MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 314

sum of two component signals,

x(t) = c 1 x 1 (t) + c 2 x 2 (t)

where c 1 and c 2 are constants, the output of the system will be

y(t) = c 1 y 1 (t) + c 2 y 2 (t)

where y n ( t ) is the output resulting from the input x n ( t )( n = 1 and 2). To gener-

alize, a linear system with input
x(t) =
n

c n x n (t)

will have output

y(t) = c n y n (t)

n

for any constants c n and where the output y n ( t ) results from the input x n ( t ).
Time invariance means that whether an input to the system is applied at t 0

or t = τ , the output will be identical except for a time delay of τ . For example,
if the output due to input x ( t )is y ( t ), the output due to input x ( t − τ )is y ( t − τ ).
Simply put, a time delay at the input should produce a corresponding time delay
at the output .

=

8.1.2 Time- and Frequency-Domain Equivalencies

Any time-domain waveform in an LTI system has an equivalent spectrum in the
frequency domain. This means that any time-domain signal, such as a digital
waveform, can also be described fully in terms of its frequency-domain parame-
ters. This concept is important because it allows the frequency-dependent nature
of electromagnetic models to be integrated into time-domain waveforms so that
the signal integrity of digital bits propagating on a bus can be analyzed. The
relationship between a time-domain signal and its frequency-domain equivalent
is described with the Fourier transform :

( 2 π) 1 − a ∞

| b |

f(t)e jbωt dt F(ω) = (8-1a)

−∞

( 2 π) 1 + a ∞

| b |

F(ω)e − jbωt dω f(t) = (8-1b)

−∞