MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 333

As described in Section 6.4.1, there is a specific relationship between ε (ω) and
ε (ω) that must be enforced to ensure realistic behavior. If the dielectric models
do not satisfy the Kramers-Kronig relations, the system will be noncausal.

Example 8-3 Use the Hilbert transform to prove that the waveform shown in
Figure 8-16a is noncausal and the waveform depicted in Figure 8-16b is casual.

SOLUTION
Step 1a: Calculate the Fourier transform of the noncausal waveform
(Figure 8-16a) using the signal processing convention ( a = 0, b =− 2 π )of

equation (8-1a):

f(t) = u s (t + 1 )u s ( − t + 2 )

cos πω sin 3 πω
πω

sin πω sin 3 πω
πω

F(ω) =

− j

Step 2a: Calculate the Hilbert transform of the real part of F ( ω ) and compare
it to the imaginary part:

F − 1 F cos πω sin 3 πω
πω

F 1
πω

1
πω =

F Re (ω)

= Re[ F(ω) ] ∗

2 cos 2 πω) sin (πω) 2
πω

( 3 +

=

=− F Re (ω) , the waveform depicted in Figure 8-16a is noncausal.
Of course, simple observation of the waveform for this example is proof of
noncausality since it exhibits nonzero values for t< 0.

Since Im[ F(ω) ]

f ( t ) f ( t )

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

− 1

0 1 2 3 − 1 0 1 2 3

Time Time

(a)

(b)

Figure 8-16 (a) Noncausal and (b) causal waveforms for Example 8-3.