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ECE 459: Communications I Problem Set 4 - Angle Modulation and FM Modulation, Assignments of Digital Communication Systems

University of Illinois - Urbana-ChampaignDigital Communication Systems

A problem set from the university of illinois at urbana-champaign, department of electrical and computer engineering, ece 459: communications i course, fall 2005. It includes problems on angle modulation and frequency modulation, covering topics such as finding frequency deviation, modulation index, approximating narrowband fm signals, and finding power. Students are expected to solve problems 4.1 through 4.6, with problem 4.4 being optional. The document also includes information about the mid-semester exam and allowed materials.

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University of Illinois at Urbana-Champaign
Department of Electrical and Computer Engineering
ECE 459: Communications I
Fall 2005
Problem Set 4
Angle Modulation, Phase and Frequency Modulation
Issued: Thursday, Sept. 22nd. Due: Tuesday, October 4th (beginning of lecture).
Reading from Lathi: Chapter 5 (excluding Section 5.5).
Announcement: The first Mid-Semester Exam will be held on Thursday, October 6th, from
7-9pm in 165 Everitt. The exam will cover all material from the beginning of the term up to
and including the lecture on Tuesday, September 27th. The corresponding material includes
Problem Sets 1 through 4 and Chapters 1, 2, 3, 4 (excluding Sections 4.7–4.9) and 5 (excluding
Section 5.5) from Lathi.
During the exam, you can bring an 8.5×11-inch double-sided sheet of handwritten notes.
Calculators are allowed but will not be necessary.
A copy of an old exam is available for download from the course website. This exam does
not necessarily resemble this year’s exam (also notice that the time allowed and the material
covered in this old exam are slightly different from this year’s exam).
Problem 4.1
(a) Problem 5.1-1 from Lathi, p. 248.
(b) Problem 5.2-2 from Lathi, p. 249.
Problem 4.2
Consider the FM signal
s(t) = Accos ωct+kfZt
−∞
m(τ)dτ,
where m(t) is the tone signal m(t) = Amsin(ωmt). The following parameters are given:
Ac= 200V, ωc= 160πMrad/s, kf= 20πKrad/(s Volt), Am= 1V, ωm= 16πKrad/s .
(a) Find the frequency deviation ∆ω.
(b) Find the modulation index β.
(c) The narrowband FM signal approximation uses
se(t) = Accos(ωct)−AckfZt
−∞
m(τ)dτsin(ωct)
as an approximation to s(t). Sketch |Se(ω)|and find its bandwidth.
pf2

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Download ECE 459: Communications I Problem Set 4 - Angle Modulation and FM Modulation and more Assignments Digital Communication Systems in PDF only on Docsity!

University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering

ECE 459: Communications I

Fall 2005

Problem Set 4 Angle Modulation, Phase and Frequency Modulation

Issued: Thursday, Sept. 22nd. Due: Tuesday, October 4th (beginning of lecture).

Reading from Lathi: Chapter 5 (excluding Section 5.5).

Announcement: The first Mid-Semester Exam will be held on Thursday, October 6th, from 7-9pm in 165 Everitt. The exam will cover all material from the beginning of the term up to and including the lecture on Tuesday, September 27th. The corresponding material includes Problem Sets 1 through 4 and Chapters 1, 2, 3, 4 (excluding Sections 4.7–4.9) and 5 (excluding Section 5.5) from Lathi. During the exam, you can bring an 8. 5 × 11-inch double-sided sheet of handwritten notes. Calculators are allowed but will not be necessary. A copy of an old exam is available for download from the course website. This exam does not necessarily resemble this year’s exam (also notice that the time allowed and the material covered in this old exam are slightly different from this year’s exam).

Problem 4.

(a) Problem 5.1-1 from Lathi, p. 248.

(b) Problem 5.2-2 from Lathi, p. 249.

Problem 4.

Consider the FM signal

s(t) = Ac cos

[ ωct + kf

∫ (^) t

−∞

m(τ )dτ

] ,

where m(t) is the tone signal m(t) = Am sin(ωmt). The following parameters are given:

Ac = 200V, ωc = 160π Mrad/s, kf = 20π Krad/(s Volt), Am = 1V, ωm = 16π Krad/s.

(a) Find the frequency deviation ∆ω.

(b) Find the modulation index β.

(c) The narrowband FM signal approximation uses

se(t) = Ac cos(ωct) − Ac

[ kf

∫ (^) t

−∞

m(τ )dτ

] sin(ωct)

as an approximation to s(t). Sketch |Se(ω)| and find its bandwidth.

(d) Find the power of se(t) and compare it to the power of s(t).

Problem 4.

Problem 5.2-6 from Lathi, p. 249.

Problem 4.4 (Optional)

Recall that the complex envelope s˜(t) of a real signal s(t) is generally complex and can be written in the form s^ ˜(t) = sI (t) + jsQ(t) ,

where sI (t) and sQ(t) are the in-phase and quadrature components of s(t). Find the expressions for the in-phase and quadrature components of the following signals:

(a) PM signal s(t) = A cos[ωct + kpm(t)];

(b) FM signal s(t) = A cos[ωct + kf

∫ (^) t

0

m(t)dt].

Problem 4.

The sinusoidal message signal m(t) = Am cos(ωmt) is frequency modulated so that the trans- mitted signal s(t) is given by

s(t) = Ac cos

[ ωct + kf

∫ (^) t

−∞

m(τ )dτ

]

= Ac cos [ωct + β sin ωmt] ,

where β =

kf Am ωm

. Given that

ωc = 20 Mrad/s, Am = 10V, kf = 6 Krad/s/V ,

specify the range of possible ωm such that the bandwidth of the resulting s(t) is below 200 Krad/s.

Problem 4.6 (Optional)

Problem 5.4-2 from Lathi, p. 250.