Solving for the system unit impulse response
- Signal and System Spring 2023 Homework Requirements and Reference Answer Summary
- Signals and Systems 2023 (Spring) Assignment Requirements - Eleventh Assignment
- Signals and Systems 2023 (Spring) Homework Reference Answers - 11th Homework
01 The eleventh homework
1. Introduction to Exercises
In signal and systems disciplines, a system function is defined as the ratio of the zero-state output signal to the input signal of a linear time-invariant system in the transform domain. According to this definition, let us find the system function. In the eleventh homework, there is a practice question, given the expression of the input signal of the system, get the zero-state response of the system, and find the unit impulse response of the system. Let us now analyze the solution to this problem.
2. Problem solving
In the time domain, the relationship between the zero-state response of a linear time-invariant system and the input signal can be described using a convolution operation. That is, the output is equal to the unit impulse response of the input signal convolution system. In the transform domain, such as Laplace transform, the convolution operation relationship becomes the score relationship. From this we get an important conclusion about the system function, that is, the system function corresponds to the Laplace transform of the system unit impulse response. Therefore, the unit impulse response of the system to be solved in this problem, h(t), can be obtained by inverse Laplace transformation of the obtained system function. Therefore, the first step in solving the exercise is to find the system function H(s) of the system. The zero-state output of the system under the action of the exponential signal has been given before the exercise, so as long as the two signals are transformed into the transform domain, their ratio is the system function of the system.
In order to find the system function, the guided system input signal and output signal are firstly transformed by Laplace. The Laplace transform corresponding to the input signal is s+1 with a score of 1. The output signal contains three exponential signals, and the corresponding Laplace transform is the superposition of three rational fractions. Next, according to the definition of the system function, the above two Laplace transforms are divided to obtain the system function of the system, and the unit impulse response h(t) of the system can be obtained by using the factorization method to perform the inverse Laplace transform . Here the result of the transformation is given, and here the unit impulse response of the system is obtained.
※ Summary ※
This paper discusses an exercise in the eleventh assignment to solve the unit impulse response of a linear time-invariant system with known input and output signals. To analyze directly in the time domain, it needs to be solved by deconvolution operation. In the transform domain, the unit impulse response signal of the system can be obtained more conveniently by using the concept of system function.
#!/usr/local/bin/python
# -*- coding: gbk -*-
#============================================================
# TEST1.PY -- by Dr. ZhuoQing 2023-05-17
#
# Note:
#============================================================
from headm import *
from sympy import symbols,simplify,expand,print_latex
from sympy import *
#------------------------------------------------------------
s = symbols('s')
Hs = 1/2/(s+1) - 2/(s+2) + 1/(s+3)
Hs = Hs*(s+1)
Hs = apart(Hs)
result = Hs
#------------------------------------------------------------
print_latex(result)
_=tspexecutepythoncmd("msg2latex")
clipboard.copy(str(result))
#------------------------------------------------------------
# END OF FILE : TEST1.PY
#============================================================
H ( s ) = − 0.5 + 1.0 0.5 s + 1.0 − 0.666666666666667 0.333333333333333 s + 1.0 H\left( s \right) = - 0.5 + { {1.0} \over {0.5s + 1.0}} - { {0.666666666666667} \over {0.333333333333333s + 1.0}} H(s)=−0.5+0.5s+1.01.0−0.333333333333333 s.d+1.00.666666666666667
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