Spectral analysis on a limited time interval. Window functions

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 which is infinite in time. In the simplest case, we can define this signal as a harmonic oscillation with a frequency . The Fourier transform of this signal will be a delta pulse at the signal frequency, i.e. . 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 on which we will calculate the spectrum of the signal. This way we get the signal that matches the original on the time interval but outside the observation interval we consider . Mathematically, can be represented as a product of the original infinite signal and square pulse duration , . Signal spectrum , according to the properties of the Fourier transform will be equal to the convolution of the spectra of the original signal and the spectrum square pulse :

In expression (1), the filtering property of the delta function was used. Signal and its spectrum shown in Figure 1 in red.

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

If you increase the analysis interval to infinity, the spectrum will narrow and strive for a delta pulse. Square pulse 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 signal readings spectrum readings taken on one repetition period of the spectrum: Signal samples taken at regular intervals Where - sampling rate (rad / s). Thus, the analysis interval then the spectral readings are taken at intervals The width of the main lobe of the spectrum (see picture 1) equals then we can consider two cases. The first case is the frequency of the signal coincides with frequency spectrum (top graph of figure 3). When sampling, we only get the countdown at the frequency 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 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.

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 and , with the amplitude of the signal at the frequency A lot less signal amplitude at frequency . Limiting the analysis interval will cause the spectra to "smear" and the signal at the frequency will not be noticeable under the side lobe signal with a frequency as shown in Figure 4.

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. , that is, it is necessary to change the window function, namely to make it smoother, as shown in Figure 5.

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. . 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.

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 . To clarify the attenuation coefficient consider the constant component window function on the interval :

In the case of a rectangular window

Attenuation coefficient called the ratio of the constant component a given window function, to the constant component of a rectangular window :

The meaning of the attenuation coefficient is that the amplitudes of all spectral components after multiplying by the window function decrease in times compared to a rectangular window. The attenuation coefficient is expressed in a logarithmic scale:

In the case of digital spectral analysis there is window function count taken through the gap Then the integral in expression (4) is replaced by the sum:

In order to take into account the attenuation coefficient after the DFT, each spectral sample should be divided by .

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 window function shown in Figure 7.

Rationing amplitude produced to account for the attenuation coefficient : . 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:

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) is defined as the normalized band at which .

2. The normalized width of the main lobe of the frequency response at zero . According to Figure 5 .

3. Maximum side lobe level .

You may notice that 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 of a given window is wider than rectangular window. Denote this parameter as . Depending on the parameter windows are divided into high resolution windows and low resolution windows .

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

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

Table 2. Properties of some window functions

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.