DSP: Properties of the DTFT
Digital Signal Processing
Properties of the Discrete-Time Fourier Transform
D. Richard Brown III
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DSP: Properties of the DTFT
Symmetry/Antisymmetry Definitions
Definition
A sequence x[n] is conjugate symmetric if x∗ [−n] = x[n].
Definition
A sequence x[n] is conjugate antisymmetric if x∗ [−n] = −x[n].
If x[n] is real and conjugate symmetric, it is an even sequence. If x[n] is real and
conjugate antisymmetric, it is an odd sequence.
Definition
A function f (a) is conjugate symmetric if f ∗ (−a) = f (a).
Definition
A function f (a) is conjugate antisymmetric if f ∗ (−a) = −f (a).
If f (a) is real and conjugate symmetric, it is an even function. If f (a) is real and
conjugate antisymmetric, it is an odd function.
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DSP: Properties of the DTFT
Decompositions
Any sequence x[n] (real or complex) can be expressed as the sum of a conjugate
symmetric and conjugate antisymmetric sequence, i.e.,
x[n] =
xe [n]
| {z }
conjugate symmetric
+
xo [n]
| {z }
conjugate antisymmetric
with
xe [n] =
1
1
(x[n] + x∗ [−n]) and xo [n] = (x[n] − x∗ [−n])
2
2
By linearity, the DTFT can be written as
X(ejω ) = Xe (ejω ) + Xo (ejω )
with
Xe (ejω ) =
1
1
X(ejω ) + X ∗ (e−jω ) and Xo (ejω ) =
X(ejω ) − X ∗ (e−jω )
2
2
It is easy to confirm that Xe (ejω ) is a conjugate symmetric function of ω and
Xo (ejω ) is a conjugate antisymmetric function of ω (just plug in −ω for ω).
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DSP: Properties of the DTFT
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DSP: Properties of the DTFT
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DSP: Properties of the DTFT
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