Digital Signal Processing Reference
In-Depth Information
8.1.4 System Response
When a system is linear and time invariant, the output and input can be repre-
sented with a
convolution
, where bold face type indicates a matrix:
∞
y
(t)
=
h
(t)
∗
x
(t)
=
h
(t
−
τ)
x
(τ ) dτ
(8-9a)
−∞
where
h
(
t
) is the system impulse response matrix, and each element
h
ij
(
t
) is the
response at port
i
when Dirac's delta function (an ideal impulse) is applied at
port
j
with all other inputs set to zero. This is an important concept, because if
the impulse response of a system is known, the response of the system with any
input,
x
(
τ
), can be determined.
When the system impulse response is represented in the frequency domain, it
is referred to as the
transfer function H
(
ω
):
Y
(ω)
=
H
(ω)
X
(ω)
(8-9b)
where
Y
(
ω
) is the frequency response matrix of the of the system output and
X
(
ω
)
is the frequency response of the system input. It is usually much more convenient
to analyze systems using transfer functions in the frequency domain rather than
impulse responses in the time domain. However, due to the equivalency between
a time-domain waveform and a frequency-domain spectrum,
the inverse Fourier
transform of the transfer function is the impulse response
:
h(t)
= F
−
1
{
H(ω)
}
(8-10)
The relationship of (8-10) is convenient because it is generally much easier to
measure the transfer function in the laboratory using a vector network analyzer
(VNA) than to measure the impulse response because an ideal impulse is
impossible to produce. In fact, in Chapter 9 we describe methods to obtain the
impulse response from
S
-parameters which can be measured in the laboratory
using a vector network analyzer. It should be noted that since real laboratory
instruments do not provide measured values for negative frequencies, the
relationship of equation (8-4) must be used to construct the negative frequency
response from the complex conjugate of the positive (measured) frequency
response.
Another useful property is that
convolution in the time domain is equivalent to
multiplication in the frequency domain
. Generally, it is much simpler to handle
the convolution of an input and an impulse response in the frequency domain
by multiplying the spectrums and performing an inverse Fourier transform to
convert back to the time domain.
Example 8-1
Determine the wave shape of an ideal, 2-ns-wide square wave
propagating through the low-pass
RC
filter shown in Figure 8-8a. Assume that
R
=
50
,
C
=
5 pF, and the amplitude of the square wave is 1 V.
Search WWH ::
Custom Search