MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 318

F ( w )

f ( t )

1.0 2.0

square
trapezoid

0.8

1.5

0.6

1.0

0.4

0.5

0.2

− 20 − 10 10 20

− 2 − 1 1 2

Time Frequency, ω

Figure 8-4 Frequency response of a trapezoidal wave compared to an ideal square wave.

8.1.3 Frequency Spectrum of a Digital Pulse
Although a square pulse is a first-order approximation of a digital bit, a bet-
ter approximation is a trapezoid. Figure 8-4 shows the frequency response of
a trapezoidal wave compared to an ideal square wave. Notice the shape of the
trapezoid's frequency response is very similar to that of a square wave, except
that the magnitude of the harmonics are smaller, especially at high frequencies.
This demonstrates another important concept: The rise and fall times of a dig-
ital waveform determine the magnitude of the high-frequency harmonics in the
frequency-domain spectrum .
Figure 8-4 also indicates that the spectrum of a trapezoid can be derived by
applying a low-pass filtering function to the spectrum of a square wave. This
allows a relationship to be defined between the spectral content of a digital
waveform and the rise and fall times. To begin the derivation of this relation-
ship, some unique properties of the square-wave spectrum need to be observed.
Equation (8-3) shows that the harmonics of a square wave are a sinc func-
tion. Taking the limits of the sinc function allows the frequency spectrum to be
generalized:

1

when ω is small

sin ω
ω

≈ (8-5)

1 /ω when ω is large

The quantity 2 /ω is compared to the spectrum calculated with equation (8-3) in
Figure 8-5. Note that the quantity 1 /ω is equivalent to a slope of 20 dB/decade
on a log-log plot:

20 log 1 / 10 ω
1 /ω

=− 20 dB/decade

meaning that the spectral content of a square wave will fall off at a rate of
− 20 dB/decade.

One way to approximate the spectrum of a trapezoidal wave is to apply a
low-pass filtering function to the harmonics of a square wave until the desired