常用数学公式14

∫e
o
∞
− at
−e 2j
− j 2 πft
dt =
1 1 1 1 1 1 + + 2 a − j 2 πft 2 a + j 2 πft 2 j a − j 2 πft
1 1 a + 2 πf a 2 πf e o = 2 = 2 + 2 = Hx ( f ) + Hx (f) 2 2 2 2 2 j a + j 2 πft a + 4 π f a + 4π f a + 4π 2 f 2 a 2 πf −j 2 2 2 a + 4π f a + 4π 2 f 2
14
The Hartley Transform
14.1 Introduction to the Hartley Transform 14.2 Relationship with the Fourier Transform 14.3 Power and Phase Spectra 14.4 Properties of the Hartley Transform 14.5 Examples of Hartley Transforms 14.6 Hartley Series 14.7 Tables of Fourier and Hartley Transforms 14.8 Two-Dimensional Hartley Transform 14.9 Discrete Hartley Transform References
© 1999 by CRC Press LLC
Example (Exponential Function): Hx ( f ) = + −
∫
∞
−∞
e − at u(t ) cas(2 πft ) dt = e
j 2 πft
∫
∞
o
e − at cas(2 πft ) dt =
∫
∞
e − at
o
e j 2 πft + e − j 2 πft dt 2
14.1.5 Signs of the cas Function
TABLE 14.2 Signs of the cas Function
Quadrant 1st 2nd 3rd 4th cas + + and – – + and –
© 1999 by CRC Press LLC
14.1.6 Variations of the cas Function
e Hx (f) =
H x ( f ) + H x (− f ) = 2 H x ( f ) − H x (− f ) = 2
∫
∞
−∞ ∞
x(t ) cos(2 πft ) dt x(t ) sin(2 πft ) dt
o Hx (f) =
∫
−∞
14.1.4 Properties of the cas Function
14.1 Introduction to the Hartley Transform
14.1.1 Definition of the Pair with Symmetry and the Use of ω (units: rad s–1)
H x (ω ) = 1 2π 1 2π
∫
∞
−∞
x(t ) cas(ωt ) dt
o Im{X ( f )} = − H x (f)
e o Hx ( f ) = Hx ( f ) + Hx ( f ) = Re{X ( f )} − Im{X ( f )}
Note: The Fourier transform is the even part of the Hartley transform minus j times the odd part. The Hartley transform is the real part of the Fourier transform minus the imaginary part.
casξ = cot ξ sin ξ + tan ξ cos ξ casτ casν = cos(τ − ν) + sin( τ + ν) casξ = cot ξ sec ξ + tan ξ csc ξ csc ξ sec ξ
cas2ξ = cas 2 ξ − cas 2 ( − ξ)
∫ cas(τ) dτ = −cas(−τ) = −cas′τ
d cas τ = cas( − τ) = cas ′τ dt cas( τ + ν) = cosτ casν + sin τ cas ′ν cas( τ − ν) = cosτ cas ′ν + sin τ casν
1 casτ + casν = 2cas 1 2 ( τ + ν) cos 2 ( τ − ν) 1 casτ − casν = 2cas ′ 1 2 ( τ + ν)sin 2 ( τ − ν)
2
X ( f ) = F {e − at u(t )} = Example (Delta Function): Hx ( f ) =
∫ ∫
∞
−∞
δ (t − to ) cas(2 πft ) dt = cas2 πfto
Example (Pulse Function): Hx ( f ) = +
∞
−∞
cas(ωt ) = cos(ωt ) + sin(ωt ) π cas(ωt ) = 2 sin ωt + 4 π cas(ωt ) = 2 cos ωt − 4 ω = 2 πf
© 1999 by CRC Press LLC
14.1.3 Odd and Even Parts of the Hartley Transform
TABLE 14.3 Variations of the cas Function
Quadrant 1st 2nd 3rd 4th cas +1 → +1 with a maximum at π/4 +1 → –1 –1 → –1 with a maximum at 5π/4 –1 → +1
14.1.7 Values of the cas at Special Angles
14.3.1 Power Spectrum
If X( f ) is the Fourier transform of x(t), then Px ( f ) = power spectrum = X ( f ) X ∗ ( f ) = [Re{X ( f )}]2 + [Im{X ( f )}]2 From (14.2.1) H ( f ) + H x (− f ) H x ( f ) − H x (− f ) e o Px ( f ) = [ H x ( f )]2 + [− H x ( f )]2 = x + 2 2 = [ H x ( f )]2 + [ H x ( − f )]2 2
∫
∞
−∞
x(t ) cos(2 πft ) dt ) sin(2 πft ) dt
e o = Hx ( f ) − j Hx (f) =
H x ( f ) + H x (− f ) H ( f ) − H x (− f ) −j x 2 2
e Re{X ( f )} = H x ( f ),
TABLE 14.4 Values of the cas at Special Angles
Angle 0° = 0 π 30° = 6 45° = 60° = 90° = π 4 π 3 π 2 2π 3 5π 6
1 2
cas 0
1 2
( 3 + 1) 2 (1 + 3 )
1
1 2
120° = 150° =
pa (t ) cas(2 πft ) dt =
∫
a
−a
cas(2 πft ) dt =
∫
e j 2 πft + e − j 2 πft dt 2 −a
a
∫
e j 2 πft − e − j 2 πft sin 2 πfa dt = 2j πf −a
a
14.3 Power and Phase Spectra
TABLE 14.1 Trigonometric Properties of the cas Function
The cas function The cas function The complementary cas function The complementary cas function Relation to cos Relation to sin Reciprocal relation Product relation Function product relation Quotient relation Double angle relation Indefinite integral relation Derivative relation Angle-sum relation Angle-difference relation Function-sum relation Function-difference relation casξ = cos ξ + sin ξ casξ = 1 2 [(1 + j ) exp( − jξ ) + (1 − j ) exp( jξ )] cas ′ξ = cas( − ξ) = cos ξ − sin ξ 3π π 2 sin ξ + = 2 cos ξ + 4 4 cosξ = 1 2 [casξ + cas( − ξ )] sinξ = 1 2 [casξ − cas( − ξ )] casξ = csc ξ + sec ξ sec ξ csc ξ