신호와 시스템 1장 솔루션.doc Report





신호와 시스템 1장 솔루션.doc

신호와 시스템 1장 솔루션.doc

Solutions to Exercises of Chapter 1
Exercises with Solutions
Exercise 1.1
Write the following complex signals in polar form, i.e., in the form for continuous-time signals, for discrete-time signals.
(a)
Answer:

(b)
Answer:


Exercise 1.2
Determine if the following systems are: 1. Memoryless, 2. Time-invariant, 3. Linear, 4. Causal, 5. BIBO Stable. Justify your answers.
(a)
Answer:
1. Memoryless No. For example, the output depends on a future value of the input.
2. Time-invariant No.

3. Linear Yes. Let . Then, the output of the system with is given by:

4. Causal No. For example, the output depends on a future value of the input.
5. Stable Yes. .
(b)
Answer:

1. Memoryless No. The system has memory since at time , it uses the past value of the input .
2. Time-invariant Yes.
3. Linear No. The system is nonlinear since it does not have the superposition property:

4. Causal Yes. The system is causal as the output is a function of the past and current values of the input and only.
5. Stable No. For the bounded input , i.e. the output is unbounded.
(c)
Answer:
1. Memoryless Yes. The output at time depends only on the current value of the input .
2. Time-invariant No.
3. Linear Yes. Let . Then,

4. Causal Yes. The output at time depends on the present value of the input only.
5. Stable No. Consider the constant input , namely, , i.e., the output is unbounded.
(d)
Answer:
1. Memoryless No. The output is computed using all future values of the input.
2. Time-invariant Yes.
3. Linear Yes. Let . Then, the output of the system with is given by:

4. Causal No. The output depends on future values of the input .
5. Stable No. For the input signal , i.e., the output is unbounded.
Exercise 1.3
Find the fundamental periods ( for continuous-time signals, for discrete-time signals) of the following periodic signals.
(a)
Answer:

will equal if ,


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