514 APPENDIX A
APPENDIX B
TRIGONOMETRIC IDENTITIES
In this section, we present several useful trigonometric identities used in communication system design.
tanSA B
2 T
sin (A) sin (B)
cos (A) cos (B)
sin (A) cos (B) cos (A) sin (B) sin (A B)
sin (A) sin (B) cos (A) cos (B) cos (A B)
2 sin (A) cos (B) sin (A B) sin (A B)
2 sin (A) sin (B) cos (A B) cos (A B)
2 cos (A) cos (B) cos (A B) cos (A B)
4 sin 3(A) 3 sin (A) sin (3A)
4 cos 3(A) 3 cos (A) cos (3A)
sin (2A) 2 sin (A) cos (A)
2 sin 2(A) 1 cos (2A)
2 cos 2(A) 1 cos (2A)
cos 2(A) sin 2(A) cos (2A)
cos 2(A) sin 2(A) 1
sin (A) cos (A 90)
cos (A) sin (A 90)
sin (A)
ejA ejA
2j
cos (A)
ejA ejA
2
e
ju cos (u)
j sin (u)
515
ln10 2.302585092
log10 e 0.434294481
e 2.718281828
cos (nA) Re5( cos (A) j sin (A))n6 sin (nA) Im5( cos (A) j sin (A))n6 tan (A
B)
tan (A)
tan (B)
1 7 tan (A) tan (B)
516 APPENDIX B
APPENDIX C
DEFINITE INTEGRALS
3
`
0
ea2x2dx
1
2a
2p
3
`
0
xneaxdx
n!
an1 (n 1)
3
`
0
ea2x2 # cos (bx)dx
1
2a
2pe
Ab
2aB2
3
`
0
eax # sin (x)dx
1
1 a2
3
`
0
eax # cos (x)dx
a
1 a2
3
`
0
x # exp (a2x2) # erfc(bx)dx
1
2a2
# c1
b 2a2 b2 d
3
`
0
x2 # exp(ax2)dx
1
4
# Ap
a3
3
`
0
eax # Ios22bxddx
1
a # e
ba
3
`
0
I0satd # ebt dt
1 2b2 a2
3
`
`
e
ax2
2 dx A2p
a
3
`
`
ex2dx 2p
517
(x ZyZ) 3
`
0
exp (xz) # Iv( yz)dz
yv 2x2 y2 (x 2x2 y2)v
In(x) a`
k0 Qx
2Rn2k
k! # (n k 1)
In(x)
1p
# 3
p
0
e x cos (u) # cos (nu)du
3
`
0
sin (px)
px dx
1
2
3
`
0
cos(nx)
1 x2 dx
p
2
eZnZ
3
`
0
sin 2(x)
x2 dx
p
2
3
`
0
cos (x2)dx
1
2Ap
2
3
`
0
sin (x2)dx
1
2Ap
2
3
`
0
tan (x)
x dx
p
2
3
`
0
sin (x)
x dx
p
2
3
`
`
x2ebx2dx
1
2
2p
b2b
3
z
0
a2xeaxdx 1 (1 az)eaz
3
`
0
x2ex2dx
1
42p
518 APPENDIX C
where u w p qand .
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