Signal representation using Fourier series - Continuous and Discrete Time Signals and Systems - page 184

factor of n ω 0 t 0 , the sign of the phase change depending on the direction
of the shift. This property is referred to as the time-shifting property.
(3) The exponential CTFS coefficients of a time-reversed periodic signal are
the time-reversed CTFS coefficients of the original signal.
(4) If a periodic signal is time-scaled , the exponential CTFS coefficients are
inversely time-scaled.
(5) The exponential CTFS coefficients of a time-differentiated periodic signal
are obtained by multiplying the CTFS coefficients of the original signal by
a factor of j n ω 0 .
(6) The exponential CTFS coefficients of a time-integrated periodic signal are
obtained by dividing the CTFS coefficients of the original signal by a factor
of j n ω 0 .
(7) For real-valued periodic signals, the exponential CTFS coefficients D n and
D − n are complex conjugates of each other.
(8) Based on Parseval's property , the power of a periodic signal x ( t ) with the
fundamental period of T 0 is computed directly from the exponential CTFS
coefficients as follows:

∞

= 1
T 0

x ( t ) 2 d t 2 . P x = D n

n =−∞

T 0

The plot of the magnitude of the exponential CTFS coefficients versus the

coefficient number n is referred to as the magnitude spectrum, while the plot of
the phase < D n of the exponential CTFS coefficients versus the coefficient num-
ber n is referred to as the phase spectrum of the periodic signal x ( t ). Section 4.6
covers the conditions for the existence of the CTFS representations, and Section
4.7 concludes the chapter by calculating the output response y ( t )ofanLTIC
system to a periodic input x ( t ). In such cases, the output y ( t )isgivenby

D n

n =−∞ D n e j n ω 0 t H ( ω )

∞

y ( t ) =

ω= n ω 0 ,

where the transfer function H ( ω ) is obtained from the impulse response h ( t )of
the LTIC system as follows:

∞

− j ω t d t .

H ( ω ) = h ( t )e

−∞

The above expression also defines the continuous-time Fourier transform
(CTFT) for aperiodic signals, which is covered in depth in Chapter 5.

Problems

4.1 Express the following functions in terms of the orthogonal basis functions
specified in Example 4.2 and illustrated in Fig. 4.3.