Digital Signal Processing Reference
In-Depth Information
[1.611, (1.133 - 0.291i ), 0.120, (1.133 + 0.291i)]
We note conjugate symmetry for Bins 1 and 3, where Bin 3 is an alias of Bin -1. Note that Bins 0
and 2 (i.e.,
N/
2) are unique, read-only bins within a single period of
k
and do not have negative frequency
counterparts.
3.9.2 PERIODICITY IN N AND K
For the sake of simplicity, the following discussion assumes that
N
is even (the DFT of an odd-length
sequence differs chiefly in that there is no Bin
N/
2, and the range of
k
must be adjusted accordingly).
The DFT is periodic in both
n
and
k
, meaning that if
n
assumes, for example, the range
N
to 2
N
-1,or2
N
to 3
N
- 1, etc., the result will be the same as it would with
n
running from 0 to
N
-1.Ina
similar manner,
k
running from
N
to 2
N
- 1 yields the same result as
k
running from 0 to
N
-1.Or,
using symmetrical
k
-indices, with
k
from -
N/
2 +1to
N/
2,
k
could instead run from
N/
2 +1to3
N/
2,
or from 3
N/
2 +1to5
N/
2, etc.
Stated formally, we have
x
[
n
+
N
]=
x
[
n
]
and
X
[
k
+
N
]=
X
[
k
]
These two relationships are true for all real integers
n
and
k
. This is the result of the periodicity of
the complex exponential, exp
(j
2
πnk/N)
over 2
π
. Thus, we have
exp
(j
[
2
πk(n
+
N)/N
]
)
=
exp
(j
[
2
πnk/N
+
2
πk
]
)
=
exp
(j
[
2
πnk/N
]
)
and likewise
exp
(j
[
2
πn(k
+
N)/N
]
)
=
exp
(j
[
2
πnk/N
+
2
πn
]
)
=
exp
(j
[
2
πnk/N
]
)
• Figure 3.8 illustrates the periodicity of the DFT. The DFT of a signal of length 32 has been
computed for bins -15 to 32. A complete set of DFT bins may be had by using either
k
=0toN
- 1 (standard MathScript method) or
k
=-
N/
2 +1to
N/
2. By computing 48 bins, as was done
here, instead of the usual 32, both arrangements may be seen in Fig. 3.8.
• In plotting a MathScript-generated DFT, a simple command (for example) such as
y = ones(1,32); stem(abs(fft(y)))
will plot the bins with indices from 1 to
N
. In order to plot the bins with indices from 0 to
N
- 1, write
x = 0:1:length(y) - 1; stem(x, abs(fft(y)))
• For the case of
k
from -
N/
2 +1to
N/
2,
X
show conjugate symmetry, i.e., conjugate
symmetry is shown about Bin[0]. For example, at plot (a) of Fig. 3.8, Re
(X
[
[
k
]
and
X
[−
k
]
1
]
)
=Re
(X
[−
1
]
),
showing even symmetry, and at plot (b), Im
(X
[
k
]
)
= -Im
(X
[−
k
]
)
, showing anti-symmetry.
