Digital Signal Processing Reference
In-Depth Information
which relates the output voltage of the system to the input voltage.
We define a function
jvL
R + jvL
H(v) =
(5.44)
and write the input-output relationship for the system as
V
2
(v) = H(v)V
1
(v),
(5.45)
or
V
2
(v)
V
1
(v)
.
H(v) =
(5.46)
Because the quantity determines the output of the circuit for any given
input signal, it is commonly called the
transfer function
of the system. The relation-
ship of (5.45) is illustrated in Figure 5.25(b).
The function in (5.44) gives mathematically the variation of the
input-output relationship of the circuit with frequency. Therefore,
H(v)
H(v)
H(v)
is also
called the
frequency response function
of the system.
The frequency response function
H(v)
is the same as the transfer function of
(3.75) when is substituted for
s
.
We can determine the frequency response experimentally, and somewhat la-
boriously, by applying a sinusoidal signal to the input of the circuit and measuring
the magnitude and phase of the input and output signals. This process is repeated
for different frequencies so that a large set of measurements is acquired over a wide
range of frequencies. Since (5.46) can be expressed in polar form as
jv
ƒ
V
2
(v)
ƒ
l
V
2
ƒV
1
(v) ƒ
l
V
1
ƒV
2
(v) ƒ
ƒV
1
(v) ƒ
l
V
2
- V
1
,
H(v) = ƒH(v) ƒ ∠
f(v) =
=
a plot of the ratio of the magnitudes of the input and output signals
versus frequency yields a plot of
ƒV
2
(v) ƒ / ƒV
1
(v) ƒ
ƒH(v) ƒ .
A plot of the difference between the
recorded phase angles
(
∠
V
2
- ∠
V
1
)
versus frequency yields a plot of
f(v).
The frequency response of a system
EXAMPLE 5.15
An engineering professor required a student to determine the frequency response of the cir-
cuit shown in Figure 5.26(a). The student decided to spend some time in the laboratory col-
lecting data, using the experimental system shown in Figure 5.26(b). The student proceeded
by measuring the input and output signals with a dual-trace oscilloscope, as shown in Fig-
ure 5.26(c). A series of measurements was taken at a progression of frequency settings on the
function generator that produced the input signal. For each frequency setting, the student set
the amplitude of
v
1
(t)
to 1 V. Therefore, the input waveform at each frequency setting,
v
x
,
could be written as
v
1
(t) = cos
v
x
t.
