Hardly noticeable - малоприметный -

cumbersome - громоздкий, объёмный

gist - суть, сущность,             score - партитура

DFT is often used for the observation and analysis of the signal.

It is usually the most interesting are the only amplitudes Ck of individual harmonic functions, not their phase. In this case, the spectrum is displayed as a graph of amplitude versus frequency.

Often the frequency scale is calibrated in decibels. The decibelsmeasure not the amplitude, and their relationship. For example, a difference of 20 dB means the ratio of the amplitudes of 10, a difference of 40 dB is the ratio of the amplitudes of 100.

The formula to calculate the difference in decibels is: . Here a1 and a2 - amplitudes, which are compared. Axis of frequency is often graded on a logarithmic scale also.

Before calculating the spectrum of the signal we must select a segment of signal, on which will be calculated spectrum. Length of the segment must be a power of two (for FFT). Otherwise the signal is to be completed with zeros to required length. After that, to selected section of the signal is applied the FFT. Crest factor considered by the formula:

С_к = .

If we will calculate the spectrum in this way, then we can obtain the following undesirable effect. at the expansion of the Fourier series, we believe that the function is periodic, with period equal to the size of the FFT. The spectrum a such function is calculated namely (rather than the one from which we have taken a segment). In this case, on the borders of periods the function will certainly have breaks (the original function is not periodic). But gaps in the function strongly reflected in its spectrum, distorting it. To eliminate this effect apply the so-called weighting window. They smoothly nullify function near the edges of the analyzed area. Selected for the analysis an area of signal is multiplied by the weight window that eliminates gaps function. . Virtual "looping" occurs at the DFT, since the DFT algorithm assumes that the function is periodic. There are many weight windows, which are named after their creators. They all have a similar shape and largely eliminate the distortion of the spectrum. We give the formulas of two good windows: Hamming window and Blackman window

(fig. 7):

Here the window applied to the signal of the index from 0 to N. Hamming window is most often used. Blackman has a stronger effect on the elimination of distortions, but it has its drawbacks. An important property of spectral analysis is that there is no one single correct of signal spectrum. The spectrum can be calculated with the use of different sizes FFT and different weight windows. For each application it's necessary to use their preferred methods. From the choice of the size of the FFT depends the spectral decomposition in frequency and time. If you choose a long section of the signal for spectral decomposition, we get a good frequency resolution, but bad timing (because the spectrum will reflect the behavior of the signal averaged over the entire section of taking the FFT). If for the decomposition spectrum choose a short section of the signal, we obtain a more precise localization of the time, but poor frequency resolution (because in the Fourier transform is too little basic frequencies). This is the fundamental principle uncertainty relation when calculating the spectrum: it is impossible to get a good decomposition spectrum on frequency and time: these resolutions are inversely proportional. .Another important property of spectral analysis is that the decomposition of the spectrum we find not the sinusoidal components, which constituted of the original signal. We only can find: with how of amplitudes you need to take multiple frequencies to get the original signal.

In other words, the expansion is carried out not on the "frequencies of the original signal," but on the "base frequencies of FFT algorithm's." But usually (especially with weight windows) this hardly noticeable in the the graph of spectrum, i.e. the graph of the spectrum is adequately reflects the frequencies of the original signal.

Fast convolution

Convolution - one of the most important processes in the digital signal processing. For this reason, it is important to be able to effectively calculate it. A direct calculation of convolution requires N * M multiplications, where N - the length of the original signal, and M - the length of the convolution kernel. Often, the length of the convolution kernel consists in the thousands of points, and the number of multiplications is huge. However, there is an algorithm which calculates the convolution much faster. This algorithm is based on the following important theorem, which we present in a non-rigorous formulation:

Convolution theorem: convolution in the time domain is equivalent to multiplication in the frequency domain, multiplication in the time domain is equivalent to convolution in the frequency domain. This means that for the convolution of two signals: them can converted into the frequency domain, multiply their spectra and put them back into the time domain. This operation looks a cumbersome. However, with the FFT algorithms for rapidly calculate of the Fourier transform, calculation of convolution in the frequency domain has been widely used.

At considerable length of the convolution kernel, this approach allows hundreds of times to reduce the time convolution.

We briefly describe the algorithm for fast one-time calculation of convolution.

First, the original signal of length N, and the convolution kernel of length M is padded to the length L (L - power of two), and so that L> N + M-1. Then it are calculated the DFT of these two signals. Then the signal spectrums we will to multiply as consisting of complex numbers, i.e. we will form a new spectrum of the coefficients  and , obtained by the formula

where k=0, , , old1 –signal; old2 - convolution kernel.

Then, from the spectrum obtained using the inverse DFT calculated signal consisting of L points. This signal contains the result of the convolution of the N + M-1 points, zero-padded to the L points.

Often we have the need to calculate the convolution of the very long signal, which can not fit in memory with a relatively short convolution kernel. In such cases, so-called sectional assembly is employed. The gist of it is that the long signal is broken up into shorter parts and each of these parts is convolved with the kernel separately. Then the obtained parts are combined to produce the final result.

Filtering.

The effect of multiplying the spectrums of signals at the convolution is called filtering. When the spectra are multiplied as complex numbers, then is going on the multiplication of the amplitudes of harmonics of the original signal and the convolution kernel. Thus, we are able to change the spectrum of the signal. This is a very useful operation.

For example, in the sound recording a change of the signal allows to clear record of noise, distortion compensate various audio devices, change the sounds of tools to focus the audience's attention on the individual scores. In image processing filtering lets you apply different effects to the picture: blur, the emphasizing of borders, embossing, and many others. .In other areas, filtering is often used to separate different signals , cleaning the signal from the noise. Filtering also is a component of many other, more complex processes.

When filtering, the convolution kernel is often called a filter.

the filter also is called all the devices that perform the filtering process.

Often the filter is called as all the device that performs the filtering process. Length (size) of the filter - is the length of the convolution kernel.In general, the filter changes in the signal spectrum and amplitude of the harmonics and their phase. However, the filters can be designed so that they did not change the phase. Such filters are called filters with linear phase. This means that if they change the phase of the signal, then make it so that all the harmonics of the signal shifted in time by the same amount. Thus, filters with linear phase does not distort the phase of the signal, but only shift the entire signal in time. Convolution kernel of such a filter is strictly symmetrical about its center point (although there are types of filters with linear phase, where the kernel is antisymmetric).The main property of any filter - is its frequency (frequency response) and phase characteristics. They show what effect the filter has on the amplitude and phase of the various harmonics processed signal. If the filter has a linear phase, it is considered only the frequency response of the filter. Usually the frequency response is depicted as a graph for <> on the frequency dependence of the amplitude (in dB). For example, if the filter passes all signals in the band 0 ... 10 kHz with no changes, and all the signals in the band above 10 kHz inhibits factor of 2 (6 dB), the frequency response will be: (Fig. 9). Frequency response of 0 dB indicates that the data frequency filter passes unchanged. Those frequencies, whose amplitude is attenuated filter 2 times, must have an amplitude of 6 dB less. Therefore, their times, must have an amplitude of 6 dB less. Therefore, their amplitude is -6 dB. If the filter is amplified some frequencies,