NETWORK ANALYSIS FOR DIGITAL ENGINEERS - Advanced Signal Integrity for High-Speed Digital Design - page 407

2.0

Noncausal
Causal

1.5

1.0

0.5

Precursor
error

2.5 5.0 7.5 10

Time, ns

Figure 9-41 Classic impulse response behavior of a severely noncausal S -parameter.
Note the precursor error, where part of the energy is arriving too early.

The most straightforward way to calculate the Hilbert transform is to use the
Fourier transform (usually in the form of the fast Fourier transform), as demon-
strated in Examples 8-3 and 8-4.
S Re ,ij (f ) = F − 1 F ( Re[ S Re ,ij (f ) ] ) F 1
πf

(9-76)

If the fast Fourier transform is used, the negative frequency values must be
appended to the end of the positive values, as demonstrated in Example 9-7.
A less rigorous way to evaluate the causality is to transform the S -parameters
into an impulse response, as described in 9.2.2. If there is no distinct point at
which the pulse arrives, it is an indication of a noncausal response, as shown in
Figure 9-41.

9.3.4 Subjective Examination of S -Parameters

Often, it is desirable to recognize the trustworthiness of S -parameter data without
the rigor of the mathematical analysis above. Understanding the basic properties
of S -parameter data will allow an intuitive analysis. For example, a gross passivity
violation can often be seen by observing the magnitude of the insertion loss.
Figure 9-42 shows an example of measured insertion loss for a transmission line.
Note that the scale often used with VNA measurements is, in decibels,

dB = 20 log ( mag ) (9-77)

where mag is the magnitude of the complex S-parameter value. Notice how
S 21 in Figure 9-42 rises above zero at the lower frequencies, indicating that
the transmission line has gain and is therefore producing energy and nonpassive