PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Time Domain Representation of Continuous and Discrete Signals 5

R.1.3 Analytically, the sampling process is accomplished by multiplying f(t) by a se-

quence of impulses. The concept of the unit impulse δ(t), also known as the Dirac

function, is introduced and discussed next.

R.1.4 The unit impulse, denoted by δ(t), also known as the Dirac or the Delta function, is

defi ned by the following relation:

()tdt



1

∞

∞

∫

meaning that the area under [δ(t)] = 1 ,

where

(t) = 0 , for t ≠ 0


(t) = 1 , for t = 0

δ(t) is an even function, that is, δ(t) = δ(−t).

The impulse function δ(t) is not a true function in the traditional mathematical

sense. However, it can be defi ned by the following limiting process:

by taking the limit of a rectangular function with an amplitude 1/τ and width τ,

when τ approaches zero, as illustrated in Figure 1.2.

The impulse function δ(t), as defi ned, has been accepted and widely used by engi-

neers and scientists, and rigorously justifi ed by an extensive literature referred as

the generalized functions, which was fi rst proposed by Kirchhoff as far back as

1882. A more modern approach is found in the work of K.O. Friedrichs published
      in 1939. The present form, widely accepted by engineers and used in this chapter is
      attributed to the works of S.L. Sobolov and L. Swartz who labeled those functions
      with the generic name of distribution functions.
      Teams of scientists developed the general theory of generalized (or distribu-
      tion) functions apparently independent from each other in the 1940s and 1950s,
      respectively.
      R.1.5 Observe that the impulse function δ(t) as defi ned in R.1.4 has zero duration, unde-
      fi ned amplitude at t = 0 , and a constant area of one. Obviously, this type of func-
      tion presents some interesting properties when analyzed at one point in time, that
      is, at t = 0.

1

lim { { t

0

0 + 0

t

FIGURE 1.

The impulse function δ(t).