Image Processing Reference
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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
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