Introduction to signals - Continuous and Discrete Time Signals and Systems - page 11

The sinusoidal signal x ( t ) has a fundamental period T 0 = 2 π/ω 0 as we prove

next. Substituting t by t + T 0 in the sinusoidal function, yields

x ( t + T 0 ) = A sin( ω 0 t + ω 0 T 0 + θ ) .

Since

x ( t ) = A sin( ω 0 t + θ ) = A sin( ω 0 t + 2 m π + θ ) , for m = 0 , 1 , 2 ,...,

the above two expressions are equal iff ω 0 T 0 = 2 m π . Selecting m = 1, the

= 2 π/ω 0 .
The sinusoidal signal x ( t ) can also be expressed as a function of a complex
exponential. Using the Euler identity,

fundamental period is given by T 0

e j( ω 0 t +θ ) = cos( ω 0 t + θ ) + j sin( ω 0 t + θ ) ,

(1.6)

we observe that the sinusoidal signal x ( t ) is the imaginary component of a
complex exponential. By noting that both the imaginary and real components
of an exponential function are periodic with fundamental period T 0 = 2 π/ω 0 ,
it can be shown that the complex exponential x ( t ) = exp[j( ω 0 t + θ )] is also a
periodic signal with the same fundamental period of T 0 = 2 π/ω 0 .

Example 1.3
(i) CT sine wave: x 1 ( t ) = sin(4 π t ) is a periodic signal with period T 1

=

2 π/ 4 π = 1 / 2;
(ii) CT cosine wave: x 2 ( t ) = cos(3 π t ) is a periodic signal with period T 2

=

2 π/ 3 π = 2 / 3;
(iii) CT tangent wave: x 3 ( t ) = tan(10 t ) is a periodic signal with period T 3 =

π/ 10;
(iv) CT complex exponential: x 4 ( t ) = e j(2 t + 7) is a periodic signal with period
T 4

= 2 π/ 2 = π ;

sin 4 π t − 2 ≤ t ≤ 2

(v) CT sine wave of limited duration: x 6 ( t ) = is an

0 otherwise

aperiodic signal;
(vi) CT linear relationship: x 7 ( t ) = 2 t + 5 is an aperiodic signal;
(vii) CT real exponential: x 4 ( t ) = e

− 2 t

is an aperiodic signal.

Although all CT sinusoidals are periodic, their DT counterparts x [ k ] =
A sin( Ω 0 k + θ ) may not always be periodic. In the following discussion, we
derive a condition for the DT sinusoidal x [ k ] to be periodic.
Assuming x [ k ] = A sin( Ω 0 k + θ ) is periodic with period K 0 yields

x [ k + K 0 ] = sin( Ω 0 ( k + K 0 ) + θ ) = sin( Ω 0 k +

Ω 0 K 0 ) + θ ) .

Since x [ k ] can be expressed as x [ k ] = sin( Ω 0 k + 2 m π

+ θ ), the value of the

= 2 π m / 0 for m = 0 , 1 , 2 ,... Since
we are dealing with DT sequences, the value of the fundamental period K 0 must
be an integer. In other words, x [ k ] is periodic if we can find a set of values for

fundamental period is given by K 0