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A–8 The Dirac Delta Function 675
A–7 HILBERT TRANSFORM PAIRS †
Definition of Hilbert Transform: xN(t) ! x(t) * 1 pt
= 1 p L
q
-q
x(l)
t - l dl
Function
Hilbert Transform
1. x(at b) xN (at + b)
2. x(t) y(t) xN (t) + yN (t)
3.
d n x(t)
dt n
d n
dt n xN(t)
4. A constant 0
5.
1
-pd(t)
t
6. sin (v 0 t + u) -cos (v 0 t + u)
7.
sin at
= Sa(at)
at
- 1
2p at Sa2 (at)
8. e ;jv 0t
< je ;jv 0t
9. d (t) 1
10.
pt
a
t
p(t 2 + a 2 )
pAt 2 + a 2 B
11. a t 1, ƒ t ƒ … T>2
b ! e
T 0, t elsewhere
1
p ln 2t + T
2t - T
A–8 THE DIRAC DELTA FUNCTION
DEFINITION. The Dirac delta function δ(x), also called the unit impulse function,
satisfies both of the following conditions:
L
q
-q
d(x) dx = 1 and d(x) = e q, x = 0
0, x = 0 f
Consequently, δ (x) is a “singular” function. ‡
† Fourier transform theorems are given in Table 2–1, and Fourier transform pairs are given in Table 2–2.
‡ The Dirac delta function is not an ordinary function since it is really undefined at x = 0. However, it is
described by the mathematical theory of distributions [Bremermann, 1965].