NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 376

positive frequency bandwidth of the measurement is defined by

1
2 t

f max = (9-37a)

To support a time-domain step size of t , the frequency sample size must be

1
2 nt

f = (9-37b)

and the maximum valid time-domain signal would be

1
2 f

t max = (9-37c)

where n is the number of samples for the positive frequency values .
For example, if a time-domain granularity of 5 ps is required, the maxi-
mum bandwidth required using 1000 positive frequency sample points would
be 100 GHz:

1

10 9 Hz f max =

10 − 12 ) =

100 ×

2 ( 5 ×

Consequently, a time-domain waveform constructed from a frequency-domain
measurement requires significant bandwidth if fine granularity is required. Unfor-
tunately, it becomes very difficult to perform VNA measurements above about
20 GHz. Both the calibration techniques and equipment costs become prohibitive.
Fortunately, there are mathematic ways to sidestep the lack of high-frequency
measured data without losing time-domain granularity. As long as the spectral
bandwidth of the digital waveforms propagating on the system interconnects
are significantly lower than f max , extrapolation or zero padding of the measured
scattering matrix can be used to increase granularity with only a small degradation
in accuracy. For example, consider a driving digital waveform with rise and fall
times of 25 ps. The spectral bandwidth of this waveform is approximated by
equation (8-8):

0 . 35

10 9 Hz

f 3dB ≈ = 14 ×

25 × 10 − 12

Common bandwidths currently available on VNAs range from 20 to 110 GHz,
which are equivalent to pulses with rise and fall times of about 3 to 18 ps.

0 . 35
20

10 − 12

= 17 . 5 × s

× 10 9

t 10 - 90% =

0 . 35
110

10 − 12

= 3 . 18 × s

× 10 9