Question

digital signal processing

Question 3 2.5 pts What are the properties of a Linear Time-Invariant (LTI) system? HTML Editor

Answer 1

Linear Time Invariant siganls are characterized in the time domain simply by their response to a unit sample sequence.

As a consequence of the linearity and time-invariance properties of the system, the response of the system to any arbitrary input signal can be expressed in terms of the unit sample response of the system.

The general form of the expression that relates the unit sample response of the system and the arbitrary input signal
to the output signal, is called the convolution sum.

Thus we are able to determine the output of any linear, time-in variant system to any arbitrary input signal.

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

1. Commutative property: For the input function and and LTI system responses   and the following expressions are valid:  
   f(t) * h(t) = h(t) + f(t) = | ft-yh(y)do -00
   and
   [Yોષs – u][3= [૫]\ * [u]૫ = []\ + [a]
   
2. Distributive property: For the input function and and LTI system responses  $h_{1}(t)$ , $h_{2}(t)$ and $h_{1}[n],h_{2}[n]$ the following expressions are valid:      and   
3. Associative property: For the input function and and LTI system responses  $h_{1}(t)$ , $h_{2}(t)$ and $h_{1}[n],h_{2}[n]$ the following expressions are valid:  ​​​​​​​ and   
4. Inversion property: If the LTI system is an inverting function, then $h_{1}(t)*h_{2}(t) = \partial (t)$ and  $h_{1}[n]*h_{2}[n] = \partial [n]$, where $h_{1}(t),h_{2}(t),h_{1}[n],h_{2}[n]$ 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$\left | f(t) \right | <A, \left | f[n] \right |<B$ then considering the convolution sum and integral we can understand if the output function is stable. $\left | g(t) \right | <\int_{-\infty }^{\infty} A\left | h(\varphi ) \right | d\varphi$ so the output function   will be stable if the integral $\int_{-\infty }^{\infty} \left | h(\varphi ) \right | d\varphi < \infty$ Similarly we can show for the discrete-time function  . $\left | g[n]< \sum_{-\infty }^{\infty} B\left | h(k ) \right | \right |$ is stable if the sum $\sum_{-\infty }^{\infty} \left | \left | h(k ) \right | \right | < \infty$