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168 Chapter 5 THE DISCRETE FOURIER TRANSFORM
3. Conjugation: Similar to the above property we have to introduce the
circular folding in the frequency domain.
DFT [x ∗ (n)] = X ∗ ((−k)) N
(5.30)
4. Symmetry properties for real sequences: Let x(n) be a realvalued
N-point sequence. Then x(n) =x ∗ (n). Using (5.30)
X(k) =X ∗ ((−k)) N
(5.31)
This symmetry is called a circular conjugate symmetry. Itfurther implies
that
Re [X(k)] = Re [X ((−k)) N
] =⇒ Circular-even sequence
Im [X(k)] = − Im [X ((N − k)) N
]=⇒ Circular-odd sequence
|X(k)| = |X ((−k)) N
| =⇒ Circular-even sequence
̸ X(k) =−̸ X ((−k)) N
=⇒ Circular-odd sequence
(5.32)
Comments:
1. Observe the magnitudes and angles of the various DFTs in Examples
5.6 and 5.7. They do satisfy the above circular symmetries. These symmetries
are different than the usual even and odd symmetries. To visualize
this, imagine that the DFT samples are arranged around a circle
so that the indices k =0and k = N overlap; then the samples will
be symmetric with respect to k =0,which justifies the name circular
symmetry.
2. The corresponding symmetry for the DFS coefficients is called the periodic
conjugate symmetry.
3. Since these DFTs have symmetry, one needs to compute X(k) only for
k =0, 1,..., N 2 ;
N even
or for
k =0, 1,..., N − 1 ; N odd
2
This results in about 50% savings in computation as well as in storage.
4. From (5.30)
X(0) = X ∗ ((−0)) N = X ∗ (0)
which means that the DFT coefficient at k =0must be a real number.
But k =0means that the frequency ω k = kω 1 =0,which is the DC
frequency. Hence the DC coefficient for a real-valued x(n) must be a
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