MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 327

1.0

0.8 0.4

Re{ X ( w )}

0.6

0.2

Im{ X ( w )}

0.4

5 10 15 20 25

0.2

− 0.2 frequency, GHz

50 100 150 200

− 0.4

time, ps

(a) (b)

Figure 8-12 (a) Waveform driving the transmission line for example 8-2; (b) Fourier
transform of the input waveform.

Step 3: Calculate the Fourier transform of the input waveform shown in
Figure 8-12a:

X(ω) =

F { x pulse (t) }

The Fourier transform of the input waveform is plotted for positive frequencies
in Figure 8-12b. Note that it takes the familiar form of a sinc function. Again,
the FFT was used in this example.
Step 4: Calculate the pulse response. First, the spectrum of the pulse response
Y ( ω ) is calculated using equation (8-9b):

Y(ω) = X(ω)H (ω)

Finally, the pulse response is calculated by taking the inverse Fourier transform
of Y(ω) :

y(t) = F − 1

{ Y(ω) }

The pulse response, which depicts the input waveform after it has propagated
from one end of the transmission line to the other, is plotted in Figure 8-13.
Note that the pulse arrives at approximately 3.1 ns. This result can be verified
by estimating the total delay from the quasistatic values of the inductance given
in Example 6-4 and capacitance using equation (3-107):
τ d = l √ LC = 0 . 5 ( 2 . 5 10 − 9

10 − 7 )( 1 . 5 10 − 10 ) =

× × 3 . 06 × s

The quasistatic approximation will not be identical to the delay calculated with
frequency-dependent parameters, but it should be close. Therefore, the delay of
the waveform in Figure 8-13 passes the sanity check.

It is interesting to note how the transmission line distorts the pulse as it prop-
agates down the transmission line. For example, the amplitude of the waveform