The Fourier series representation of a periodic signal makes it
easy to determine how a linear, time-invariant filter reshapes
such signals in general. The fundamental
property of a linear system is that its input-output relation
obeys superposition:
L
a
1
s
1
t+
a
2
s
2
t=
a
1
L
s
1
t+
a
2
L
s
2
t
L
a
1
s
1
t
a
2
s
2
t
a
1
L
s
1
t
a
2
L
s
2
t
.
Because the Fourier series represents a periodic signal as a
linear combination of complex exponentials, we can exploit the
superposition property. Furthermore, we found for linear
circuits that their output to a complex exponential input is
just the frequency response evaluated at the signal's frequency
times the complex exponential. Said mathematically, if
xt=ei2πktT
x
t
2
k
t
T
,
then the output
yt=HkTei2πktT
y
t
H
k
T
2
k
t
T
because
f=kT
f
k
T
.
Thus, if
xt
x
t
is periodic thereby having a Fourier series, a linear circuit's
output to this signal will be the superposition of the output to
each component.
yt=∑
k
=−∞∞
c
k
HkTei2πktT
y
t
k
c
k
H
k
T
2
k
t
T
(1)
Thus, the output has a Fourier series, which means that it too
is periodic. Its Fourier coefficients equal
c
k
HkT
c
k
H
k
T
.
To obtain the spectrum of the output, we simply multiply the
input spectrum by the frequency response.
The circuit modifies the magnitude and phase of each Fourier
coefficient. Note especially that while the Fourier
coefficients do not depend on the signal's period, the circuit's
transfer function does depend on frequency, which means that the
circuit's output will differ as the period varies.
The periodic pulse signal shown on the left above serves as
the input to a
RC
R
C
-circuit that has the transfer function (calculated
elsewhere)
Hf=11+i2πfRC
H
f
1
1
2
f
R
C
(2) shows the output changes
as we vary the filter's cutoff frequency. Note how the
signal's spectrum extends well above its fundamental
frequency. Having a cutoff frequency ten times higher than
the fundamental does perceptibly change the output waveform,
rounding the leading and trailing edges. As the cutoff
frequency decreases (center, then left), the rounding becomes
more prominent, with the leftmost waveform showing a small
ripple.
What is the average value of each output waveform? The
correct answer may surprise you.
Because the filter's gain at zero frequency equals one, the
average output values equal the respective average input
values.
This example also illustrates the impact a lowpass filter can
have on a waveform. The simple
RC
R
C
filter used here has a rather gradual frequency response, which
means that higher harmonics are smoothly suppressed. Later, we
will describe filters that have much more rapidly varying
frequency responses, allowing a much more dramatic selection of
the input's Fourier coefficients.
More importantly, we have calculated the output of a circuit to
a periodic input without writing, much less
solving, the differential equation governing the circuit's
behavior. Furthermore, we made these calculations entirely in
the frequency domain. Using Fourier series, we can calculate
how any linear circuit will respond to a
periodic input.
"Electrical Engineering Digital Processing Systems in Braille."