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Discrete Fourier Transform (DFT) Properties

Lecture



Content
  • Introduction
  • Property 1. Linearity
  • Property 2. The time shift
  • Property 3. DFT cyclic convolution of signals
  • Property 4. Spectrum of the product of two signals
  • Property 5. Frequency Shift Property
  • Property 6. Inversion of the spectrum of a valid signal
  • Duality property
  • findings
Introduction
Previously, expressions for the direct and inverse discrete Fourier transforms were obtained. We give them again:
Discrete Fourier Transform (DFT) Properties (one)
Where Discrete Fourier Transform (DFT) Properties - DFT operator, and Discrete Fourier Transform (DFT) Properties - operator of the inverse DFT. Everywhere later in this article it is believed that Discrete Fourier Transform (DFT) Properties
In this article we will consider the properties of the DFT.
Property 1. Linearity
The spectrum of the sum of the signals is equal to the sum of the spectra of these signals. If a Discrete Fourier Transform (DFT) Properties then the spectrum Discrete Fourier Transform (DFT) Properties equals:
Discrete Fourier Transform (DFT) Properties (2)
Where Discrete Fourier Transform (DFT) Properties and Discrete Fourier Transform (DFT) Properties - signal spectra Discrete Fourier Transform (DFT) Properties and Discrete Fourier Transform (DFT) Properties respectively.
When a signal is multiplied by a constant, the signal spectrum is also multiplied by a constant:
Discrete Fourier Transform (DFT) Properties (3)
Property 2. The time shift
Let the signal Discrete Fourier Transform (DFT) Properties has a spectrum Discrete Fourier Transform (DFT) Properties . If you shift the signal Discrete Fourier Transform (DFT) Properties cyclically on Discrete Fourier Transform (DFT) Properties counts, i.e. Discrete Fourier Transform (DFT) Properties , then the spectrum of the shifted signal is equal to:
Discrete Fourier Transform (DFT) Properties (four)
Introduce variable substitution Discrete Fourier Transform (DFT) Properties then Discrete Fourier Transform (DFT) Properties and expression (4) can be rewritten:
Discrete Fourier Transform (DFT) Properties (five)
Thus the cyclic shift of the signal by Discrete Fourier Transform (DFT) Properties 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.
Discrete Fourier Transform (DFT) Properties
Figure 1: Example of a cyclic shift signal
The source signal is shown in red in the upper graph. Discrete Fourier Transform (DFT) Properties on average Discrete Fourier Transform (DFT) Properties with a shift Discrete Fourier Transform (DFT) Properties countdown (ahead), and in the lower graph Discrete Fourier Transform (DFT) Properties shifted by Discrete Fourier Transform (DFT) Properties countdown (with delay). It can be seen that with cyclic shift when advancing the first Discrete Fourier Transform (DFT) Properties reports are transferred from the beginning to the end of the sample, and when they are late, the last Discrete Fourier Transform (DFT) Properties reports are transferred from the end of the sample to the beginning.
Property 3. DFT cyclic convolution of signals
Let the signal Discrete Fourier Transform (DFT) Properties is the result of cyclical convolution of signals Discrete Fourier Transform (DFT) Properties and Discrete Fourier Transform (DFT) Properties :
Discrete Fourier Transform (DFT) Properties (6)
Calculate the spectrum of the signal Discrete Fourier Transform (DFT) Properties :
Discrete Fourier Transform (DFT) Properties (7)
Swap the summation operations:
Discrete Fourier Transform (DFT) Properties (eight)
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.
Property 4. Spectrum of the product of two signals
Let the signal Discrete Fourier Transform (DFT) Properties equal to the product of signals Discrete Fourier Transform (DFT) Properties and Discrete Fourier Transform (DFT) Properties , and Discrete Fourier Transform (DFT) Properties and Discrete Fourier Transform (DFT) Properties - signal spectra Discrete Fourier Transform (DFT) Properties and Discrete Fourier Transform (DFT) Properties respectively.
Discrete Fourier Transform (DFT) Properties (9)
Substitute in the expression (9) the expression for the signal Discrete Fourier Transform (DFT) Properties in the form of ODPP from the spectrum Discrete Fourier Transform (DFT) Properties :
Discrete Fourier Transform (DFT) Properties (ten)
Swap the operations of summation in expression (10) and get:
Discrete Fourier Transform (DFT) Properties (eleven)
Thus, the spectrum of the product of signals is a cyclical convolution of the spectra of these signals.
Property 5. Frequency Shift Property
Let the signal Discrete Fourier Transform (DFT) Properties has a spectrum Discrete Fourier Transform (DFT) Properties . Perform a cyclic shift of the spectrum Discrete Fourier Transform (DFT) Properties and consider the IDPF, then:
Discrete Fourier Transform (DFT) Properties (12)
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.
Property 6. Inversion of the spectrum of a valid signal
Inversion of the frequency spectrum of the actual signal is shown in Figure 2.
Discrete Fourier Transform (DFT) Properties
Figure 2: Inversion of the signal spectrum
If a Discrete Fourier Transform (DFT) Properties - signal spectrum Discrete Fourier Transform (DFT) Properties , the inverse spectrum Discrete Fourier Transform (DFT) Properties equals:
Discrete Fourier Transform (DFT) Properties (13)
Figure 3 shows that the spectrum inversion corresponds to a cyclic frequency shift of the spectrum by Discrete Fourier Transform (DFT) Properties spectral counts in the direction of advance, or lag.
Discrete Fourier Transform (DFT) Properties
Figure 3: Inversion of the spectrum of the actual signal due to frequency shift
Then the signal with an inverse spectrum, according to (12) the property about the frequency shift of the spectrum is equal to:
Discrete Fourier Transform (DFT) Properties (14)
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 Discrete Fourier Transform (DFT) Properties spectral samples, and multiplying odd samples corresponds to a cyclic shift of the spectrum to the left by Discrete Fourier Transform (DFT) Properties spectral counts.
Duality property
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 Discrete Fourier Transform (DFT) Properties counts. In this case, from the expression (5) the signal spectrum with a cyclic shift on Discrete Fourier Transform (DFT) Properties equals:
Discrete Fourier Transform (DFT) Properties (15)
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.
findings
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.

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Digital signal processing

Terms: Digital signal processing