Digital Signal Processing Reference
In-Depth Information
Property 6 is illustrated by the square wave of Table 4.3 and Example 4.2,
where the Fourier coefficients were calculated to be
2V
jpk
,
k odd
C
k
=
c
.
0,
k even
The square wave has discontinuities, and the harmonics approach zero as 1/
k
, as
k
approaches infinity.
However, the Fourier coefficients for the train of impulse functions of Exam-
ple 4.5 and Table 4.3 do not satisfy Property 6. These coefficients are calculated in
Example 4.5 to be for all
k
and do not approach zero as
k
approaches in-
finity. This is not surprising, since an impulse function is not bounded. Hence, the
Dirichlet condition 3 is violated, and Property 6 does not apply for this periodic
function.
In this section, seven properties of Fourier series are given without proof.
These properties are useful in the application of Fourier series in both analysis and
design.
C
k
= 1/T
0
4.5
SYSTEM ANALYSIS
In this section, we consider the analysis of stable LTI systems with periodic inputs. In
the Fourier series representation of the input signal, the sinusoidal components are
periodic for all time and it is assumed that the initial input was applied at time
. Therefore, it is assumed that the transient response has reached steady-
state, and since the systems are stable, the natural responses can be ignored; only
the
steady-state responses
are determined.
The system linearity allows the use of superposition. Since a periodic input sig-
nal can be represented as a sum of complex exponential functions, the system re-
sponse can be represented as the sum of steady-state responses to these complex
exponential functions. We can also represent the periodic input signal as a sum of
sinusoidal functions; the system response is then a sum of steady-state sinusoidal re-
sponses. In either case, it will be shown that we must consider the variation of the
system sinusoidal response with frequency. This variation is called the
system fre-
quency response
. The analysis procedure developed in this section does not give a
good indication of a plot of the steady-state response as a function of time; instead,
it yields the frequency spectrum of the output signal.
We begin by considering the LTI system of Figure 4.16, and we use the stan-
dard notation for systems:
t : -
q
x
(
t
)
y
(
t
)
h
(
t
)
Figure 4.16
LTI system.
