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
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The inverse discrete Fourier transform (IDFT) is used to transform the X(k) back into
the original sequence x(n). Given the frequency samples X(k), the IDFT is defined as
xn
1
N
NÀ1
k0
Xke
j2p=Nkn
1
N
NÀ1
k0
XkW
Àkn
N
, n 0, 1, . . . , N À 1:
7:1:6
This is identical to the DFT with the exception of the normalizing factor 1/N and
the sign of the exponent of the twiddle factors. The IDFT shows that there is no loss
of information by transforming the spectrum X(k) back into the original time sequence
x(n). The DFT given in (7.1.3) is called the analysis transform since it analyzes the signal
x(n) at N frequency components. The IDFT defined in (7.1.6) is called the synthesis
transform because it reconstructs the signal x(n) from the frequency components.
Example 7.2: Consider the finite-length sequence
xn a
n
, n 0, 1, . . . , N À 1,
where 0 < a < 1. The DFT of the signal x(n) is computed as
xk
NÀ1
n0
a
n
e
Àj2pk=Nn
NÀ1
n0
ae
Àj2pk=N
n
1 À ae
Àj2pk=N
N
1 À ae
Àj2pk=N
1 À a
N
1 À ae
Àj2pk=N
, k 0, 1, . . . , N À 1:
The DFT and IDFT defined in (7.1.3) and (7.1.6), can be expressed in matrix±vector
form as
X Wx
7:1:7a
and
x
1
N
W
Ã
X ,
7:1:7b
where x x0 x1 . . . xN À 1
T
is the signal vector, the frequency-domain DFT
coefficients are contained in the complex vector X X0 X1 . . . XN À 1
T
, and
the NxN twiddle-factor matrix (or DFT matrix) W is given by
W W
kn
N
Â
Ã
0 k
,
n NÀ1
1
1
Á Á Á
1
1
W
1
N
Á Á Á
W
NÀ1
N
.
.
.
.
.
.
..
.
.
.
.
1 W
NÀ1
N
Á Á Á W
NÀ1
2
N
P
T
T
T
T
T
T
T
R
Q
U
U
U
U
U
U
U
S
,
7:1:8
306
FAST FOURIER TRANSFORM AND ITS APPLICATIONS
Summary :
The DFT of the signal x(n) is computed as xk NÀ1 n0 a n e Àj2pk=Nn NÀ1 n0 ae Àj2pk=N n 1 À ae Àj2pk=N N 1 À ae Àj2pk=N 1 À a N 1 À ae Àj2pk=N , k 0, 1, . XN À 1 T , and the NxN twiddle-factor matrix (or DFT matrix) W is given by W W kn N Â Ã 0 k , n NÀ1 1 1 Á Á Á 1 1 W 1 N Á Á Á W NÀ1 N .
Tags :
nà1,dft,transform,idft,àj2pkn,signal,gien,frequency,716,sequence,defined,back,ector