Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
Document source : notes.ump.edu.my
z
1
z
2
x
1
x
2
y
1
y
2
jx
2
y
1
À x
1
y
2
x
2
2
y
2
2
A:19a
r
1
r
2
e
jy
1
Ày
2
:
A:19b
Note that addition and subtraction are straightforward in rectangular form, but is
difficult in polar form. Division is simple in polar form, but is complicated in rectangu-
lar form.
The complex arithmetic of the complex number x can be listed as
z
Ã
x À jy re
Àjy
,
A:20
where * denotes complex-conjugate operation. In addition,
zz
Ã
jzj
2
,
A:21
z
À1
1
z
1
r
e
Àjy
,
A:22
z
N
r
N
e
jNy
:
A:23
The solution of
z
N
1
A:24
are
z
k
e
jy
k
e
j2pk=N
, k 0, 1, . . . , N À 1:
A:25
As illustrated in Figure A.2, these N solutions are equally spaced around the unit circle
jzj 1. The angular spacing between them is y 2p=N.
Re[z]
Im[z]
|z|=1, unit circle
e
j(2p /N)
Figure A.2 Graphical display of the Nth roots of unity, N 8
448
APPENDIX A: SOME USEFUL FORMULAS
Summary :
z 1 z 2 x 1 x 2 y 1 y 2 jx 2 y 1 À x 1 y 2 x 2 2 y 2 2 A:19a r 1 r 2 e jy 1 Ày 2 : A:19b Note that addition and subtraction are straightforward in rectangular form, but is difficult in polar form.
Tags :
form,figure,addition,but,àjy,unit,complex,polar,jzj,circle,straightforward,jny,useful