Digital Signal Processing Reference
In-Depth Information
If the DFT of a sequence
x
[
n
]
is
X
[
k
]
, the DFT of a circularly folded version of
x
[
n
]
is a circularly
folded version of the DFT of the sequence, i.e.,
DFT (x
[
(
−
n)
]
N
)
=
X
[
(
−
k)
]
N
The following code demonstrates this property.
n = 0:1:7; x = [0:1:7]; xret = x(mod(-n,8)+1);
subplot(3,2,1);stem(n,x);y=fft(x); yret = fft(xret);
subplot(3,2,2);stem(n,xret); subplot(3,2,3); stem(n,real(y));
subplot(3,2,4);stem(n,real(yret));subplot(3,2,5); stem(n,imag(y));
subplot(3,2,6); stem(n,imag(yret))
3.
Shift (circular) in Time Domain
Since the DFT is periodic in
n
, shifting the sequence some number of samples
m
to the right can
be equivalently achieved by a circular shift, and
X(k)e
−
j
2
πmk/N
The following code verifies this for a short sequence. The DFT
prxshfft
is constructed according
to the formula from
xf t
, the DFT of the original sequence
x
.
x=[1 2 3 4]; xsh1 = [4 1 2 3]; xft = fft(x), xshfft = fft(xsh1),
k=0:1:length(x)-1; prxshfft = xft.*(exp(-j*2*pi*1*k/4))
4.
Circular Convolution of Two Time Domain Sequences
If the circular convolution of two sequences
x
1
[
DFT (x
[
n
−
m
]
N
)
=
n
]
and
x
2
[
n
]
is defined as
N
−
1
x
1
[
n
]
x
2
[
n
]=
x
1
[
m
]
x
2
[
(n
−
m)
]
N
m
=
0
≤
≤
−
for 0
n
N
1, then
]
This property and its usefulness to perform ordinary (or linear) convolution will be discussed in
detail later in the chapter.
5.
Multiplication of Time Domain Sequences
The DFT of the product of two time domain sequences is 1
/N
times the circular convolution of
the DFTs of each:
DFT (x
1
[
n
]
x
2
[
n
]
)
=
X
1
[
k
]
X
2
[
k
1
N
(X
1
[
DFT (x
1
[
n
]
x
2
[
n
]
)
=
k
]
X
2
[
k
]
)
6.
Parseval's Relation
The energy of a sequence is the sum of the squares of the absolute values of the samples, and this
has an equivalent computation using DFT coefficients:
N
−
1
N
−
1
1
N
2
2
0
|
x
[
n
]|
=
0
|
X
[
k
]|
n
=
k
=
