Digital Signal Processing Reference
In-Depth Information
TABLE 12.7
Key Equations of Chapter 12
Equation Title
Equation Number
Equation
X(Æ) =
(x[n]) =
q
n=-q
x[n]e
-jnÆ
Discrete-time Fourier transform
(12.1)
Æ
1
+ 2p
1
2p
L
1
2p
L
2p
X(Æ)e
jnÆ
dÆ
-1
[X(Æ)] =
X(Æ)e
jnÆ
dÆ=
Inverse discrete-time Fourier transform
(12.2)
x[n] =
Æ
1
Relation of Fourier transform of
sampled signal to DTFT
q
(12.6)
[x
s
(t)]
vT=Æ
=
B
x(nT)d(t - nT)
R
vT=Æ
=
[x(nT) = x[n]]
a
n=-
q
Periodicity of DTFT
(12.7)
X(Æ+2p) = X(Æ)
x[n],
0 F n F N - 1
One period of a periodic signal
(x
0
[n])
(12.17)
x
0
[n] =
c
0,
otherwise
q
Periodic signal in terms of
x
0
[n]
(12.19)
x[n] = x
0
[n]*
a
d[n - kN]
k=-
q
q
k=-
q
N
q
k=-
q
2p
2pk
N
d[n - kN]
Î
"
¢
≤
DTFT of DT impulse train
(12.20)
d
Æ-
N
q
k=-
q
2p
2pk
N
2pk
N
¢
≤
¢
≤
DTFT of periodic signal
(12.23)
X(Æ) =
X
0
d
Æ-
N- 1
x[n]e
-j2pkn
>
N
,
DFT and IDFT
(12.30)
X[k] =
[x[n]] =
a
k = 0, 1, 2, Á , N - 1
n= 0
N- 1
1
N
a
>
-1
[X[k]] =
X[k]e
j2pkn
N
,
x[n] =
n = 0, 1, 2, Á , N - 1
k= 0
DFT and IDFT with shorthand
notation
N- 1
x[n]W
kn
,
(12.33)
X[k] =
[x[n]] =
a
k = 0, 1, Á , N - 1
n= 0
N- 1
1
N
a
-1
[X[k]] =
X[k]W
-kn
,
x[n] =
n = 0, 1, Á , N - 1
k= 0
N- 1
N- 1
W
0n
W
kn
Orthogonality of DT exponentials
(12.35)
= N and
a
= 0,
k = 1, 2, Á , N - 1
a
n= 0
n= 0
N- 1
Circular convolution
(12.46)
y[n] = x[n]
*
h[n] =
a
x[l]h[n - l]
l = 0
x[n]
*
h[n]
Î
"
X[k]H[k]
DFT of circular convolution
(12.47)
