APPENDIX B
to VOLUME A1
SOME USEFUL EXPANSIONS
Appendix to Volume A1
B- 2
Some useful expansions
Appendix to Volume A1
SOME USEFUL EXPANSIONS
cosA.cosB = 1/2 [ cos(A-B) + cos(A+B) ]
sinA.sinB = 1/2 [ cos(A-B) - cos(A+B) ]
sinA.cosB = 1/2 [ sin(A-B) + sin(A+B) ]
sin(A+B) = sinA cosB + cosA sinB
sin(A-B) = sinA cosB - cosA sinB
cos(A+B) = cosA cosB - sinA sinB
cos(A-B) = cosA cosB + sinA sinB
cos2Α = 1/2 + 1/2 cos2Α
cos3Α = 3/4 cosΑ + 1/4 cos3Α
cos4Α = 3/8 + 1/2 cos2Α + 1/8 cos4Α
cos5Α = 5/8 cosΑ + 5/16 cos3Α + 1/16 cos5Α
cos6Α = 5/16 + 15/32 cos2Α + 3/16 cos4Α + 1/32 cos6Α
sin2Α = 1/2 - 1/2 cos2Α
sin3Α = 3/4 sinΑ - 1/4 sin3Α
sin4Α = 3/8 - 1/2 cos2Α + 1/8 cos4Α
sin5Α = 5/8 sinΑ - 5/16 sin3Α + 1/16 sin5Α
sin6Α = 5/16 - 15/32 cos2Α + 3/16 cos4Α - 1/32 cos6Α
•
During envelope waveform evaluations one or other of the following expansions is often
needed:
arctan [
r sin z
1
1
1
] = r sin z + r 2 sin 2 z + r 3 sin 3z + r 4 sin 4 z +......
(1 − r ) cos z
2
3
4
1
2 r sin z
1
1
arctan [
] = r sin z + r 3 sin 3z + r 5 sin 5 z +.....
2
2
3
5
1− r
1 − r cos z
1 − 2 r cos z + r 2
arctan x = x −
Some useful expansions
= 1 + r cos z + r 2 cos 2 z + r 3 cos 3z +....
x3 x5
+
−..... for |x|< 1
3
5
B-3
Appendix to Volume A1
•
The binomial expansion, for x < 1:
(1 + x )n = 1 + nx +
n( n − 1) x 2 n( n − 1)( n − 2 ) x 3
+
+.....
2!
3!
is especially useful for the case n = ½ and n = -½
•
A zero-mean square wave, peak-to-peak amplitude 2E, period
make it an even function:
square wave =
•
(
2π
) , time axis chosen to
ω
4E
1
1
[cos ωt − cos 3ωt + cos 5ωt −.....
π
3
5
Required for FM spectral analysis are the following:
cos(β sinφ) = J0(β) + 2 [ J2(β) cos2φ + J4(β) cos4φ + ..................]
sin(β sinφ) = 2 [ J1(β) sinφ + J3(β) sin3φ + J5(β) sin5φ + ............]
cos(β cosφ) = J0(β) - 2 [ J2(β) cos2φ - J4(β) cos4φ + ....................]
sin(β cosφ) = 2 [ J1(β) cosφ - J3(β) cos3φ + J5(β) cos5φ - ..............]
where Jn(β) is a Bessel function of the first kind, argument β, and order n.
•
B- 4
You will also need to know that:
J− n (β ) = ( −1)n Jn (β )
Some useful expansions
Appendix to Volume A1
Some useful expansions
B-5