Digital Signal Processing Reference
In-Depth Information
matrix is known, it can be convolved with any arbitrary input (such as a pulse
or a bit stream) and the signal integrity can be evaluated. It is impossible to
measure the impulse response in the laboratory because it requires a driver
capable of driving a Dirac delta function that has infinitely fast rise and fall
times. Furthermore, even when a fast pulse is generated in the laboratory, induc-
tance and capacitive loading of the probes introduces into the measured response
unwanted noise, filtering, and resonances that are not associated with the electrical
behavior of the device under test. Consequently, experimental evaluation of the
impulse response in the laboratory using time-domain techniques is an impractical
endeavor.
For most practical purposes, the impulse response of the system interconnects
can be measured indirectly using a vector network analyzer (VNA), which is a
device used to evaluate the scattering matrix as a function of frequency in the
laboratory. Standard techniques to remove the parasitic inductance and capaci-
tance effects of the probes and test fixtures from the measured scattering network
are achieved through proper instrument calibration. Once the scattering matrix is
measured, the impulse response can be calculated by taking the inverse Fourier
transform, described in Section 8.1.4:
h
(t)
= F
−
1
{
S
(ω)
}
(9-36a)
where
S
(
ω
) is the scattering matrix measured with a VNA.
To calculate the impulse response, the scattering matrix must contain values for
negative frequencies that obey the complex-conjugate rule to ensure a real-valued
time-domain response, as described in Section 8.2.1. Since VNA measurements
only provide values for the positive frequencies, the negative frequency values
must be calculated from the positive values as
S(
−
f)
=
S(f )
∗
(9-36b)
When calculating the impulse response using the fast Fourier transform (FFT), the
negative values are appended to the end of the positive values, as demonstrated
in Example 9-7.
Example 9-7
Calculate the impulse response from the measured values of
S
21
shown in Figure 9-22a.
SOLUTION
Step 1:
Calculate the negative frequency values of
S
21
using (9-36b) as shown
in Figure 9-22b.
S(
−
f)
=
S(f )
∗
Step 2:
Append the negative frequency values to the positive frequency values
to create a continuous spectrum with both positive and negative frequencies.
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