Digital Signal Processing Reference
In-Depth Information
The following code verifies this property using a random sequence:
N=9;x=randn(1,N); td = sum(abs(x).ˆ2),
fd = (1/N)*sum(abs(fft(x)).ˆ2)
7.
Conjugate Symmetry
The DFT of a real input sequence is
Conjugate-Symmetric
, which means that
Re
(X
[
k
]
)
=
Re
(X
[−
k
]
)
and
[
]
=−
[−
]
Im
(X
k
)
Im
(X
k
)
This property implies that (for a real sequence
x
[
n
]
) it is only necessary to compute the DFT for a limited
number of bins, namely
k
=
0:1:
N/
2
Neven
k
=
0:1:
(N
−
1
)/
2
Nodd
Example 3.6.
Demonstrate conjugate symmetry for several input sequences.
We'll use the input sequence
x
= [16:-1:-15]. Since the length is even, there are two bins that will
not have complex conjugates, namely Bins 0 and 16 in this case, or Bins 0 and
N/
2 generally.The following
code computes and displays the DFT; conjugate symmetry is shown except for the aforementioned Bins
0 and 16.
n=[0:1:31]; x=[16:-1:-15]; y=fft(x); figure
subplot(2,1,1); stem(n,real(y)); subplot(2,1,2); stem(n,imag(y))
Figure 3.6 shows the result of the above code.
If we modify the sequence length to be odd, then there is no Bin
N
/2, so conjugate symmetry is
shown for all bins except Bin 0. The result from running the code below is shown in Fig. 3.7.
n=[0:1:32]; x=[16:-1:-16]; y=fft(x); figure
subplot(2,1,1); stem(n,real(y)); subplot(2,1,2); stem(n,imag(y))
8.
Even/Odd TD-Real/Imaginary DF T Parts
The circular even and odd decompositions of a time domain sequence are defined as
x
[
0
]
n
=
0
x
cE
[
n
]=
(x
[
n
]+
x
[
N
−
n
]
)/
2
1
≤
n
≤
N
−
1
and
0
n
=
0
x
cO
[
n
]=
[
]−
[
−
]
≤
≤
−
(x
n
x
N
n
)/
2
1
n
N
1
In such a case, it is true that
DFT (x
cE
[
n
]
)
=
Re
(X
[
k
]
)
(3.11)
