Digital Signal Processing Reference
In-Depth Information
e
−
jω
5
which can be recognized by inspection as the DTFT of the time domain sequence
1
e
−
jω
0
0
e
−
jω
1
e
−
jω
2
e
−
jω
3
0
e
−
jω
4
+
+
−
+
−
[
1
,
0
,
1
,
−
1
,
0
,
−
1
]
To confirm, we make the following call:
y = conv([1,0,1],[1,0,0,-1])
which produces the same result.
1.6.6 EVEN AND ODD COMPONENTS
If a real sequence
x
is decomposed into its even and odd components, and the DTFT taken of each,
it will be found that the DTFT of the even part is all real and equal to the real part of
DT FT (x
[
n
]
)
,
while the DTFT of the odd part will be found to be imaginary only and equal to the imaginary part of
DT FT (x
[
n
]
)
. Using
X(e
jω
)
,
XE(e
jω
), XO(e
jω
)
as the DTFT's of
x
[
n
]
[
n
]
,
xe
[
n
]
, and
xo
[
n
]
, we would
have
Re
(X(e
jω
))
=
Re
(XE(e
jω
))
=
XE(e
jω
)
and
Im
(X(e
jω
))
Re
(XO(e
jω
))
XO(e
jω
)
=
=
Example 1.7.
Write a script that will demonstrate the above even-odd properties.
We can perform the even-odd decomposition using the script
LVEvenOddSymmZero
presented
earlier, and then use the script
LVxDTFT
in three separate calls to open three separate windows to show
X(e
jω
)
,
XE(e
jω
)
, and
XO(e
jω
)
; the following script also reconstructs the original DTFT as the sum
of its even and odd components, subtracts this from the DTFT of the original signal, and obtains the
RMS error, which should prove to be essentially zero, within the limits of roundoff error.
x = [1:1:9]; [xe,xo,m] = LVEvenOddSymmZero(x,[0:1:8]);
d = LVxDTF T(x,[0:1:8],200,2,2,10);
de = LVxDTF T(xe,m,200,2,2,11);
do = LVxDTF T(xo,m,200,2,2,12);
RMS = sqrt((1/200)*sum((d - (de+do)).ˆ2))
The value of RMS reported after running the above code was [1.0190e-016, -1.6063e-015i], which
is essentially zero, within the limits of roundoff error.
1.6.7
MULTIPLICATION BY A RAMP
DT F T (nx
j
dX(e
jω
)
dω
[
n
]
)
=
is
X(e
jω
)
, then the DTFT of
nx
[
n
]
This property states that if the DTFT of
x
[
n
]
is the derivative
with respect to
ω
of
X(e
jω
)
multiplied by
j
.
