NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 406

Very often, VNA calibration problems or incorrect model assumptions generate
S -parameters that are not causal. This error becomes especially critical when
S -parameters are used to represent the behavior of physical channel components
when simulating system performance. Causality errors will introduce delay and
amplitude errors in the time-domain responses, as well as distort waveforms to
such a degree that signal integrity may be falsely predicted, which can lead to
false solution spaces, incorrect equalization settings, and incorrect understanding
of bus performance. Many commercially available tools create models that are
noncausal simply because of the assumptions of a frequency-invariant dielectric
model where ε r and tan δ are not properly related. Figure 8.14 is a good example
of how a noncausal model can distort the waveform of a single bit. The errors
depicted in Figure 8.14 will be amplified for very long channel lengths and high
frequencies. Many of the digital designs created in the past were designed using
noncausal models that assume frequency-invariant dielectric parameters. At low
data rates (below about 1 to 2 Gbit/s) and short lengths (a few inches), causality
errors do not appreciable affect signal integrity. However, at high frequencies
and/or long lengths, they can completely destroy the model's ability to predict
realistic behavior.
Causality requirements are often difficult to judge by looking directly at
S -parameters because the phase errors are small compared to the propagation
delay of the structure. Furthermore, even if the causality errors of a single com-
ponent such as a connector or via are small in isolation, when numerous models
are cascaded in a complete channel simulation the errors become cumulative.
Consequently, it is important to check the causality of each component as well
as the system response for the entire channel.
As described fully in Section 8.2.1 and demonstrated in Examples 8-3 and
8-4, causality can be tested by performing the Hilbert transform of the real part
and ensuring that it is identical to the imaginary part . If it is not identical, the
model is not causal. However, care should to be taken when performing this test
because for bandlimited data, the Hilbert transform can exhibit truncation errors.
Equation (8-19) is adapted to calculate the Hilbert transform of the real part
of the S -parameter data,

1
πf

S Re ,ij (f ) =

Re[ S ij (f ) ] ∗ (9-74)

where S Re ,ij (f ) denotes the Hilbert transform of the real portion of the
S -parameter, S ij (f ) is an S -parameter where port j is driving and port i is
receiving, and

∗ denotes convolution. The S -parameter is causal when (9-75) is

satisfied:

S Re ,ij (f ) =

Im[ S ij (f ) ] (9-75)

To implement (9-74), it is often necessary to reconstruct the negative frequency
components of the response using

S( − f) = S(f ) ∗

(9-36b)