112

WAVEFORMS AND SIGNALS

[CHAP. 6

Fig. 6-11 lim I ¼ lim 12 ½ f ðt0 Þ þ f ðt0 þ TÞ

T!0

T!0

We assumed f ðtÞ to be continuous between t0 and t0 þ T.

Therefore,

lim I ¼ f ðt0 Þ

T!0 ð1

But

lim I ¼

T!0

ð34dÞ

ðt  t0 Þ f ðtÞ dt

ð34eÞ (34f)

1

ð1 and so

ðt  t0 Þ f ðtÞ dt ¼ f ðt0 Þ

(34g)

1

The identity (34g) is called the sifting property of the impulse function.

It is also used as another deﬁnition for

ðtÞ.

6.10

THE EXPONENTIAL FUNCTION

The function f ðtÞ ¼ est with s a complex constant is called exponential. It decays with time if the real part of s is negative and grows if the real part of s is positive. We will discuss exponentials eat in which the constant a is a real number. The inverse of the constant a has the dimension of time and is called the time constant  ¼ 1=a. A decaying exponential et= is plotted versus t as shown in Fig. 6-12. The function decays from one at t ¼ 0 to zero at t ¼ 1. After  seconds the function et= is reduced to e1 ¼ 0:368. For  ¼ 1, the function et is called a normalized exponential which is the same as et= when plotted versus t=. EXAMPLE 6.19 Show that the tangent to the graph of et= at t ¼ 0 intersects the t axis at t ¼  as shown in Fig. 6-12. The tangent line begins at point A ðv ¼ 1; t ¼ 0Þ with a slope of det= =dtjt¼0 ¼ 1=. The equation of the line is vtan ðtÞ ¼ t= þ 1. The line intersects the t axis at point B where t ¼ . This observation provides a convenient approximate approach to plotting the exponential function as described in Example 6.20. EXAMPLE 6.20 Draw an approximate plot of vðtÞ ¼ et= for t > 0. Identify the initial point A (t ¼ 0; v ¼ 1Þ of the curve and the intersection B of its tangent with the t axis at t ¼ . Draw the tangent line AB. Two additional points C and D located at t ¼  and t ¼ 2, with heights of 0.368 and

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An