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Spectral analysis on a limited time interval. Window functions

Lecture



Content
Introduction
Spectrum of a time-limited signal
DFT of a time-limited signal. Using window smoothing
Window attenuation coefficient
The main frequency characteristics of the spectrum window function
The main properties of the window function and their characteristics
findings

Introduction
The digital spectral analysis is based on the discrete Fourier transform (DFT) apparatus. In this case, the DFT has high-performance fast algorithms (FFT). However, when using the DFT, difficulties often arise due to the finiteness of the treatment interval. In this article we will set a goal to analyze the effects arising from the limitation of the analysis interval. It is assumed that the reader has previously studied the Fourier transform and its properties, and also understands the meaning of the DFT.

Spectrum of a time-limited signal
Let there be a signal   Spectral analysis on a limited time interval.  Window functions which is infinite in time. In the simplest case, we can define this signal as a harmonic oscillation with a frequency   Spectral analysis on a limited time interval.  Window functions . The Fourier transform of this signal will be a delta pulse at the signal frequency, i.e.   Spectral analysis on a limited time interval.  Window functions . The original signal and its spectrum are shown in blue in the figure. In practice, we cannot calculate the spectrum by numerical integration over the entire time axis (of course, except when we can obtain an analytical expression for the signal spectrum, as in the above example), so we fix the time interval   Spectral analysis on a limited time interval.  Window functions on which we will calculate the spectrum of the signal. This way we get the signal   Spectral analysis on a limited time interval.  Window functions that matches the original on the time interval   Spectral analysis on a limited time interval.  Window functions but outside the observation interval we consider   Spectral analysis on a limited time interval.  Window functions . Mathematically,   Spectral analysis on a limited time interval.  Window functions can be represented as a product of the original infinite signal   Spectral analysis on a limited time interval.  Window functions and square pulse   Spectral analysis on a limited time interval.  Window functions duration   Spectral analysis on a limited time interval.  Window functions ,   Spectral analysis on a limited time interval.  Window functions . Signal spectrum   Spectral analysis on a limited time interval.  Window functions , according to the properties of the Fourier transform will be equal to the convolution of the spectra of the original signal and the spectrum   Spectral analysis on a limited time interval.  Window functions square pulse   Spectral analysis on a limited time interval.  Window functions :
  Spectral analysis on a limited time interval.  Window functions (one)
In expression (1), the filtering property of the delta function was used. Signal   Spectral analysis on a limited time interval.  Window functions and its spectrum   Spectral analysis on a limited time interval.  Window functions shown in Figure 1 in red.

  Spectral analysis on a limited time interval.  Window functions
Figure 1: Spectrum of a time-limited signal

Thus, instead of a delta pulse, the spectrum   Spectral analysis on a limited time interval.  Window functions turned into a type function   Spectral analysis on a limited time interval.  Window functions (spectrum of a square wave function   Spectral analysis on a limited time interval.  Window functions ) and the width of the petal depends on the duration of the analysis interval, as is clearly shown in Figure 2.

  Spectral analysis on a limited time interval.  Window functions
Figure 2: Spectrum changes with an increase in the analysis interval

If you increase the analysis interval   Spectral analysis on a limited time interval.  Window functions to infinity, the spectrum will narrow and strive for a delta pulse. Square pulse   Spectral analysis on a limited time interval.  Window functions let's call the window function.

DFT of a time-limited signal. Using window smoothing
Now consider the case of DFT. DFT sets in compliance   Spectral analysis on a limited time interval.  Window functions signal readings   Spectral analysis on a limited time interval.  Window functions   Spectral analysis on a limited time interval.  Window functions spectrum readings taken on one repetition period of the spectrum:   Spectral analysis on a limited time interval.  Window functions Signal samples taken at regular intervals   Spectral analysis on a limited time interval.  Window functions Where   Spectral analysis on a limited time interval.  Window functions - sampling rate (rad / s). Thus, the analysis interval   Spectral analysis on a limited time interval.  Window functions then the spectral readings are taken at intervals   Spectral analysis on a limited time interval.  Window functions The width of the main lobe of the spectrum   Spectral analysis on a limited time interval.  Window functions (see picture 1) equals   Spectral analysis on a limited time interval.  Window functions then we can consider two cases. The first case is the frequency of the signal coincides with   Spectral analysis on a limited time interval.  Window functions frequency spectrum   Spectral analysis on a limited time interval.  Window functions (top graph of figure 3). When sampling, we only get the countdown at the frequency   Spectral analysis on a limited time interval.  Window functions the amplitude of the corresponding amplitude of the signal, the remaining spectral samples will be zero, since the spectral sampling times coincide with the zeros of the spectrum of the window function. The second case is when the frequency   Spectral analysis on a limited time interval.  Window functions does not coincide with any frequency from the grid of spectral samples (lower graph of figure 3). In this case, the spectrum of the signal "blurred." Instead of a single spectral sample, we get a set of samples, since the discretization is no longer done at the zeros of the spectrum of the window function, and all side lobes appear in the spectrum. In addition, the amplitude of the spectral samples also decreases.

  Spectral analysis on a limited time interval.  Window functions
Figure 3: DFT with coincidence and mismatch of the signal frequency and spectrum frequency grid

The coincidence of the frequency with the grid of spectral samples will be in the case if the processing interval fits a whole number of periods of the signal. Otherwise, the spectrum will "smear."
Spectrum spreading is a negative effect that needs to be addressed. Let's show it by example. Let there be two harmonic signals at frequencies   Spectral analysis on a limited time interval.  Window functions and   Spectral analysis on a limited time interval.  Window functions , with the amplitude of the signal at the frequency   Spectral analysis on a limited time interval.  Window functions A lot less signal amplitude at frequency   Spectral analysis on a limited time interval.  Window functions . Limiting the analysis interval will cause the spectra to "smear" and the signal at the frequency   Spectral analysis on a limited time interval.  Window functions will not be noticeable under the side lobe signal with a frequency   Spectral analysis on a limited time interval.  Window functions as shown in Figure 4.

  Spectral analysis on a limited time interval.  Window functions
Figure 4: A small amplitude signal is not noticeable under the side lobe of another signal.

Obviously, in order to detect a weak signal, it is necessary to eliminate the side lobes in the spectrum that arise when we have limited the signal to a rectangular window. So in order to eliminate these petals it is necessary to eliminate them in the spectrum of the window function.   Spectral analysis on a limited time interval.  Window functions , that is, it is necessary to change the window function, namely to make it smoother, as shown in Figure 5.

  Spectral analysis on a limited time interval.  Window functions
Figure 5: Smooth weight function

With a smooth window function, no side lobes are observed in the spectrum (or their level decreases significantly), however, there is an expansion of the main spectrum lobe compared to a rectangular window.   Spectral analysis on a limited time interval.  Window functions . Thus, we seem to have overcome the side lobes, and were able to detect weak signals (see Figure 6), which were previously lost in the side lobes, but paid for this by extending the main lobe.

  Spectral analysis on a limited time interval.  Window functions
Figure 6: With a smooth weight function, weak signals are not lost in the side lobes

It should be noted that the greater the suppression of the side lobes of the spectrum of the window function, the wider the main lobe is obtained. This contradiction led to the development of a large number of window functions with different suppression of side lobes and different widths of the main lobe. The main common windows will be discussed below.

Window attenuation coefficient
We will consider another property of the window function, namely, the attenuation coefficient   Spectral analysis on a limited time interval.  Window functions . To clarify the attenuation coefficient   Spectral analysis on a limited time interval.  Window functions consider the constant component   Spectral analysis on a limited time interval.  Window functions window function on the interval   Spectral analysis on a limited time interval.  Window functions :
  Spectral analysis on a limited time interval.  Window functions (2)
In the case of a rectangular window
  Spectral analysis on a limited time interval.  Window functions (3)
Attenuation coefficient   Spectral analysis on a limited time interval.  Window functions called the ratio of the constant component   Spectral analysis on a limited time interval.  Window functions a given window function, to the constant component of a rectangular window   Spectral analysis on a limited time interval.  Window functions :
  Spectral analysis on a limited time interval.  Window functions (four)
The meaning of the attenuation coefficient is that the amplitudes of all spectral components after multiplying by the window function decrease in   Spectral analysis on a limited time interval.  Window functions times compared to a rectangular window. The attenuation coefficient is expressed in a logarithmic scale:
  Spectral analysis on a limited time interval.  Window functions (five)
In the case of digital spectral analysis there is   Spectral analysis on a limited time interval.  Window functions window function count   Spectral analysis on a limited time interval.  Window functions taken through the gap   Spectral analysis on a limited time interval.  Window functions Then   Spectral analysis on a limited time interval.  Window functions the integral in expression (4) is replaced by the sum:
  Spectral analysis on a limited time interval.  Window functions (6)
In order to take into account the attenuation coefficient after the DFT, each spectral sample should be divided by   Spectral analysis on a limited time interval.  Window functions .

The main frequency characteristics of the spectrum window function
Let us generalize the main frequency characteristics of the spectrum of the window function, allowing to compare different windows with each other. To do this, consider the normalized amplitude-frequency characteristic   Spectral analysis on a limited time interval.  Window functions window function shown in Figure 7.

  Spectral analysis on a limited time interval.  Window functions
Figure 7: Normalized AFC of the window function

Rationing amplitude produced to account for the attenuation coefficient   Spectral analysis on a limited time interval.  Window functions :   Spectral analysis on a limited time interval.  Window functions . Thus, all frequency response will have a maximum equal to one (0 dB) at zero frequency. Since the width of the main lobe depends on the window duration in time (see Figure 2), frequency normalization is introduced:
  Spectral analysis on a limited time interval.  Window functions (7)
Thus, the form of the normalized frequency response of the window function will not change when the window duration is changed. Then you can enter the following normalized parameters:
1. The normalized width of the main lobe of the frequency response to the level of 0.5 (-3 dB)   Spectral analysis on a limited time interval.  Window functions is defined as the normalized band at which   Spectral analysis on a limited time interval.  Window functions .
2. The normalized width of the main lobe of the frequency response at zero   Spectral analysis on a limited time interval.  Window functions . According to Figure 5   Spectral analysis on a limited time interval.  Window functions .
3. Maximum side lobe level   Spectral analysis on a limited time interval.  Window functions .
You may notice that   Spectral analysis on a limited time interval.  Window functions rectangular window is 2. Then you can enter a parameter that shows how many times the normalized width of the main lobe of the frequency response at zero level   Spectral analysis on a limited time interval.  Window functions of a given window is wider than   Spectral analysis on a limited time interval.  Window functions rectangular window. Denote this parameter as   Spectral analysis on a limited time interval.  Window functions . Depending on the parameter   Spectral analysis on a limited time interval.  Window functions windows are divided into high resolution windows   Spectral analysis on a limited time interval.  Window functions and low resolution windows   Spectral analysis on a limited time interval.  Window functions .

The main properties of the window function and their characteristics
Table 1 lists the expressions for some window functions.

Table 1. Expressions for some window functions
Window Name Discrete expression:   Spectral analysis on a limited time interval.  Window functions Note
Rectangular window (rectangle window)   Spectral analysis on a limited time interval.  Window functions A high-resolution window is the minimum width of the main lobe, but the maximum level of side lobes
Sine window   Spectral analysis on a limited time interval.  Window functions High resolution window
Lanczos window, or sinc - window   Spectral analysis on a limited time interval.  Window functions High resolution window
Bartlett window, or triangular window   Spectral analysis on a limited time interval.  Window functions High resolution window
Hanna window   Spectral analysis on a limited time interval.  Window functions High resolution window
Barlett - Hanna window (Bartlett – Hann window)   Spectral analysis on a limited time interval.  Window functions High resolution window
Hamming window   Spectral analysis on a limited time interval.  Window functions High resolution window. The best window when   Spectral analysis on a limited time interval.  Window functions
Blackman window   Spectral analysis on a limited time interval.  Window functions High resolution window.
Blackman – Harris window   Spectral analysis on a limited time interval.  Window functions Low resolution window
Natalla window (Nuttall window)   Spectral analysis on a limited time interval.  Window functions Low resolution window
Blackman – Nuttall window   Spectral analysis on a limited time interval.  Window functions Low resolution window
Flat Top Window   Spectral analysis on a limited time interval.  Window functions Low resolution window
Gaussian window (Gaussian window)   Spectral analysis on a limited time interval.  Window functions Window properties are parameter dependent.   Spectral analysis on a limited time interval.  Window functions

Properties of window functions listed in Table 1 are shown in Table 2.

Table 2. Properties of some window functions
Window Name   Spectral analysis on a limited time interval.  Window functions   Spectral analysis on a limited time interval.  Window functions   Spectral analysis on a limited time interval.  Window functions   Spectral analysis on a limited time interval.  Window functions db   Spectral analysis on a limited time interval.  Window functions db
Rectangular window (rectangle window) 2 0.89 one -13 0
Sine window 3 1.23 1.5 -23 -3.93
Lanczos window, or sinc - window 3.24 1,3 1.62 -26,4 -4.6
Bartlett window, or triangular window four 1.33 2 -26,5 -6
Hanna window four 1.5 2 -31,5 -6
Barlett - Hanna window (Bartlett – Hann window) four 1.45 2 -35,9 -6
Hamming window four 1.33 2 -42 -5.37
Blackman window 6 1.7 3 -58 -7.54
Blackman – Harris window eight 1.97 four -92 -8.91
Natalla window (Nuttall window) eight 1.98 four -93 -9
Blackman – Nuttall window eight 1.94 four -98 -8,8
Flat Top Window ten 3.86 five -69 0
Gaussian window (Gaussian window)   Spectral analysis on a limited time interval.  Window functions eight 1.82 four -65 -8,52
Gaussian window (Gaussian window)   Spectral analysis on a limited time interval.  Window functions 3.4 1.2 1.7 -31,5 -4,48
Gaussian window (Gaussian window)   Spectral analysis on a limited time interval.  Window functions 2.2 0.94 1.1 -15,5 -0.96

You can see the view of the above window functions and their frequency characteristics here.

findings
Thus, the issue of calculating the signal spectrum when observing on a limited time interval was considered. It is shown that the limitation of the analysis time is equivalent to the use of a rectangular window function, the frequency characteristic of which has maximum side lobes. A mechanism for reducing the level of side lobes by smoothing with a window is presented, which in turn worsens the resolution of the spectral analysis due to the expansion of the main lobe. The basic properties of the frequency response of window functions are shown, and expressions are given for the most common windows.

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Digital signal processing

Terms: Digital signal processing