Digital Signal Processing Reference
In-Depth Information
1
0
−1
0
0.2
0.4
0.6
0.8
(a) Time, Seconds
0.5
0
−0.5
250
300
350
400
(b) Sample
400
200
0
0
10
20
30
40
50
60
(c) Frequency, Hz
Figure 3.45:
(a) A zero-order-hold-reconstructed 1 Hz sine wave prior to being lowpass filtered; (b) A
150-sample portion of the waveform at (a) simulated with 1000 samples; (c) The 1000-point DFT of
the entire simulated sequence representing the waveform at (a).
e
j
2
πnk/N
x
[
0
:
N
−
1
]
k
=
X
[
k
]
or using the rectangular notation,
[
:
−
]
k
=
[
]
[
]+
[
]
x
0
N
1
X
k
(
cos
2
πkn/N
j
sin
2
πnk/N
)
where the vector
n
= 0
:
N
−
1, and the complete IDFT is the accumulation of all
X
[
k
]
-weighted
harmonic basis vectors
x
[
0
:
N
−
1
]
k
N
−
1
x
[
0
:
N
−
1
]=
x
[
0
:
N
−
1
]
k
k
=
0
Notice how similar the IDFT is to the DFT; they differ only by using
x
[
n
]
in the forward transform
vice
X
[
k
]
in the reverse transform, a positive sign in the exponential for the reverse transform as opposed
to a negative sign for the forward transform, and the scaling constant 1
/N
. The negative sign may be used
in the exponential in either one of the forward or reverse transform, providing the positive sign is used
in the other transform, and the scaling constant may be used on either the forward or reverse transform.
