FIGURE 2.1 Unit-step function. FIGURE 2.2 Graphical representation of the impulse Ko(n) δ(t)=0, t≠0 6O)dX=1, for any real numbere>0 The first condition states that d(n is zero for all nonzero values of t, while the second condition states that the area under the impulse is 1, so 8(n) has unit area. It is important to point out that the value 8(0)of &(r)at t 0 is not defined; in particular, 8(0)is not equal to infinity. For any real number k, ka(t) is the impulse with area K. It is defined by K6(t)=0, t≠0 Kδ(^λ)d^=K, for any real numbere The graphical representation of K8(n is shown in Fig. 2.2. The notation K in the figure refers to the area of the impulse Ko(n The unit-step function u(t) is equal to the integral of the unit impulse 8(t); more precisely, we have (t) δOλ)d入, ll t except t =0 Conversely, the first derivative of u(n), with respect to t, is equal to 8(n), except at t=0, where the derivative of u( t) is not defined Ramp function The unit-ramp function r( r)is defined mathematically by t≥0 r(t) Note that for t20, the slope of r(r) is 1. Thus, r(t) has unit slope, which is the reason r( n)is called the unit-ramp function. If K is an arbitrary nonzero scalar(rea ber), the ramp function Kr( n)has slope K for t20. The FIGURE 2.3 Unit-ramp function unit-ramp function is plotted in Fig. 2.3 The unit-ramp function r(t) is equal to the integral of the unit-step function id r); that is (t) u(a)dn e 2000 by CRC Press LLC© 2000 by CRC Press LLC The first condition states that d(t) is zero for all nonzero values of t, while the second condition states that the area under the impulse is 1, so d(t) has unit area. It is important to point out that the value d(0) of d(t) at t = 0 is not defined; in particular, d(0) is not equal to infinity. For any real number K, Kd(t) is the impulse with area K. It is defined by The graphical representation of Kd(t) is shown in Fig. 2.2. The notation K in the figure refers to the area of the impulse Kd(t). The unit-step function u(t) is equal to the integral of the unit impulse d(t); more precisely, we have Conversely, the first derivative of u(t), with respect to t, is equal to d(t), except at t = 0, where the derivative of u(t) is not defined. Ramp Function The unit-ramp function r(t) is defined mathematically by Note that for t ³ 0, the slope of r(t) is 1. Thus, r(t) has unit slope, which is the reason r(t) is called the unit-ramp function. If K is an arbitrary nonzero scalar (real num￾ber), the ramp function Kr(t) has slope K for t ³ 0. The unit-ramp function is plotted in Fig. 2.3. The unit-ramp function r(t) is equal to the integral of the unit-step function u(t); that is, FIGURE 2.1 Unit-step function. FIGURE 2.2 Graphical representation of the impulse Kd(t) u (t) t 123 1 0 Kd (t) t 0 (K) d d l l e e e ( ) , ( ) , t t d = ¹ = -Ú 0 0 1 for any real number > 0 K t t K d K d d l l e e e ( ) , ( ) , = ¹ = -Ú 0 0 for any real number > 0 u t d t t t ( ) = ( ) , -• Ú d l l all except = 0 FIGURE 2.3 Unit-ramp function r(t) t 123 1 0 r t t t t ( ) , , = ³ < Ï Ì Ó 0 0 0 r t u d t ( ) = ( ) -• Ú l l