Signal and system 5: analog and digital signal processing

This section describes the unique properties of Linear Time-varying systems (Note: Only the LTI system can meet the requirements, and other systems may not !)

Three algebraic properties: Exchange Law, combination law, allocation Law

In a series cascade, positions can be exchanged freely to meet the exchange law. For example, if it is not a linear system, positions cannot be changed randomly in a series cascade, for example, the square after the first request is different from the square before the first request. Therefore, we can look at whether the system is LTI in another way. Check whether the exchange law is used ,(But is LTI a sufficient condition to meet the exchange law and system requirements?)

As we mentioned above, the LTI system can be fully expressed by the unit impulse response, so we try to explain the nature of the system, such as whether there is memory, whether it is reversible, stable, and causal, is it associated with the system's impulse response?

1. There is no memory, there is a convolution formula, available, in order to meet the non-memory requirements, that is, the Unit Impulse Response H (n) must be K delta (N) (K is a constant ).

2. reversible. In order to meet the reversible requirement, another impulse response H1 (n) is required, and the convolution result between H (N) and H1 (n) is delta (n)

3. Stability: the impulse response must meet the absolute and (absolute product) conditions, that is:

Input x (n) is bounded x (n) <C, and output must also be bounded.

Therefore, the impulse response is absolutely sum (absolutely product ).

4. Causal. When n is less than 0, H (n) = 0 indicates that the most essential cause is that the system cannot predict the future output (if the first two inputs are the same at a time point, the output before the current time is the same, because the output after the current time cannot be predicted by the listener ).

Finally, I talked about an operational definition. To be honest, I didn't quite understand it. I vaguely understood it as defining something hard to define from another perspective (evaluate the Unit Impulse Function ).