the length and width of the rectangle. When the width of this rectangle becomes infinitely small, then its length is infinity, that is, when the width of this function is equal to 0, its strength is equal to infinity, which is called the impact function.
The shock function is defined as follows:
The first integral equation indicates how the rectangular signal changes, its area is always 1 (the integral is the area), and when the width approaches 0, the height approaches infinity. The second equation is simpler , indicating that the width of the ideal shock function ( i.e. time t ) is equal to 0 , so this function has only one value at the moment of t = 0 , and 0 at the rest. So why define such a function? What can it do? Let's look at the following equation:
This equation obviously holds, because only if t=0
it is not 0, so the original integral becomes
Look at the equation below
This equation indicates that over time, the shock function can take out the value of the function f(t) at different times, which is the most important use of the impact function: numerical sampling. The proof can be modeled on the above figure, and t is close to 0 to close to t0. This effect can be represented by a simple circuit:
The sampling process in the figure above is equivalent to the switch S constantly closing and disconnecting, and closing once to get a value of the input voltage Ui. This switch is equivalent to the shock function, expressed in the form of a signal:
Obviously, the smaller the sampling pulse width, the better, the smaller the sampling of the more accurate the data, the ideal sampling results as shown in the figure:
This process is what we often call the process of converting an analog signal into a digital signal, that is, A/D conversion. In order to make this A/D device as accurate as possible, many large companies in the world are now making devices in this area, such as ADI, TI, BB, PHILIP, MOTOROLA and so on.