Digital Signal Processing Reference
In-Depth Information
x0 = 1.0000 - 0.0000i
x1 = -4.9127e-017
x2 = 1.0000 + 0.0000i
1.6 A FEW PROPERTIES OF THE DTF T
1.6.1 LINEARITY
The DTFT of a linear combination of two sequences
x
1
[
]
and
x
2
[
]
n
n
is equal to the sum of the individual
responses, i.e.,
DT F T (ax
1
[
n
]+
bx
2
[
n
]
)
=
aDTFT(x
1
[
n
]
)
+
bDTFT(x
2
[
n
]
)
1.6.2
CONJUGATE SYMMETRY FOR REAL
x
[
n
]
, the real part of the DTFT shows even symmetry (
X(e
jω
)
=
X(e
−
jω
)
), and the imaginary
part shows odd symmetry (
X(e
jω
)
=
For real
x
[
n
]
X(e
−
jω
)
).
−
1.6.3 PERIODICITY
The DTFT of a sequence
x
[
]
n
repeats itself every 2
π
:
X(e
j(w
+
2
πn)
)
X(e
jw
)
=
where
n
=0,
2
..
.
To illustrate this principle, consider the following: for a given sequence length, the Nyquist limit is
half the sequence length, and this represents a frequency shift of
π
radians. To shift 2
π
radians therefore
is to shift by a frequency equal to the sequence length. The following m-code, the results of which are
illustrated in Fig. 1.5, verifies this property.
SR = 100; nN = (0:1:SR)/SR;
LVxDTF T_MS([cos(2*pi*25*nN)],0,exp(j*2*pi*SR*nN),200,2,2,1)
You can gain insight by noting that the code
±
1
,
±
nN = (0:1:100)/100;y=exp(j*2*pi*100*nN)
yields
y
=
ones(
1
,
101
)
, which clearly transforms the original sequence into itself, i.e., the new sequence
is the same as the old, and hence the DTFT is the same. In other words, the DTFT of a sequence repeats
itself for every frequency shift of 2
π
radians of the original sequence.
1.6.4 SHIF T OF FREQUENCY
If the signal
x
[
n
]
is multiplied by a complex exponential of frequency
F
0
, the result is that the DTFT of
x
[
n
]
is shifted.
X(e
j(ω
−
ω
0
)
)
To demonstrate this property, we can use the script
LVxDTFT_MS
. We pick the short sequence
[1,0,1] as
x
[
n
]
e
jω
0
n
)
DT FT (x
[
n
]
=
, and specify no sample offset, but a frequency offset of 2
π/
16 radians. We thus make the
call
LVxDTF T_MS([1,0,1],0,exp(j*2*pi*1*(0:1:2)/16),100,2,1,2)
