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신호와 시스템 3장 솔루션(한티 미디어).doc

신호와 시스템 3장 솔루션(한티 미디어).doc

Solutions to Exercises of Chapter 3
Exercises with Solutions
Exercise 3.1
Consider the following first-order, causal LTI differential system initially at rest:

(a) Calculate the impulse response of the system . Sketch it for .
Answer:
Step 1: Set up the problem to calculate the impulse response of the left-hand side of the equation:
.
Step 2: Find the initial condition of the corresponding homogeneous equation at by integrating the above differential equation from ...Solutions to Exercises of Chapter 3
Exercises with Solutions
Exercise 3.1
Consider the following first-order, causal LTI differential system initially at rest:

(a) Calculate the impulse response of the system . Sketch it for .
Answer:
Step 1: Set up the problem to calculate the impulse response of the left-hand side of the equation:
.
Step 2: Find the initial condition of the corresponding homogeneous equation at by integrating the above differential equation from to . Note that the impulse will be in the term , so will have a finite jump at most. Thus we have , and hence is our initial condition for the homogeneous equation for :
.
Step 3: The characteristic polynomial is and it has one zero at , which means that the homogeneous response has the form for . The initial condition allows us to determine the constant : , so that
.
Step 4: LTI systems are commutative, so we can apply the right-hand side of the differential equation to in order to obtain :

This impulse response is plotted in Figure 3.1 for :

Figure 3.1: Impulse response of the first-order differential system.
(b) Is the system BIBO stable Justify your answer.
Answer: Yes, it is stable. The single real zero of its characteristic polynomial is negative: .
Exercise 3.2
Consider the following second-order, causal LTI differential system initially at rest:

Calculate the impulse response of the system .
Answer:
Solution 1:
Step 1: Set up the problem to calculate the impulse response of the left-hand side of the equation

Step 2: Find the initial conditions of the corresponding homogeneous equation at by integrating the above differential equation from to . Note that the impulse will be in the term , so will have a finite jump at most. Thus we have
,
hence is one of our two initial conditions for the homogeneous equation for :
.
Since has a finite jump from to , the othe


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