470
Practical MATLAB
® Applications for Engineers
and
tan[ ( )]
W
W
W
ϭ
imag F e
real F e
j
j
{ [ ]}
{ [ ]}
R.5.34 It can be shown that if f(n) is a real sequence, then F(e
jW ) and |F(e
JW )| are real functions of W.
R.5.35 Since θ(W) is a periodic function with a period of 2π rad, then its principal value is
defi ned over the domain −π ≤ W ≤ +π.
R.5.36 A suffi cient condition for the existence and convergence of DTFT F(e
jW ) is that the
following relation must be satisfi ed:
F e
f n
jW
n
( )
( )
ϭ
ϱ
ϭϪϱ
ϩϱ
∑
Ͻ
R.5.37 It can be shown that if f(n) is a real sequence, then F(e
jW ) = real {F(e
jW )} is an even
function of W, whereas imag {F(e
jW )} is an odd function of W.
Recall that the spectrum equations indicate that the magnitude spectrum is even,
whereas the phase spectrum is odd, indicated by
| F(e
jW )| = |F(e
−jW )|
and
∠F(e
jW ) = −∠F(e
−jW )
R.5.38 Let f 1 (n) ↔ F 1 (e
jW ) and f 2 (n) ↔ F 2 (e
JW ) be transform pairs. Then, some important properties that relate the time and frequency domains are summarized as follows:
a. Linearity (where a 1 and a 2 are arbitrary constants)
a 1 f 1 (n) + a 2 f 2 (n) ↔ a 1 F 1 (e
jW ) + a 2 F 2 (e
jW )
b. Time shifting
f n n
e
F ejW
jWn
(
)
(
)
Ϫ
Ϫ
0
0
↔
c. Time reverse
f(−n) ↔ F(e
−jW )
d. Differentiation in frequency
nf n
j
d
dw
F e
jw
( )
{ ( )}
↔ Ϫ
e. Freq uency shifting (modulation theorem)
f n e
F e
j n
j W W
( )[
]
(
)
(
)
Ϫ
Ϫ
W 0
0
↔
CRC_47760_CH005.indd 470
CRC_47760_CH005.indd 470
7/23/2008 1:47:18 PM
7/23/2008 1:47:18 PM