NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 377

Therefore, measurements taken using a standard 20-GHz VNA would have ade-
quate bandwidth to resolve rise and fall times as fast as 17.5 ps. Although a
bandwidth of 20 GHz is sufficient to resolve the edge rate, the granularity of
the time-domain waveform calculated using the FFT is only 25 ps, as calculated
by (9-37a):

1
2 f max

1

10 − 12 t = =

10 9 ) =

25 × s

2 ( 20 ×

Consequently, extrapolation or zero padding is required to ensure reasonable
granularity in the time-domain waveform, as demonstrated in Example 9-8.

Example 9-8 Assume that the complex values of S 21 used in Example 9-7
have been measured to 20 GHz. Calculate the impulse response with a resolution
of 5 ps.

SOLUTION
Step 1: Calculate f max using equation (9-37a):

1
2 t =

1

10 9 Hz

f max =

10 − 12 ) =

100 ×

2 ( 5 ×

Step 2: Calculate the negative frequency values from − 20 GHz to dc using

equation (9-36b):

S( − f) = S(f ) ∗

Step 3 : Calculate the sample interval of the frequency-domain data assuming
1000 samples of the measurable positive frequency values:

1
2 nt =

1
2 ( 1000 )( 5

10 6 Hz

f =

10 − 12 ) =

100 ×

×

Step 4: Calculate the number of zero points that need to be added to the
spectrum. At a sample rate of 100 MHz, the number of samples up to 20 GHz is

10 9 Hz 20 ×

10 6 Hz / sample =

200 samples

100 ×

Therefore, 800 zero points need to be added to both the positive and negative
spectra to achieve 100-GHz bandwidth.

Step 5: Append the positive and negative spectrums together as shown in
Figure 9-25.
Step 6: Perform an inverse FFT on the zero-padded spectrum to get the impulse
response, as shown in Figure 9-26.

±