2.3 LTI Continuous-Time Systems 127The FM system is nonlinear. Suppose that we scale the message toγm(t), for some constantγ, the
corresponding output is given by
cos(ct+γ∫t−∞m(τ)dτ)which is not the previous output scaled (i.e.,γz(t)); thus FM is a nonlinear system.The Beginnings of Radio
The names of Nikola Tesla (1856–1943) and Reginald Fessenden (1866–1932) are linked to the invention of radio and ampli-
tude modulation [3, 58, 75]. Radio was initially called “wireless telegraphy” and then “wireless.” Tesla was a mechanical
as well as an electrical engineer, but mostly an inventor. He has been credited with significant contributions to electricity
and magnetism in the late 19th and early 20th centuries. His work is the basis of the alternating-current (AC) power system
and the induction motor. His work on wireless communications using the “Tesla coils” was capable of transmitting and
receiving radio signals. Although Tesla submitted a patent application for his basic radio before Guglielmo Marconi, it was
Marconi who was initially given the patent for the invention of the radio (1904). The Supreme Court in 1943 reversed the
decision in favor of Tesla [45].
Fessenden has been called the “father of radio broadcasting.” His early work on radio led to demonstrations in December
1906 of the capability of point-to-point wireless telephony, and what appears to be the first radio broadcasts of entertainment
and music ever made to an audience (in this case, shipboard radio operators in the Atlantic). Fessenden was a professor of
electrical engineering at Purdue University and the first chairman of the electrical engineering department of the University
of Pittsburgh in 1893.Vocal System
A remarkable system that we all have is the vocal system (see Figure 2.4). The air pushed out from
the lungs in this system is directed by the trachea through the vocal cords, making them vibrate and
create resonances similar to those from a wind musical instrument. The generated sounds are then
muffled by the mouth and the nasal cavities, resulting in an acoustic signal carrying a message. Given
the length of the typical vocal system, it is modeled as a distributed system and represented by partial
differential equations. Due to the complexity of this model, it is the speech signal along with the
understanding of the speech production that is used to obtain models of the vocal system. Speech
processing is one of the most fascinating areas of electrical engineering.
A typical linear time-invariant model for speech production considers segments of speech of about
20 msec, and for each develops a low-order LTI system. The input is either a periodic pulse for the
generation of voiced sounds (e.g., vowels) or a noiselike signal for unvoiced sounds (e.g., the /sh/
sound). Processing these inputs gives speechlike signals. A linear time-varying model would take into
consideration the variations of the vocal system with time and it would thus be more appropriate.
nExample 2.4
Characterize time-varying resistors, capacitors, and inductors. Assume zero initial conditions in the
capacitors and inductors.