This document contains the solutions to an homework assignment on linear and nonlinear systems. It examines several examples and determines whether they are linear or nonlinear by applying the superposition principle. It also identifies examples as causal or non-causal. Finally, it analyzes some circuit examples and determines properties like memoryless, causal, linear, and time-invariant.

1. 1
EE 221
HW 2
Due Date: 16/04/2015
1. Which of the following systems are linear and nonlinear
a. 𝑦(𝑡) = 10𝑥(𝑡 + 2) + 5
b. 𝑦(𝑡) = 𝑥(𝑡2
)
c. 𝑦(𝑡) = 𝑥2
(𝑡)
d. 𝑦(𝑡) = ∫ 𝑒−(𝑡−𝜏)
𝑥(𝜏)𝑑𝜏
𝑡
0
SOLUTION
a) Lets apply superposition to test linearity.
let input to the system by a sum of signals say, 𝑥1(𝑡) + 𝑥2(𝑡), to the output will be
𝑦(𝑡) = 10(𝑥1(𝑡 + 2) + 𝑥2(𝑡 + 2)) + 5 (1)
which is clearly not equal to the sum of the outputs as
𝑦1(𝑡) = 10𝑥1(𝑡 + 2) + 5
𝑦2(𝑡) = 10𝑥2(𝑡 + 2) + 5 𝑎𝑛𝑑
𝑦1(𝑡) + 𝑦2(𝑡) = 10(𝑥1(𝑡 + 2) + 𝑥2(𝑡 + 2)) + 10 (2)
so as (1) ≠ (2), superposition does not hold so the system is not linear.
b) Let input to the system by a sum of signals say, 𝑥1(𝑡) + 𝑥2(𝑡), to the output will be
𝑦(𝑡) = 𝑥1(𝑡2) + 𝑥2(𝑡2) (1)
and sum of outputs
𝑦1(𝑡) + 𝑦2(𝑡) = 𝑥1(𝑡2) + 𝑥2(𝑡2) (2)
so as (1) = (2), superposition holds so the system is linear.
c) Let input to the system by a sum of signals say, 𝑥1(𝑡) + 𝑥2(𝑡), to the output will be
𝑦(𝑡) = 𝑥1
2(𝑡) + 𝑥2
2(𝑡) + 2𝑥1(𝑡)𝑥2(𝑡) (1)
and sum of outputs
𝑦1(𝑡) + 𝑦2(𝑡) = 𝑥1
2(𝑡) + 𝑥2
2(𝑡) (2)
so as (1) ≠ (2), superposition does not hold so the system is not linear.
d) Let input to the system by a sum of signals say, 𝑥1(𝑡) + 𝑥2(𝑡), to the output will be
𝑦(𝑡) = ∫ 𝑒−(𝑡−𝜏)
(𝑥1(𝜏) + 𝑥2(𝜏))𝑑𝜏
𝑡
0
= ∫ 𝑒−(𝑡−𝜏)
𝑥1(𝜏)𝑑𝜏
𝑡
0
+ ∫ 𝑒−(𝑡−𝜏)
𝑥2(𝜏)𝑑𝜏
𝑡
0
= 𝑦1(𝑡) + 𝑦2(𝑡)
so as superposition holds, the system is linear.

2. 2
2. Determine which signals are causal and non-causal
a. 𝑦[𝑛] = 𝑥[−𝑛]
b. 𝑦(𝑡) = 𝑥(𝑛 + 1)
Non Causal
c. 𝑦 ( 𝑡 ) = 𝑥 ( 𝑡 ) 𝑥 ( 𝑡 + 1)
Non Causal
d. 𝑦 ( 𝑡 ) = 𝑥 ( 𝑡 ) + 1
Causal As I is a dc value which is a constant and does not affect x(t)
3. Consider an RC circuit below. Find the input x(t) and output y(t) relationship for this circuit in
a. If 𝑥(𝑡) = 𝑉𝑠(𝑡) 𝑎𝑛𝑑 𝑦(𝑡) = 𝑉𝑐(𝑡)
b. If 𝑥(𝑡) = 𝑉𝑠(𝑡) 𝑎𝑛𝑑 𝑦(𝑡) = 𝑖(𝑡)
Figure 1

3. 3

4. 4
4. Consider the system shown in Fig. 2. Determine whether it is
(a) memoryless,
(b) causal,
(c) linear,
(d ) time-invariant.
Figure 2
The system is linear.

5. 5
5. The discrete-time system shown in Fig. 3 is known as the unit delay element. Determine whether
the system is
(a) Memoryless,
(b) Causal,
(c) Linear,
(d) Time invariant
Figure 3