Signals and Systems - Electrical Engineering

128 C H A P T E R 2: Continuous-Time Systems

Nose

Lips

Nasal cavity

Tongue

Vellum

Epiglottis

Vocal

chords

Lungs

(a)

e LTI system

v(t)

Voiced

s(t)

Unvoiced

eu(t)

Speech

(b)

FIGURE 2.4

(a) Vocal system: principal organs of articulation. (b) Model for speech production.

Solution

If we generalize the characteristic equations for the resistor, capacitor, and inductor to be

v(t)=R(t)i(t)

q(t)=C(t)vc(t)

φ(t)=L(t)iL(t)

as straight lines with time-varying slope, we have linear but time-varying elements. Usingi(t)=

dq(t)/dtandv(t)=dφ(t)/dt, we obtain the following voltage–current relations:

v(t)=R(t)i(t)

i(t)=C(t)

dvc(t)

dt

+

dC(t)

dt

vc(t)

v(t)=L(t)

diL(t)

dt

+

dL(t)

dt

iL(t)

AsR(t)is a function of time, the resistor is a time-varying system. The second and third equation

are linear differential equations with time-varying coefficients representing time-varying capacitors

and inductors. n