Time Domain Representation of Continuous and Discrete Signals 7
R.1.11 Figure 1.4 indicates that the unit doublet cannot be represented as a conventional
function since there is no single value, fi nite or infi nite, that can be assigned to δ(t)’
at t = 0.
R.1.12 Additional useful properties of the impulse function δ(t) that can be easily proven
are stated as follows:
a. ft t t dt ft() ( oo) ( )
∞∞
∫
b. ft t()() ()ootdt f t
∞∞
∫
c. ft t()() ( )o t t dt ftt
110
∞∞
∫
R.1.13 The unit impulse δ(t), the unit doublet δ(t)’, and the higher derivatives of δ(t) are
often referred as the impulse family. These functions vanish at t = 0 , and they
all have the origin as the sole support. At t = 0 , all the impulse functions suffer
discontinuities of increasing complexity, consisting of a series of sharp pulses going
positive and negative depending on the order of the derivative.
As was stated δ(t) is an even function of t, and so are all its even derivatives, but
all the odd derivatives of δ(t) return odd functions of t.
The preceding statement is summarized as follows: () (), ’()ttt t ()
or in general() 22 nn()tt( ) (even case)21 nn()tt (^21) ( ) ( odd case)
R.1.14 A train of impulses denoted by the function Imp[(t)T] defi nes a sequence consisting of
an infi nite number of impulses occurring at the following instants of time nT, ..., −T,
T, 2T, 3T, ..., nT, as n approaches ∞. This sequence can be expressed analytically by
Imp[( ) ]ttnT ( T)
n
∞∞
∑illustrated in Figure 1.5.FIGURE 1.
Plot of Imp[(t)T].Imp[(t)T]1t
− 2 T −T 0 T 2 T 3 T 4 T