0195136047.pdf

630 SIGNAL PROCESSING

Solution

(a) This is a continuous-time signal (called a sinusoidal signal) that is real and periodic with

periodT= 1 /f 0 , as sketched in Figure E14.1.1(a).

(b) This is a complex periodic exponential signal. Its real part is

xr(t)=Acos( 2 πf 0 t+θ)

and its imaginary part is

xi(t)=Asin( 2 πf 0 t+θ)

This signal could also be described in terms of its modulus and phase. The absolute value

ofx(t)is

|x(t)|=

√

x^2 r(t)+x^2 i(t)=A

and its phase is

 x(t)= 2 πf 0 t+θ

Sketches of these functions are shown in Figure E14.1.1(b). In addition, the following

relations apply:

xr(t)=|x(t)|cos[x(t)]

xi(t)=|x(t)|sin[ x(t)]

x(t)=arctanxi(t)

xr(t)

(c) The unit step signal is a nonperiodic signal, sketched in Figure E14.1.1(c).

(d) A sketch of this discrete-time signal is shown in Figure E14.1.1(d).

This is not periodic for all values off 0. The condition for it to be periodic is

2 πf 0 (n+kN)+θ= 2 πf 0 n+θ+ 2 mπ

x(t)

(a)

3

2

1

0

− 1

− 2

− 3

t

Figure E14.1.1(a)Sinusoidal signal.(b)Real–imaginary and

magnitude–phase graphs of the complex exponential signal.(c)

Unit step signal.(d)Discrete sinusoidal signal.