E E 238 B1 - Winter 2012 HW #2

Homework Assignment # 2Due: Monday, 2012 January 30, 16:00pm

Instructions:

1. Please ensure that your name and ID number are clearly written on your assignment.

2. Please submit your assignment before 16:00 pm on the due date. The assignment box is located in the

ETLC Atrium (second floor), and is marked by E E 238 B1.

Assignment Problems:

1. (10 points) Simplify the expressions and calculate the integrals.

i)(

sin(t2)t+2

)δ(t); ii)

(sin(kω)

ω

)δ(ω); iii)

∫∞−∞ δ(t − 2)e−jωtdt;

iv)∫∞−∞ f(2 − t)δ(3 − t)dt; v)

∫∞−∞ ex−1 cos

[π2(x − 5)

]δ(x − 3)dx.

For ii), use L’Hopital’s rule.

2. (5 points) (Lathi 1.4-7 (a)) Find and and sketch∫ t

−∞ f(x)dx for the signalf(t) in Figure 1.

0 1

1

-1

3 t

f(t)

1

Figure 1: Signal for Problem 2.

3. (16 points)Classify the systems described below, with the inputf(t) and outputy(t), as linear or

non-linear systems, time-invariant or time-varying systems, systems with memory or memoryless

systems, and invertible or non-invertible systems.

i) y(t) = f(t) + 1; ii) y(t) =∫ 5

−5f(τ)dτ ; iii) dy(t)

dt= f 2(t); iv) y(t) = f(at), a > 1.

4. (6 points)For the systems described below, with the inputf(t) and outputy(t), determine which

are causal systems and which are non-causal systems.

i) y(t) = f(t − 2); ii) y(t) = f(1 − t); iii) y(t) = f (2t).

5. (10 points)(Lathi 1.8-1) For the circuit depicted in Figure 2, find the differential equations relat-

ing outputsy1(t) andy2(t) to the inputf(t).

−

+

+

−

3Ω

1H

f(t) y1(t) y2(t)

Figure 2: Circuit for Problem 5.

1