Page 78 - Fundamentals of Communications Systems

2.30 Chapter Two

(a) The Fourier transform, U i (f ), i = 1, 2, 3.
(f ).
(b) The energy spectral density, G u i
?
(c) The correlation function, V u i (τ). What is the energy, E u i
Hint: Some of the computations have an easy way and a hard way so think
before turning the crank!

Problem 2.13. This problem looks at several simplifying approximations for com-
plex numbers that will be useful in analyzing the characteristics of analog
modulations and demodulators. Assume z is a complex number of the form

(2.92)
z = 1 + 1 + j 2

where i is a real number with i 1 for i = 1, 2.

(a) Show a logical argument to justify the approximation |z|≈ 1 + 1 .
(b) Assume 1 = 2 and find the values of 1 , where this approximation results
in a error of less than 1%.
(c) Show a logical argument to justify the approximation arg{z}≈ 2 .
(d) Assume 1 = 2 and find the values of 1 , where this approximation results
in a error of less than 1%.

Problem 2.14. Pulse shapes will be important in digital communication systems.
A Gaussian pulse shape arises in several applications and is given as

1 t 2
x(t) = √ exp − 2 (2.93)
2πσ 2 2σ
(a) Plot x(t) for σ = 0.25, 1, 4.
(b) Calculate X(f ). Hint:

1 ∞ x 2
a 2
√ exp − + ax dx = exp (2.94)
2π −∞ 2 2

(c) Plot G X (f ) for σ = 0.25, 1, 4 using a dB scale on the y-axis.
Problem 2.15. A common signal has a Fourier series representation of

∞ k
4 (−1)
x(t) = cos(10π(2k + 1)t) (2.95)
π 2k + 1
k=0