Signal and system 3: Unit Step, unit response, and system nature

There are two main points: First, two types of functions, unit step and unit response. Second, some basic properties of the system.

1. Naturally, the Unit Step and unit response must be divided into continuous time and discrete time. Pay attention to the relationship between unit step and unit response.

Delta (n) = u (N)-U (n-1), u (n) = sum of negative infinity to n (delta (n )).

The continuous time is similar, but the trouble is the unit step response of the continuous time, because it is troublesome to reach the limit. My understanding is the nature of the problem: the flexible use of an area of 1 is more conducive to processing.

In the middle of the course, the professor inserted the basic combination of system interconnection, and interconnected series feedback.

2. Focuses on the several properties of the system, which are non-memory, reversible, causal, stable, time-unchanged, and linear.

(1) There is no memory. The official definition of output is only related to input. It is a simple nature;

(2) reversible and irreversible. Many definitions can be: whether the output can be unique to get input; or: two different inputs must be two different outputs, which is convenient for application.

If you want to prove irreversible, you only need to check whether the previous definition is not satisfied, but to prove reversible, you cannot use the definition, if you can find a system of this system, the system will be OK (for example, the integral circuit is reversible, because its inverse system is a differential circuit, but in turn it will not be true, because the constant differentiation cannot be unique to get the input ).

(3) result: the output is only related to the current and previous input. With the professor's understanding, the future is unpredictable in the macro sense! It can also be learned that none of memory is causal.

In addition, considering the cause and effect, we can also have a system impact response to determine that if n is <= 0, H (n) = 0, it is stable;

There is a problem. In the book of Oppenheim, an example is Y (t) = x (t) Cos (t + 1). This is causal! Even non-memory. Cos (t + 1) is a time-varying function, so only input x (t) affects the output. I don't understand!

(4) Stability: The input and output are also bounded. It is also quite understandable, but the proof is the same as reversible and irreversible, and the inverse method can be used for instability, but the proof of stability requires rigorous mathematical proof. In addition, the professor uses a well-understood experiment to explain that feedback can make instability stable. At this time, it is suggested that the negative feedback is basically conducive to making stability, and positive feedback is more likely to damage stability.

(5) constant, linear: Two super important properties. There is not much nonsense, and you can use the mathematical proof to deduce it. (You must be extremely careful when the proof remains unchanged !)

The professor also gave an example of many systems. The following describes three types of systems: accumulators, differential circuits, and integrated circuits.

A. accumulators, using expressions

It is good to know that there is a memory, a causal, and unstable (from the negative infinity), and there is an integral circuit that is reversible to guess that the accumulators are also reversible, the reversible system is the score (? ). The last two important things are constant and linear. It is proved that they are linear and also time-unchanged.

B. integral circuit, time unchanged, linear proof is still smooth, and memory, causal, reversible, unstable (such as x (t) = E ^ (-3 T ), when there are credits, we can know that the x (t) divergence is infinite when the negative infinity occurs ).

C. differential circuit, because the formula for differentiation is very abstract, it is easier to judge with the definition of Differentiation

With memory, irreversible, and causal,Unstable (this is not quite understandable)

According to the above formula, it can be more convenient to prove that the system is linear and constant.