代写ECE 380、代写Java/Python编程

日期：2023-02-22 09:36

ECE 380 Introduction to Communication Systems

Homework Assignment 2

Due: 16:00pm Tuesday, Feb. 14, 2023

Problem 1. Given g1(t) ? G1(f), g2(t) ? G2(f), please use the definitions of FT and inverse FT to

proof the the following FT properties.

a) The differentiation property: d

dt

g1(t)? j2πfG1(f).

b) The convolutional property: g1(t) ? g2(T )? G1(f)G2(f).

c) Parseval’s theorem: Eg =

∫∞

?∞ |g1(t)|2dt =

∫∞

?∞ |G1(f)|2df .

Problem 2. a) Find the energy spectral density of the signal g(t) = e?|t|.

b) Show that the signal g1(t) = e?|t?2| has the same energy spectral density as g(t).

Problem 3. Let gT0(t) be a periodic signal with period π. Over the period 0 ≤ t < π, it is defined by

gT0(t) = cos t. Find the Fourier transform of gT0(t) and draw the frequency spectrum.

Note: cosx cos y = 1

2

[cos(x? y) + cos(x+ y)],

sinx cos y = 1

2

[sin(x? y) + sin(x+ y)],∫

eax cos(bx)dx = e

ax

a2+b2

[a cos(bx) + b sin(bx)].

ECE 380 Introduction to Communication Systems

Homework Assignment 1

Due: 16:00pm Tuesday, Feb. 7, 2023

Problem 1. Find the inverse Fourier transforms of G(f) for the spectra in Figure 1 (a) and (b).

Note: G(f) = |G(f)|ej∠G(f).

Figure 1: Signals for Problem 1.

Problem 2. Find the Fourier transforms of the signals g1(t) and g2(t) in Figure 2 using either the

definition of Fourier Transform or the properties of Fourier Transform together with the table of

Fourier transform pairs posted on the course website.

Figure 2: Signals for Problem 2.

Problem 3. a) Prove the following result via properties of Fourier transform: For any signal g(t) with

Fourier transform G(f), we have

g(t) sin(2πfct)?

1

2j

[G(f ? fc)?G(f + fc)].

b) Using the result in (a), please find the Fourier transform of the time-domain signal

s(t) = [2 + cos(2πf0t)] sin(200πt),

where f0 > 0. Draw the spectra.

Note: Consider different ranges of f0.

1

ECE 380 Introduction to Communication Systems

Problem 4. (Haykin & Moher Problem 2.20 with revision) Any function g(t) can be split unambigu-

ously into an even part, ge(t), and an odd part, go(t), as shown by

g(t) = ge(t) + go(t),

where

ge(t) =

1

2

[g(t) + g(?t)] , go(t) = 1

2

[g(t)? g(?t)] .

a) Evaluate the even and odd parts of u(t).

b) What are the Fourier transforms of these two parts and the Fourier transform of u(t)?