Signals and Systems - Electrical Engineering

100 C H A P T E R 1: Continuous-Time Signals

ρ(t) dρ(t)

dt

1

1

1

− 1

223

3

0 0

t t

(1) (2)

(−2) (−2)

· · · · · ·

FIGURE 1.13

Causal train of pulsesρ(t)and its derivative. The number enclosed in()is the area of the corresponding delta

function and it indicates the jump at the particular discontinuity—positive when increasing and negative when

decreasing.

is the desired signal. Notice thatρ(t)equals zero fort<0, thus it is causal. Given that the derivative

of a sum of signals is the sum of the derivative of each of the signals, the derivative ofρ(t)is

dρ(t)

dt

=

∑∞

k= 0

ds(t− 2 k)

dt

=

∑∞

k= 0

[δ(t− 2 k)− 2 δ(t− 1 − 2 k)+δ(t− 2 − 2 k)]

which can be simplified to

dρ(t)

dt

=[δ(t)− 2 δ(t− 1 )+δ(t− 2 )]+[δ(t− 2 )− 2 δ(t− 3 )+δ(t− 4 )]+[δ(t− 4 )···

=δ(t)+ 2

∑∞

k= 1

δ(t− 2 k)− 2

∑∞

k= 1

δ(t− 2 k+ 1 )

whereδ(t), 2δ(t− 2 k), and− 2 δ(t− 2 k+ 1 )fork≥1 occur att=0,t= 2 k, andt= 2 k−1 for

k≥1, or the times at which the discontinuities ofρ(t)occur. The value associated with theδ(t)

corresponds to the jump of the signal from the left to the right. Thus,δ(t)indicates there is a

discontinuity inρ(t)at zero as it jumps from 0 to 1, while the discontinuities at 2, 4,...have a

jump of 2 from−1 to 1, increasing. The discontinuities indicated byδ(t− 2 k− 1 )occurring at 1,

3, 5,...are from 1 to−1 (i.e., decreasing, so the value of−2). See Figure 1.13. n

1.4.3 Special Signals—the Sampling Signal and the Sinc

Two signals of great significance in the sampling of continuous-time signals and their reconstruction

are the sampling signal and the sinc. Sampling a continuous-time signal consists in taking samples of

the signal at uniform times. One can think of this process as the multiplication of a continuous-time