The aim of this post is o summarize linear system definitions such as admissible, causality, time invariance relaxed and impulse function.
For Korean version please refer to here. If you can understand the following terms, you can skip this post.
U and V are input and output space which is set of admissible signal
The picture below is the main content will explain linear system definitions. Now let’s start step by step.
The space comes up which is a very important definition in linear system and linear algebra .Here, I would like to give a rough outline rather than a mathematical point of view. Let’s think about it simply.
Several experiments were conducted. In order to organize the experimental results, the input and output data are separately collected and organized. where, collected input data is called input space and output data is output space.
it is annoying to put it in words, so we can express it mathematically as follows. For reference, to clearly understand the term space, you need to know the basis and the Linear operator, but you don’t need to know it clearly in a Linear system.
u(t) \in U \rightarrow system \rightarrow y(t) \in Y \\U:\{u(t_0),u(t_1),\cdots,u(t_k)\} \\V:\{y(t_0),y(t_1),\cdots,y(t_k)\} \\it can be piecewise continuous and exponentially bound conditions must be satisfied. By exponentially bound we mean a signal that can partially diverge slightly, but does not diverge rapidly. That is, if bound can be expressed as an exponential function, it can be considered admissible.
A system that satisfies the linear operator is called a linear system. Linear operator means that it satisfies additivity and Homogeneity.
The time invariance system concept is very important.Time invariance is a system in which the coefficient values of each component do not change with time.
If the coefficients change with time, then it is a time varying system. Most of the posts will cover the time invariance system first, followed by the time varying system. The mathematical expression is as follows. (N is not dependent on time)
\text{if only if (iff) } \forall t \in T \\N:U \rightarrow Y \\y(t)=Nu(t) \\y(t-T)=Nu(t-T)Causality means that future do not affect the present and the past. In other words, it can be interpreted as "operator depending on the past and present results." The mathematical expression is as follows.
u_1(t)=u_2(t) \\N:U \rightarrow Y \\u_1(t)=u_2(t) \, \forall t \text{< T} \\Nu_1(t) =Nu_2(t)u(t)=\begin{cases} 0 \, t<0\\1 \text{ else}\end{cases} The step function can be defined as above, and its derivative value is called the impulse function. \frac{du(t)}{dt}=\delta(t)
The graphs of the Step function and Impulse function are as follows. If we follow it slowly, we will lead to the delta function.
Just because we have a Step that increments by 1 at a time, we can represent it as a Step function. However, if you think about this a little more realistically, if you think that it takes time for T to increase to 1, it can be expressed as a Ramp function.
When looking at the derivative values for this, the Ramp function is expressed as follows. It increases with a constant slope up to T, and then becomes 0 from T. Now, if we send this T to 0, it becomes a function called the impulse function or Dirac delta function.
let’s consider impulse response. we can examine below equation.
u\backsimeq \sum _{\ }^{\ }u\left(t_i\right)\delta \left(t-t_i\right)\Delta \\ y=Hu\ \\ y=\sum _{\ }^{\ }Hu\left(t_i\right)\delta \left(t-t_i\right)\Delta \\ y=\sum _{\ }^{\ }H\delta \left(t-t_i\right)u\left(t_i\right)\Delta \ \\ where,\ t_i=\tau \ and\ \Delta \to 0\\ \text{The overall interval is proportional to} \frac{1}{\ \Delta } \\\therefore \text{ This sum can be expressed as an integral}\\ y=\int _{\ }^{\ }H\delta \left(t-\tau \right)u\left(\tau \right)d\tau \\ H\delta \left(t-\tau \right)=g\left(t,\tau \right)This means that the state of the system can be considered 0 until any impulse is applied!! To put it simply, we can say this, but let’s say it a little more difficult.
I put the same input value several times at time t0, but the output value is different. This is not a relaxed system.
The reason is that a certain value was stored in the system (because it is not 0), so different output values may appear.
In other words, when a signal of 0 is applied to t0, it refers to a situation in which a certain signal is detected. In the relaxed system, if 0 is input, 0 should come out, which means that the value of the system is 0.
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