Contents

Introduction

Earlier equations for the direct and inverse
Discrete Fourier Transform

were got.
Give them once again:

where

is a DFT operator,
and

is an inverse DFT operator,
and

bear the name of DFT twiddle factors.
Here and elsewhere in this section it is considered that

and

index temporal
and spectrum samples respectively.

In this section some DFT properties will be considered.

DFT Linearity

DFT of the signals sum is equal to the DFT sum of these signals.
If

then:

where

and

are the DFT of signals

and

respectively.

Upon multiplying the signal by the constant

,
the signal DFT is also multiplied by the constant:

DFT of Signal with Circular Shifting in Time

Let

be the DFT signal

.
If we shift the pickup signal

cyclically to

samples,
i.e.

then the DFT of the shifted signal is equal to:

Introduce the variable change

,
then

and (4) can be written as follows:

Thus, the circular signal shifting to

samples leads
to the phase spectrum rotating while the amplitude
spectrum does not change.

Equation (5) is correct only for circular shifting
which example is shown in Figure 1.

Figure 1. Example of Circular Signal Shifting

The red color in the upper chart is for the pickup signal

,
in the average one there is

with circular shifting

samples (with advancing), and in the lower chart there is

,
shifted to

samples (with delay). It is obvious that in case
of circular shifting with advancing the first

samples are
transferred from the beginning to the end of the capture.
In case of delay the last

samples are transferred
from the end of the capture to the beginning.

DFT of Signals Circular Convolution

Let the signal

be the result of circular convolution
of signals

and

:

Calculate the DFT signal

:

Change the positions of summing operations:

Upon deriving (8) the circular temporal
shifting property was used.

Thus, the circular convolution DFT of two signals is equal
to the DFT cut product of these signals.
This property allows to use algorithms of
fast Fourier transform to calculate convolutions of signals.

DFT of Signals Circular Convolution

Let the signal

be equal to the cut
product of signals

and

,
i.e.

, at that

and

are the DFT of signals

and

respectively.
Then the signal DFT

is equal to:

Substitute

in (9) in the form of the IDFT of the spectrum

:

Change the positions of summing operations in (10) and get:

Thus, the DFT of the signals cut product represents
the DFT circular convolution of these signals.

Property of the DFT Circular Shifting in Frequency

Let

be the signal DFT

.
Perform the spectrum circular shifting

and consider the IDFT, then:

Thus, the DFT circular shifting in frequency
is carried out by multiplying the signal by
the complex exponent.
It is important to note that after multiplying
by the complex exponent of a real-valued signal,
the resulting signal will be complex,
and its spectrum will stop being symmetric.

DFT Symmetry of a Real-Valued Signal

If the pickup signal is real,
i.e.

is for

,
then for even

it is:

The spectrum sample

has also no imaginary component.

Take into account that

is for any integer

.
In this case:

i.e. the second half of spectrum samples is conjugated
with the first one in a complex way.

The form of the real and imaginary components of
the real signal complex spectrum under even

are presented in Figure 2a.
Just real-valued

and

spectrum components are marked as red.

Figure 2. Real and imaginary components of the real DFT signal

The real and imaginary components of the real signal complex
spectrum under odd

are presented in Figure 2b.
Under odd

only the first DFT spectrum sample
of the real-valued signal is real-valued.
In the general case other spectrum samples are complex.

Spectrum Frequency Inversion of a Real-Valued Signal for Even

Frequency signal spectrum inversion is shown in Figure 3 for even

.

If

is a signal spectrum

,
then the inverse spectrum

is equal to:

Due to the DFT symmetry of the real-valued signal
we can perform the frequency signal spectrum
inversion by shifting spectrum components.

Consider the inverse spectrum

for

:

Figure 3. DFT Frequency Inversion

of the Real-Valued Signal for Even

Likewise

for

is equal to:

Thus, for frequency inversion of the real-valued spectrum
signal it is necessary to multiply every second sample by

according to (16).
At the same time it is important to note that
it is necessary to multiply samples beginning with the second,
i.e. for

,
because indexation of samples begins with

.
If we multiply every second sample beginning
with the first by

, we will get the inverse
spectrum with a negative sign

.

Note that (18) is correct only for even

.

It is shown in Figure 4 that frequency spectrum inversion
corresponds to the circular spectrum shifting in frequency
to

spectrum samples by advancing or delay.

Figure 4. Frequency Spectrum Signal Inversion due to the DFT Shifting in Frequency

Then according to the frequency spectrum shifting property (12)
the signal with the inverse spectrum is equal to:

The zero DFT sample

The zero DFT sample is the sum of signal samples.

Duality Property

We have considered the main DFT properties.
The DFT have one more remarkably property:
the duality property the essence of which is that all DFT
properties are correct either for temporal
or for frequency signal representation.

For example, it is possible to consider the DFT property
of circular convolution which says:
the DFT of signal circular convolution is the DFT
cut product of convolved signals.
At the same time it can be formulated invertedly:
the DFT cut product of signals is DFT
circular convolution of these signals.

It is also possible to reformulate the frequency shifting property.
So, the shifting in time leads to multiplying the spectrum by
the complex exponent while multiplying the signal by the complex
exponent leads to the circular spectrum shifting in the frequency area.

Conclusions

In this section we considered the main DFT properties:
linearity, properties of temporal and frequency shiftings,
convolution spectrum and cut product of signals, and we
analyzed spectrum and signal inversion.
There is also shown the DFT duality allowing to formulate
properties simultaneously for frequency and
for temporary signal representation.

Reference

[1]
Bracewell R.N.
The Fourier Transform and its Applications.
McGraw Hill, Singapor, 2000.

[2]
Oppenheim Alan V. and Schafer Ronald W.
Discrete-Time Signal Processing
Second Edition.
Prentice-Hall, New Jersey, 1999.

[3]
Robert J. Marks II
The Joy of Fourier:
Analysis, Sampling Theory, Systems, Multidimensions, Stochastic
Processes, Random Variables, Signal Recovery, Pocs, Time
Scales, & Applications.
Baylor University, 2006.

[4]
Nussbaumer Henri J.
Fast Fourier Transform and Convolution Algorithms.
Second Corrected and Updated Edition.
Springer-Verlag, 1982.