PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

470 Practical MATLAB® Applications for Engineers

and

tan[ ( )] W

W


 W

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(ejW) and |F(eJW)| are real func-
tions 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(ejW) is that the
following relation must be satisfi ed:

FejW f n

n

()
()

 



∑ 

R.5.37 It can be shown that if f(n) is a real sequence, then F(ejW) = real {F(ejW)} is an even
function of W, whereas imag {F(ejW)} 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(ejW)| = |F(e−jW)|

and

∠F(ejW) = −∠F(e−jW)

R.5.38 Let f 1 (n) ↔ F 1 (ejW) and f 2 (n) ↔ F 2 (eJW) be transform pairs. Then, some important prop-
erties 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 (ejW) + a 2 F 2 (ejW)

b. Time shifting

fn n e FejW
() jWn ()
0 ↔^0

c. Time reverse

f(−n) ↔ F(e−jW)

d. Differentiation in frequency

nf n j

d

dw

()↔ {( )}Fejw

e. Freq uency shifting (modulation theorem)

fn e Fe
()[jnW^00 ]↔ (jWW())