In the previous section
we got equations for the direct discrete Fourier transform
(DFT) and inverse DFT (IDFT):

(1)

Calculating the DFT and IDFT is carried out on the basis
of temporal and frequency
sample indexes without frequency sampling.

Thus, it is possible to use the DFT and IDFT in case of
any sampling frequency without changing the computing procedure.

In this section we will consider how to assign the DFT
indexes to frequency values ,
stated in Hertz, or to values of the angular
frequency rad/s.

Indexing of DFT spectrum samples

Upon considering the DFT we said that the spectrum
of the discrete signal ,
, is a periodic function with
the period rad/s where
is the samplerate (Hz) of a pickup signal .

Respectively the repetition period of the spectrum
of the discrete signal , ,
for the frequency , stated in Hertz, is equal
to the sampling frequency Hz.

The DFT is got by sampling of the periodic function
in one repetition period with the step :

(2)

Thus, the -th spectrum sample corresponds to the frequency

(3)

or

(4)

Example 1. Under the sampling frequency ,
under , the first spectrum sample corresponds to the
frequency .

Example 2. Under the sampling frequency ,
under , the spectrum sample with the number ,
corresponds to the frequency .

DFT shifting

Let the input signal , ,
be a complex one and consist of one complex exponent
with the frequency Hz:

(5)

Set the sampling frequency which is equal to Hz
and take pickup signal samples.

Calculate the DFT , , of this signal and
get the amplitude spectrum as it is shown in Figure 1.

Figure 1. Amplitude signal spectrum

As it is possible to note in Figure 1, there is only one component
with the index that corresponds to the frequency
according to (4) in the signal spectrum

(6)

that can seem strange because we set the pickup signal frequency Hz.

However there is nothing special in it if we remember that the spectrum
of our discrete signal is the periodic function with the period
Hz, i.e. for our discrete signal the spectrum
consists of the infinite number of harmonicas with frequencies
, ,
as it is shown in Figure 2.

Figure 2. Periodic spectrum of the signal

The figure shows that when the spectrum sampling in one
repetition period is from 0 to Hz (in Figure 2
it is set as the DFT period) the periodic harmonica with 100 Hz
frequency occurs in the sample capture.
At the same time the spectrum in the frequency range from
Hz to Hz periodically
repeats the spectrum in the frequency range from
Hz to Hz.

Thus, we can perform the DFT shifting for the signal spectrum analysis
in the range of frequencies from to
Hz.

Letâ€™s state an important remark. The frequency component
corresponding to the frequency Hz
in view of the discrete signal spectrum periodicity also corresponds
to the frequency Hz.
Upon shifting we will refer this component to the
frequency Hz.

DFT shifting for even

Under even the spectrum sample corresponds
to the frequency Hz in accordance with (4).
As we noted above, the same sample corresponds to the frequency
Hz.
hen it is possible to write the spectrum
after shifting for even as follows:

(7)

The -th spectrum sample after shifting
corresponds to the frequency

(8)

or

(9)

Thus, the sample corresponds to the frequency
Hz, the sample
corresponds to the frequency
Hz and the sample
corresponds to the frequency
Hz.

DFT shifting for even is shown in Figure 3.

Figure 3. DFT shifting for even

DFT shifting for odd

Under odd the spectrum sample corresponds
to the frequency Hz
in accordance with (4), and the spectrum sample
corresponds to the frequency Hz.

The spectrum after shifting for odd :

(10)

The spectrum sample after shifting
corresponds to the frequency

(11)

or

(12)

After DFT shifting under odd , the spectrum sample
corresponds to the frequency
Hz
in accordance with (12), the spectrum sample
corresponds to the frequency
Hz and the last sample
corresponds to the frequency
Hz.

DFT shifting for odd is shown in Figure 4.

Figure 4. DFT shifting for odd

DFT shifting example

Perform DFT shifting for correct presenting negative
frequencies for the example given above (Figure 1).

The number of samples in the given example is ,
therefore we can use (7) for getting .
Then after shifting spectrum samples will correspond to frequencies (9).

In Figure 4 there is given the amplitude spectrum
after shifting and frequency values
(Hz).
Spectrum samples correspond to this frequency .

Figure 5. Amplitude spectrum
after shifting and corresponding frequency values

As it is possible to see in Figure 5, after shifting of spectrum samples
the component corresponds to the frequency
Hz that we set for the pickup signal.

Conclusions

In this section we considered the question of indexing and shifting of spectrum samples in the DFT output.

The equations for DFT shifting for even and odd for correct presenting negative frequencies after the DFT were given.

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

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

[3]Robert J. Marks IIThe 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.

Appendix

# -*- coding: utf-8 -*-
"""
Images 1 and 4 data calculation script.
DFT frequency indexation and shifting
@author: Sergey Bakhurin
www.dsplib.org
"""
import numpy as np
# DFT size
N = 30
# Signal frequency Hz
f0 = -20.0
# Sample-rate Hz
Fs = 120.0
n = np.linspace(0, N, N, endpoint = False, dtype = 'float64')
f = n / N * Fs
# Input signal
s = np.exp(2j * np.pi * n * f0 / Fs)
# DFT
S = np.abs(np.fft.fft(s))
# DFT shifting
S_sh = np.fft.fftshift( S )
f_sh = f - Fs/2.0
# Save data to files for plotting
np.savetxt('dat/dft_freq_fig1.txt', np.transpose([n, S]))
np.savetxt('dat/dft_freq_fig4.txt', np.transpose([n, S_sh]))
# Data plotting.
# Please uncomment this block if you want to plot data.
"""
import matplotlib.pyplot as plt
plt.figure(1)
plt.stem(f, np.abs(S))
plt.xlabel('f, Hz')
plt.ylabel('|S(f)|')
plt.figure(2)
plt.stem(f_sh, np.abs(S_sh))
plt.xlabel('fsh, Hz')
plt.ylabel('|Ssh(fsh)|')
plt.show()
"""