Digital Signal Processing Reference
In-Depth Information
• For the case of
k
from0to
N
- 1, the negative bins are aliased as bins having values of
k
greater
than
N/
2. Thus, for the example shown in Fig. 3.8, for
k>
16 (i.e.,
N/
2), note that
X
[
k
−
N
]
=
[
]
X
.For
k
= 17, the equivalent bin is Bin[17-32] = Bin[-15], Bin[18] = Bin[-14],...and finally,
Bin[31] = Bin[-1]. Note that the periodicity extends to positive and negative infinity for all real
integer values of
k
. Bin[32], for example, is equivalent to Bin[0], as is Bin[64] (not shown), etc.
k
• For the asymmetrical bin arrangement, conjugate symmetry is had between Bin[N/2 - m] and
Bin[N/2 + m]. For example, as seen in Fig. 3.8, Bin[15] is the complex conjugate of Bin[17], which
is equivalent to Bin[17-32] = Bin[-15].
1
MathScript DFT Bin Arrangement
0.5
−N/2
N/2
N
0
−0.5
−10
0
10
20
30
(a) k (Bin)
1
MathScript DFT Bin Arrangement
0.5
−N/2
N/2
N
0
−0.5
−10
0
10
20
30
(b) k (Bin)
MathScript DFT Bin Arrangement
0.5
−N/2
N/2
N
0
−10
0
10
20
30
(c) k (Bin)
Figure 3.8:
(a) The real part of the DFT of a signal, evaluated from -
N/
2+1 to
N
where
N
= 32; (b)
The imaginary part of the DFT of the same signal; (c) The magnitude of the DFT of the signal. See the
text for details of the “MathScript DFT Bin Arrangement.”
[
−
]
Example 3.10.
Demonstrate the periodicity of k for the length-three DFT of the signal
1
,
1
,
1
.
An easy way to proceed is to make this call, letting
k
assume various values.
k = 0; y = sum(exp(-j*2*pi*(0:1:2)*k/3 ).*([1,-1,1]))
Bin 0 is, of course, real only, and there is no Bin
N/
2 since
N/
2 = 1.5. Bin 2 should prove to be
the same as Bin -1. The following answers are obtained by making the calls using the values of
k
shown:
