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Sec. 2–1 Properties of Signals and Noise 35
Physically Realizable Waveforms
Practical waveforms that are physically realizable (i.e., measurable in a laboratory) satisfy
several conditions: †
1. The waveform has significant nonzero values over a composite time interval that is finite.
2. The spectrum of the waveform has significant values over a composite frequency interval
that is finite.
3. The waveform is a continuous function of time.
4. The waveform has a finite peak value.
5. The waveform has only real values. That is, at any time, it cannot have a complex value
a + jb, where b is nonzero.
The first condition is necessary because systems (and their waveforms) appear to exist
for a finite amount of time. Physical signals also produce only a finite amount of energy. The
second condition is necessary because any transmission medium—such as wires, coaxial
cable, waveguides, or fiber-optic cable—has a restricted bandwidth. The third condition is a
consequence of the second, and will become clear from spectral analysis as developed in
Sec. 2–2. The fourth condition is necessary because physical devices are destroyed if voltage
or current of infinite value is present within the device. The fifth condition follows from the
fact that only real waveforms can be observed in the real world, although properties of waveforms,
such as spectra, may be complex. Later, in Chapter 4, it will be shown that complex
waveforms can be very useful in representing real bandpass signals mathematically.
Mathematical models that violate some or all of the conditions listed previously are
often used, and for one main reason—to simplify the mathematical analysis. However, if we
are careful with the mathematical model, the correct result can be obtained when the answer
is properly interpreted. For example, consider the digital waveform shown in Fig. 2–1. The
mathematical model waveform has discontinuities at the switching times. This situation violates
the third condition—that the physical waveform be continuous. The physical waveform
is of finite duration (decays to zero before t =;q), but the duration of the mathematical
waveform extends to infinity.
In other words, this mathematical model assumes that the physical waveform has
existed in its steady-state condition for all time. Spectral analysis of the model will approximate
the correct results, except for the extremely high-frequency components. The average
power that is calculated from the model will give the correct value for the average power of
the physical signal that is measured over an appropriate time interval. The total energy of the
mathematical model’s signal will be infinity because it extends to infinite time, whereas that
of the physical signal will be finite. Consequently, the model will not give the correct value
for the total energy of the physical signal without the use of some additional information.
However, the model can be used to evaluate the energy of the physical signal over some
finite time interval. This mathematical model is said to be a power signal because it has the
property of finite power (and infinite energy), whereas the physical waveform is said to be an
energy signal because it has finite energy. (Mathematical definitions for power and energy
† For an interesting discussion relative to the first and second conditions, see the paper by Slepian [1976].