Discrete Fourier Transform (DFT) Properties

Previously, expressions for the direct and inverse discrete Fourier transforms were obtained. We give them again:

In this article we will consider the properties of the DFT.

The spectrum of the sum of the signals is equal to the sum of the spectra of these signals. If a then the spectrum equals:

When a signal is multiplied by a constant, the signal spectrum is also multiplied by a constant:

Let the signal has a spectrum . If you shift the signal cyclically on counts, i.e. , then the spectrum of the shifted signal is equal to:

Introduce variable substitution then and expression (4) can be rewritten:

Thus the cyclic shift of the signal by leads to a rotation of the phase spectrum, and the amplitude spectrum does not change.

Need to make a comment. Expression (5) is valid only for cyclic shift. An example of a cyclic shift is shown in Figure 1.

The source signal is shown in red in the upper graph. on average with a shift countdown (ahead), and in the lower graph shifted by countdown (with delay). It can be seen that with cyclic shift when advancing the first reports are transferred from the beginning to the end of the sample, and when they are late, the last reports are transferred from the end of the sample to the beginning.

Let the signal is the result of cyclical convolution of signals and :

Calculate the spectrum of the signal :

Swap the summation operations:

In deriving expression (8), the time shift property was used. Thus, it can be concluded that the spectrum of cyclical convolution of two signals is equal to the product of the spectra of these signals. This property allows you to use fast DFT algorithms to calculate convolution.

Let the signal equal to the product of signals and , and and - signal spectra and respectively.

Substitute in the expression (9) the expression for the signal in the form of ODPP from the spectrum :

Swap the operations of summation in expression (10) and get:

Thus, the spectrum of the product of signals is a cyclical convolution of the spectra of these signals.

Let the signal has a spectrum . Perform a cyclic shift of the spectrum and consider the IDPF, then:

Thus, we obtained that the shift of the spectrum is carried out by multiplying the signal by the complex exponent. It is important to note that after multiplying by a complex exponent, the signal will be complex, and its spectrum will cease to be symmetrical.

Inversion of the frequency spectrum of the actual signal is shown in Figure 2.

If a - signal spectrum , the inverse spectrum equals:

Figure 3 shows that the spectrum inversion corresponds to a cyclic frequency shift of the spectrum by spectral counts in the direction of advance, or lag.

Then the signal with an inverse spectrum, according to (12) the property about the frequency shift of the spectrum is equal to:

Thus, to invert the spectrum of a signal, it is enough to multiply every second signal reading by minus one. In this case, multiplying by minus one even sample counts corresponds to a cyclic shift of the spectrum to the right by spectral samples, and multiplying odd samples corresponds to a cyclic shift of the spectrum to the left by spectral counts.

We considered the basic properties of the DFT. The DFT has one more remarkable property: the property of duality, which consists in the fact that all the properties of the DFT are valid both for the signal and for the spectrum. For example, we can consider property 3 of the DFT, which reads as follows: the spectrum of cyclical convolution of signals is the product of the spectra. At the same time, this can be stated in the opposite direction: the spectrum of the product of signals is the cyclic convolution of the spectra of these signals (property 4). Similarly, we can reformulate property 5 from property 3: a shift in time leads to a multiplication of the spectrum by a complex exponent, while multiplying a signal by a complex exponent leads to a cyclic shift of the spectrum. Using the property of duality, it can be assumed based on property 6, that multiplying every second spectral sample by minus one will invert the signal according to (13). Indeed, from Figure 3 it follows that the inversion according to (13) will be when the signal or spectrum is shifted by counts. In this case, from the expression (5) the signal spectrum with a cyclic shift on equals:

We obtained that for the inversion of a signal in time, it is necessary to multiply every second sample of its spectrum by minus one, as we assumed using the DFT duality property.

Thus, we considered the basic properties of the DFT: linearity, properties of the time and frequency shifts, the convolution spectrum and the product of the signals, and analyzed the spectrum and signal inversion. The duality of the DFT was also shown, which makes it possible to formulate properties simultaneously for both the spectrum and the signal.