Dynamic recalculation of spectral samples at each sampling clock. Modified algorithm

Introduction

Formulation of the problem

Recurrence ratio for spectral reading

Filter for calculating the spectrum of a given harmonic

Selecting a set of fixed harmonics from the signal spectrum

Selection of amplitudes of multiple harmonics with nonlinear distortion of the signal

findings

In this article, we will look at the rather specific task of analyzing a change in a given spectral reference over time. Such a problem is relevant when decoding a DTMF signal or when monitoring a given set of multiple harmonics in time, when the use of FFT algorithms is impractical due to the small number of spectral samples.

Let us say a few words about the notation that will be used in this article. The samples of the original discrete signal (in the general case complex) will be denoted by small , - reference number. Spectral signal counts we will denote by capital letter , - number of spectral counting corresponding to frequency , - sampling frequency, - the number of points of the DFT.

Figure 1 shows a three-dimensional picture. On the vertical axis, we have the values ​​of the signal spectrum - number of the spectral reference. Over time, we get more and more new samples and spectrum is changing. So in Figure 1 blue shows the change in the spectrum for three time slices corresponding to the signal sample numbers .

Then for some specific spectral reading we can get a frequency cut (shown in red) from which we can analyze the amplitude and phase of the spectral reading in time. In this article we will deal with the analysis of changes in a given spectral sample in time when new signal samples are received.

For DTMF decoding, the Goertzel algorithm, which allowed us to calculate a given spectral count using the recurrence relation, which we represented as an IIR filter, became widespread. But we said that when using the Goertzel algorithm, we were not interested in all intermediate results of the recurrence formula, therefore we could not get the value of the spectral count more than every counts. We will consider an algorithm that allows for spectrum recalculation “on the fly,” that is, upon receipt of each successive reference. We will see that the resulting algorithm will be a modification of the Goertzel algorithm for recalculating the spectral count at each sampling clock.

Consider the discrete Fourier transform (DFT), which is given by the expression:

To calculate the DFT value for one fixed spectral reference (one harmonic) with the number DFT expression takes the form

Calculations of one fixed harmonic with a number can be done using the structure shown in Figure 2. In essence (2) this is the output of an FIR filter.

Thus, to calculate a single spectral reference, it is required complex multiplications and additions.
Let the input samples enter the buffer length and with each new reference it is necessary to recalculate the value of the fixed harmonic as shown in Fig. 3. In other words, when the next sample arrives, the buffer is shifted and the output of the FIR filter changes.

Figure 3 shows harmonic values. on the previous clock cycle, i.e. when a reference signal arrives :

For the sampling interval must be performed complex multiplications and additions, which is impossible in real time at high sampling rates. However, from Figure 3 it can be noted that the recalculation of the entire spectral reference is extremely inefficient, since the same samples are used as in the previous cycle, except for which is not involved in the calculation and which is not involved in the calculation . Thus, it can be assumed that there is a way to recurrently recalculate the spectral count at the current and previous ticks.

Consider more carefully the calculation . Figure 4 shows that upon receipt of the next reference (shown in green), the buffer is shifted and the countdown (marked in yellow) goes out of the buffer, or it can be said that it goes beyond the impulse response of the FIR filter.

Then the expression for calculating the harmonics at the next cycle can be written:

Note that

In addition, from expression (3), it can be noted that

Then you can finally write down:

Expression (7) is nothing more than a recurrence relation that relates the value of the DFT of one harmonic for the previous and next clock cycle. Recalculation of the previous clock to the next requires only one complex multiplication and two complex additions, which can be implemented in real time. It is very important that the complexity of the recalculation of the spectral reference does not depend on - always one complex multiplication and two additions.

The recurrence relation (7) can be interpreted as the difference equation of the IIR filter:

Thus, we have managed to move from the FIR structure to the recurrence relation of the corresponding IIR filter. The structure of an IIR filter corresponding to the difference equation (8) is shown in Figure 5.

The transfer characteristic of the resulting filter is:

All input, output filter counts, as well as coefficient are complex.

Note that (9) is very similar to the transfer characteristic. IIR - the filter corresponding to the algorithm of Goetzel:

Then the resulting transfer characteristic IIR - a filter that “can recalculate” the previous value of the spectral reference to the current one when the next sample of a discrete signal arrives, can be represented through the transfer characteristic of the IIR - filter of the Goertzel algorithm:

Thus, we obtained a modification of the algorithm of Hörzel in the form of a non-recursive branch , which allows to conduct a dynamic recalculation of the spectral sample when a signal is received in real time.

Let the complex input signal equals:

The spectral counts at the filter output are:

The filter coefficient is:

Then the expression (8) can be written:

Converting (15) you can get:

Then the calculation of the spectral reference can be represented by the structural scheme shown in Figure 6 using only real multiplications and additions.

We will consider an example of the selection of two harmonics with frequencies and at sampling rate . Consider how to choose the right number of points DFT as well as numbers of spectral samples and corresponding frequencies and .

In order for us to distinguish these signals in the spectrum, the number of DFT points should be such that frequency and “Did not merge” into one harmonic, i.e.

Then number spectral count for frequency will be equal to:

Let the sampling rate be , and . Then the number of DFT points should be:

Let be then , but . The location of the harmonics is shown in Figure 7.

As can be seen from Figure 7, the harmonics are located midway between the DFT points, which means their amplitudes will be distorted due to the spreading effect of the spectrum. To eliminate this, you need round up then and according to (18) . Then at practically coincides with the 35th number of the spectral reference, as shown in Figure 8.

With the coincidence of the frequency of the harmonic and the frequency of the spectral reference, we obtain that the buffer length an integer number of repetition periods of the signal falls, which will reduce the spreading effect when calculating the spectral count.
Figure 9 shows the graphs for measuring the amplitude of the harmonics. and when alternating signals with frequencies and sampling rate and . Frequencies and replace each other every second.

From Figure 9 it can be seen that when the frequencies change, a “transient process” is observed whose duration is equal to . This must be considered when processing.

In this section, we will give an example of measuring multiple harmonics using an IIR filter shown in Fig. 6. To isolate amplitudes of multiple harmonics of a non-linearly distorted signal with a frequency .

Let the signal at the output of the nonlinear element is described by the expression:

Figure 10 shows the waveform. at the output of the nonlinear element. The sampling rate is .

At the output of the nonlinear element, we get a set of multiple harmonics separated from each other by .

The buffer length required to distinguish multiple harmonics is equal to:

For better discrimination, you can take a buffer length points. Then the numbers of spectral samples corresponding to the first four harmonics are equal:

All harmonics fall on the DFT points, therefore spreading of the spectrum will not be observed. Figure 11 shows the signal spectrum. , and in figure 12, the amplitudes of the first four harmonics are estimated using the modified Goertzel algorithm.

Figure 12 shows that the first points the modified algorithm completely repeats the algorithm of Goetzel, but after the buffer is completely filled, and the non-recursive branch of the IIR filter begins to work, we obtain a dynamic recalculation of the amplitudes of the harmonics at each sampling clock. This allows us to conduct continuous tracking of the amplitudes of multiple harmonics at the output of the nonlinear element. The measured amplitudes are exactly equal to the amplitudes of the harmonics shown in Figure 11, but we can observe their change over time as shown in the three-dimensional projection (Figure 1).

In this article, we reviewed a modification of the Goertzel algorithm, which allows us to recalculate a given spectral sample at each discretization cycle using the recurrence relation (7). It was shown that a modification of the Goertzel algorithm consists of adding a non-recursive branch with a delay of samples to IIR - filter of Goertzel algorithm and allows to take into account all intermediate results when calculating a given harmonic of the signal spectrum.Also shown are examples of choosing the buffer length when analyzing two spectral components, as well as harmonics of a multiple frequency at the output of the nonlinear element.