Image Processing Reference
In-Depth Information
considered. The forward equation multiplies the periodic time domain number series from
x
(0) to
x
(
N
-1) by a sinusoid and sums the results over the complete time-period.
The frequency domain signal
X
(
k
) is an
N
-point complex periodic signal in a single
frequency interval, such as [0÷0.5ω
s
], [-0.5ω
s
÷0], [-0.25ω
s
÷0.25ω
s
], etc. (the sampling
frequency ω
s
is often used in its normalized value). The inverse equation employs all the
N
points in the frequency domain to calculate a particular discrete value of the time domain
signal. It is clear that
complex
DFT works with finite-length data.
Both the time domain
x
(
n
) and the frequency domain
X
(
k
) signals are complex numbers, i.e.
complex
DFT also recognizes negative time and negative frequencies. Complex mathematics
accommodates these concepts, although imaginary time and frequency have only a
theoretical existence so far.
Complex
DFT is a symmetrical and mathematically
comprehensive processing technology because it doesn't discriminate between negative and
positive frequencies.
Fig. 6 shows how the forward
complex
DFT algorithm works in the case of a complex time-
domain signal.
x
R
(
n
) is a real time domain signal whose frequency spectrum has an even real
part and an odd imaginary part; conversely, the frequency spectrum of the imaginary part
of the time domain signal
x
I
(
n
) has an odd real part and an even imaginary part. However,
as can be seen in Fig. 6, the actual frequency spectrum is the sum of the two individually-
calculated spectra. In reality, these two time domain signals are processed simultaneously,
which is the whole point of the Fast Fourier Transform (FFT) algorithm.
Time
Domain
x(n)= x
R
(
n
) +
j
x
I
(
n
)
x
R
(
n
)
x
I
(
n
)
Real time signal Imaginary time signal
Comp
lex
DFT
Frequency
Domain
X
(
k
)
=X
R
(
k
)+
X
I
(
k
)
Real Frequency Spectrum
(even)
Real Frequency Spectrum
(odd)
Imaginary Frequency Spectrum
(odd)
Imaginary Frequency Spectrum
(even)
Fig. 6. Forward
complex
DFT algorithm
The imaginary part of the time-domain complex signal can be omitted and the time domain
then becomes totally real, as is assumed in the numerical example shown in Fig. 7. A real
sinusoidal signal with amplitude
M,
represented in a complex form, contains a positive ω
0
and a negative frequency -ω
0
. The complex spectrum
X
(
k
) describes the signal in the