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Category: ECE 102

Description

5/5 - (4 votes)

1. (10 points) Even and odd parts.

Sketch the even and odd components of the following signal:

2. (15 points) Time scaling and shifting.

(a) (10 points) Consider the following signal.

Sketch the following:

i. x(1 − 3t)

ii. x(

t

2 − 2)

(b) (5 points) The figure below shows two signals: z(t) and y(t). Can you express y(t) in

terms of z(t)?

1

3. (22 points) Periodic signals.

(a) (14 points) For each of the following signals, determine whether it is periodic or not. If

the signal is periodic, determine the fundamental period and frequency.

i. x1(t) = sin(2t + π/3)

ii. x2(t) = cos(√

2πt)

iii. x3(t) = sin2

(3πt + 3)

iv. x4(t) = x1(t) + x2(t)

v. x5(t) = x1(πt) + x3(t)

vi. x6(t) = e

−tx1(t)

vii. x7(t) = e

j(πt+1)x2(t)

(b) (4 points) Assume that the signal x(t) is periodic with period T0, and that x(t) is odd

(i.e. x(t) = −x(−t)). What is the value of x(T0)?

(c) (4 points) If x(t) is periodic, are the even and odd components of x(t) also periodic?

4. (21 points) Energy and power signals.

(a) (15 points) Determine whether the following signals are energy or power signals. If the

signal is an energy signal, determine its energy. If the signal is a power signal, determine

its power.

i. x(t) = e

−|t|

ii. x(t) = (

√

1

t

, if t ≥ 1

0, otherwise

iii. x(t) = (

1 + e

−t

, if t ≥ 0

0, otherwise

(b) (6 points) Show the following two properties:

• If x(t) is an even signal and y(t) is an odd signal, then x(t)y(t) is an odd signal;

• If z(t) is an odd signal, then for any τ > 0 we have:

Z τ

−τ

z(t)dt = 0

Use these two properties to show that the energy of x(t) is the sum of the energy of its

even component xe(t) and the energy of its odd component xo(t) , i.e.,

Ex = Exe + Exo

Assume x(t) is a real signal.

5. (17 points) Euler’s identity and complex numbers.

(a) (9 points) Use Euler’s formula to prove the following identities:

i. d

dθ sin(θ) = cos(θ)

2

ii. sin2

(θ) = 1

2

(1 − cos(2θ))

iii. e

jα + e

jβ = 2 cos

α−β

2

e

j

α+β

2

(b) (8 points) Let x(t) = −(1 + j)e

j(1+2t)

.

i. Compute the real and imaginary parts of x(t).

ii. Compute the magnitude and phase of x(t).

6. (15 points) MATLAB tasks

(a) (2 points) Task 1

Plot the waveform

x(t) = e

−t

2

cos(2πt)

for −5 ≤ t ≤ 5, with a step size of 0.1. Label the time axis.

(b) (3 points) Task 2

Create a vector x corresponding to the function given in Problem 1. Use a sample

spacing of 0.01 over the range -2 to 2. Plot this vector. Properly label the time axis.

(c) (4 points) Task 3

Create two vectors that represent the even and odd components of the vector x created

in Task 2. A vector in MATLAB is reversed by

>> xrev = x(length(x):-1:1);

Plot the even and odd components of x.

(d) (6 points) Task 4

Consider the following three signals:

x1(t) = cos(2πt)

x2(t) = cos(60πt)

x3(t) = x1(t)x2(t)

Plot the signals separately (you can use the function subplot) for −3 ≤ t ≤ 3, with a

step size of 0.001.

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