Page 717 - The Mechatronics Handbook

0066_Frame_C23 Page 25 Wednesday, January 9, 2002 1:53 PM

TABLE 23.7 DTFT of Common DT Signals
x(n) Frequency-Domain Representation, X( f )
δ(n) 1
∞ 1
A, −∞ < n < ∞ AF s ∑ δ f – mF s ), F s = ---
(
k=∞ T
–
∞
1
(
u(n) ------------------- + F s ∑ δ f – kF s )
----
1 – e – j2πf 2 k=∞
–
 n  sin ( [ 2q + 1)πf ]
Π --------------- 1 ---------------------------------------

2q + 
πf )
(
sin
2
 sin πfq( )
n
Λ --- -----------------------
q  qsin πf( )
2
2
sgn(n) -------------------
1 – e – j2πf
n 1
α u(n) ------------------------, α < 1
1 αe– – j2πf
1 α
2
–
α |n| ---------------------------------------------------, α < 1
1 – 2α cos ( 2πf ) + α 2
– j2πf
n αe
nα u(n) ------------------------------, α < 1
)
( 1 αe – j2πf 2
–
−αn 1
e u(n) -----------------------------, α > 0
j2πf )
α +
1 – e ( –
– 2α
−α |n| 1 – e
e ---------------------------------------------------------, α > 0
α
2α
–
1 – 2e cos ( 2πf ) + e –
∞
j2πf 0 n
(
e F s ∑ δ f – f 0 + kF s )
k=∞
–
∞
(
F s jθ j – θ
cos(2πf 0 n + θ) ---- ∑ { e δ f –( f 0 + kF s ) + e δ f + f 0 + kF s )}
2
k=∞
–
sin ( 2πf c n)  f 
-------------------------- Π ------
πn  2f c 
Discrete Fourier Series 6–8
Suppose x(n) is a periodic DT signal with period N, then it is possible to obtain its discrete Fourier series
(DFS) expansion in a manner analogous to the computation of the complex Fourier series for the CT
– n j2πn/N
signals. The orthogonal basis function for the DFS is W N = e so that the decomposition of x(n)
into a sum of N harmonically related complex exponentials is expressed as
N−1 N−1
xn() = ∑ c k e j2πkn/N = ∑ c k W N – kn (23.42)
k=0 k=0
mn
where c k represents the discrete Fourier series coefficients. Multiplying both sides of (23.42) by W N , summing
over one period, and using the fact that
N−1  N, k = m
∑ W N nk−m) = 
(
k=0  0, k ≠ m
©2002 CRC Press LLC