This post covers properties of LTI system. Linear-time invariant systems, that were partially discussed before, play an important role in describing signals. Generally speaking, any process can be described with the idea of Linear-Time Invariant systems. So here we will consider LTI system properties in detail.

Let’s consider continuous-time function f(t). We can divide it into fractions as depicted in Figure 1. Each fraction can be described with δ function δ(t)=1τ, 0<t<τ0, t<0, or t>τ.

Let’s assume that T0, then f(t)=f(φ)δ(tφ)dφ. Let’s assume that h(t) is the response of LTI system to the input δfunction. So the LTI system output for the function f(t) will be g(t)=f(φ)h(tφ)dφ.

This expression is called a convolution integral for a system. Convolution is the mathematical operation of obtaining the third function from two others, describing how one of the function changes the form of another. Convolution can also be written as g(t)=f(t)*h(t).

properties of LTI system

Figure 1.Continuous-time function represented with the impulses.

Let’s represent discrete-time signal as the row of unit impulses, as depicted in Figure 2. Mathematically it will be fn=k=fkδnk . As we know, δfunction is the function equal to zero everywhere except zero. In our case, we can consider shifted δfunction with weight fk in every considered point. So here the considered function is a superposition of δfunctions with corresponding weights.

Let’s assume that the LTI system response for δfunction is hk, then the output function of the LTI system will be gn=k=fkhnk. This expression is called a convolution sum for a system. Convolution is the mathematical operation of obtaining the third function from two others, describing how one of the function changes the form of another. Convolution can also be written as gn=fn*hn.

properties of LTI systemFigure 2. The discrete-time function represented with impulses.

Here are some properties of linear-time invariant systems convolution.

  1. Commutative property. For the input functions f(t) and fn, and LTI system responses h(t) and hn the following expressions are valid:f(t)*h(t)=h(t)*f(t)=f(tφ)h(φ)dφ and fn*hn=hn*fn=fnkhk .
  2. Distributive property. For the input functions f(t) and fn, and LTI system responses h1(t)h2(t) and h1nh2n the following expressions are valid: f(t)*(h1(t)+h2(t))=f(t)*h1(t)+f(t)*h2(t) and fn*(h1n+h2n)=fnh1n+fnh2n.
  3. Associative property. For the input functions f(t) and fn, and LTI system responses h1(t)h2(t) and h1nh2n the following expressions are valid:f(t)*(h1(t)*h2(t))=(f(t)*h1(t))*h2(t), and fn*(h1n*h2n)=(fn*h1n)*h2n.
  4. Inversion property. If the LTI system is an inverting function, then h1(t)*h2(t)=δ(t) and h1n*h2n=δn, here h1(t), h2(t), h1n, h2n are responses of the input and output functions in the case of continuous-time and discrete-time functions.
  5. Stability property. Let’s assume that the input function is limited, so f(t)<A, fn<B, then considering the convolution sum and integral we can understand if the output function is stable. g(t)<Ah(φ)dφ, so the output function gt will be stable if the integral h(φ)dφ<. Similarly we can show for the discrete-time function gn<k=Bhk is stable if the sum k=hk<. [1]

More educational tutorials can be accessed via Reddit community r/ElectronicsEasy.

 

[1] “Signals and systems”, 2nd edition, 1997. Alan V. Oppenheim, Allan S. Willsky.

 

Tags:

Leave a Reply