Discrete Fourier transform - Continuous and Discrete Time Signals and Systems - page 548

12.4.1 Periodicity

The M -point DFT of an aperiodic DT sequence with length N ( M ≥ N )is

periodic with period M . In other words,

X [ r ] = X [ r + M ] , (12.22)

for 0 ≤ r ≤ M − 1.

12.4.2 Orthogonality

The column vectors F r of the DFT matrix F , defined in Section 12.2.2, form
the basis vectors of the DFT. These vectors are orthogonal to each other and,
for the M -point DFT, satisfy the following:

M

1 / M for p = q

F p F q = F p ( m , 1)[ F q ( m , 1)] ∗ =

0 for p = q ,

m = 1

where the matrix F p is the transpose of F p and the matrix F q is the complex

conjugate of F q .

12.4.3 Linearity

If x 1 [ k ] and x 2 [ k ] are two DT sequences with the following M -point DFT pairs:

DFT
←−−→ X 1 [ r ] and x 2 [ k ]

DFT
←−−→ X 2 [ r ] , x 1 [ k ]

then the linearity property states that

DFT
←−−→ a 1 X 1 [ r ] + a 2 X 2 [ r ] , a 1 x 1 [ k ] + a 2 x 2 [ k ] (12.23)

for any arbitrary constants a 1 and a 2 , which may be complex-valued.

12.4.4 Hermitian symmetry

The M -point DFT X [ r ] of a real-valued aperiodic sequence x [ k ] is conjugate-
symmetric about r = M / 2. Mathematically, the Hermitian symmetry implies
that

X [ r ] = X ∗ [ M − r ] , (12.24)

where X ∗ [ r ] denotes the complex conjugate of X [ r ].
In terms of the magnitude and phase spectra of the DFT X [ r ], the Hermitian
symmetry property can be expressed as follows:

X [ M − r ] = X [ r ] and < X [ M − r ] =−< X [ r ] , (12.25)

implying that the magnitude spectrum is even and that the phase spectrum is
odd.
The validity of the Hermitian symmetry can be observed in the DFT plotted
for various aperiodic sequences in Examples 12.2-12.11.