Digital Signal Processing Reference
In-Depth Information
1.4
Time and Frequency Domains
Pulses described using time are said to be represented in the time domain. Oscil-
loscopes are common time-domain instruments, and the SPICE circuit simulator
displays its results in the time domain by default.
Most signal integrity work is performed in the time domain, but it is important
to understand the way in which the electrical characteristics of circuit board traces
change at different frequencies. This frequency domain approach is especially use-
ful when studying how losses on transmission lines affect the shape (distortion) and
alignment of pulses (intersymbol interference).
A vector network analyzer (VNA), which displays the impedance of a com-
ponent or transmission line at various frequencies, and the spectrum analyzer, a
device that simultaneously displays the amplitudes across a range of frequencies,
are examples of frequency-domain instruments.
1.4.1 Line Spectrums
From signal theory we know that a series of sine waves can be used to recreate a
recurring stream of pulses. In fact, a Fourier analysis can be used to find the ampli-
tude and phase of each of the frequencies and determine how many are needed for
reconstructing the pulses to the desired level of precision [7, 8].
After doing this analysis, we find that there is a fundamental frequency that
has the highest amplitude and possible harmonic frequencies with lesser amplitudes
occurring at integer multiples of the fundamental frequency.
The fundamental frequency
f
o
is found with (1.1) where
T
is the pulse period.
1
(1.1)
f
=
o
T
For instance, the fundamental frequency of a 10-ns stream of pulses is 100
MHz. The second harmonic (if present) is twice the fundamental frequency (200
MHz), the third is 300 MHz, and so on indefinitely. The amplitude of each har-
monic depends on the duty cycle and rise time of the pulse, and some can have a
value of zero (that is, will not be present). None will have a value higher than the
fundamental. In fact, (1.1) shows the lowest frequency that will be present in a
pulse stream.
For example, a 10-ns square wave (a pulse with the same high and low times)
with zero rise time edges has a fundamental frequency of 100 MHz and only the
odd harmonics. This is represented in the frequency domain by a discrete line spec-
trum as shown in Figure 1.2. The amplitude of the fundamental has been set to 1,
and the harmonics have been scaled accordingly.
1.4.2 Recreating a Pulse with Sine Waves
The harmonics with the amplitudes shown in the spectrogram can be added to-
gether to recreate the pulse stream. In theory an infinite number of harmonics may
be necessary, but far fewer suffice in practice.
