DE102019207230B3 - Digital oscillator - Google Patents

Abstract

Digital oscillator, where the digital oscillator operates with integer values, with recursive processing being used, with a representation of a phase angle being used as the input variable, with a calculation of the oscillator value based on a first phase angle range to reduce rounding errors and error propagation depending on the phase angle in a first phase angle range recursive sine processing is determined, and in a second phase angle range, which is different from the first phase angle range, a calculation of the oscillator value is determined on the basis of recursive cosine processing.

Description

The invention relates to a digital oscillator.

background

The increasing digitalization goes hand in hand with the provision of processing units on different levels. It should be noted, however, that for reasons of cost / installation space, the resources of these computing units are small and inexpensive. This goes hand in hand with low performance and, as a rule, very limited resources. Another reason for resource-limited devices, especially in the case of portable, often battery-operated devices, is that the energy consumption should be low in order to generate little waste heat on the one hand, but also to be able to provide the longest possible running times.

The result of this is that more complex arithmetic tasks, which may also be more memory-intensive and / or require special mathematical operations, are generally not provided by these resource-limited arithmetic units.

Examples are e.g. certain transformations, e.g. Fourier transforms, wavelet transforms, cryptographic operations.

However, if such processing is required, the result is required promptly. I.e. forwarding to a computing unit that is external from the sensor point of view, e.g. by means of remote data transmission, is unattractive in terms of latency or excluded in terms of real-time processing. This is e.g. even if initially large amounts of data, such as continuous signal data must be sent to external processing in order to determine certain frequency components (e.g. using Fourier transformation). I.e. the latency caused by the provision of the raw data alone could destroy meaningful processing.

The so-called Fast Fourier Transformation (FFT for short) is a representative of such a more complex computing task, which is known to be memory-hungry. Even so-called smartphones, which have significant computing power and memory, reach their limits when it comes to tasks that require an FFT. Two factors play an important role here.

On the one hand, sine (cosine) shaped waves are required that have to be generated. On the other hand, the values generated in this way must be applied to a random, continuous input signal as part of an FFT.

FFT, in particular, is computationally intensive and requires more specialized mathematical operations that are not available on all platforms. Typically, microcontrollers do not have the desired / required mathematical operations. Examples of such mathematical operations are floating point operations or the associated software libraries.

In the prior art, it is known to generate sinusoidal signals or their values in digital systems in such a way that first the phase position is generated for a given (time) value and is then scaled to a desired amplitude. This method does not require any recursion as it is not based on a previous value. Recursive schemes generate the desired value of the sinusoidal signal directly, but based on one or more previous values. An example of this is known from the article “Improving performance of digital sinusoidal oscillators by means of error feedback circuits”, published in. IEEE transactions on circuits and systems 33, 4 (1986), 373-380.

Furthermore, from the article “Recursive Discrete-Time Sinusoidal Oscillators” published in: IEEE Signal Processing Magazine, May 2003. Pages 103-111, a digital oscillator is known which uses recursive cosine processing. Automatic gain control is used to correct errors.

In the non-recursive case, the computational effort can be reduced by lookup tables and / or function approximations (Chebychev, Taylor series ...), which, however, are memory-intensive. The recursive methods are less power and memory-hungry, but usually lead to increasing errors and restricted waveforms. An example of this is “A simple recursive digital sinusoidal oscillator with uniform frequency spacing” by the author MM AI-Ibrahim published in Circuits and Systems, 2001. ISCAS 2001. The 2001 IEEE International Symposium on, Vol.

To process an FFT, it is necessary to generate and save sine values. Specialized software solutions based on the use of floating point libraries have been developed for many hardware platforms.

A solution that has general use, e.g. on resource-limited platforms, but is not known.

task

However, it would be desirable to be able to make digital oscillators available on resource-limited platforms under the given boundary conditions.

Summary of the invention

The object is achieved by a digital oscillator according to claim 1. Further advantageous configurations are, in particular, the subject matter of the dependent claims, the figures and the description.

Figure list

• 1 eine Darstellung einer Sinus-Schwingung mit möglichen Unterteilungen gemäß Ausführungsformen der Erfindung,
• 2 eine Darstellung einer Cosinus-Schwingung mit möglichen Unterteilungen gemäß Ausführungsformen der Erfindung, und
• 3 eine Darstellung von Blöcken entsprechend einer möglichen Unterteilungen gemäß Ausführungsformen der Erfindung gemäß 1 und/oder 2.

The invention is explained in more detail below with reference to the figures. In this shows:

• 1 a representation of a sinusoidal oscillation with possible subdivisions according to embodiments of the invention,
• 2 a representation of a cosine oscillation with possible subdivisions according to embodiments of the invention, and
• 3 a representation of blocks according to a possible subdivision according to embodiments of the invention 1 and / or 2.

Detailed description of the invention

The invention is illustrated in more detail below with reference to the figures. It should be noted that different aspects are described, which can be used individually or in combination. I.e. any aspect can be used with different embodiments of the invention unless explicitly shown as a pure alternative.

Furthermore, for the sake of simplicity, reference will generally only be made to one entity in the following. Unless explicitly noted, the invention can also have several of the entities concerned. In this respect, the use of the words “a”, “an” and “an” is only to be understood as an indication that at least one entity is used in a simple embodiment.

As far as methods are described below, the individual steps of a method can be arranged and / or combined in any order, unless the context explicitly results in something different. Unless expressly stated otherwise, the processes can also be combined with one another.

Figures with numerical values are generally not to be understood as exact values, but also include a tolerance of +/- 1% to +/- 10%.

References to standards or specifications or norms are to be understood as a reference to standards or specifications or norms that apply at the time of filing and / or - if a priority is claimed - at the time of filing for priority. However, this is not to be understood as a general exclusion of applicability to the following or replacing standards or specifications or norms.

The invention offers, inter alia, the advantage that, with its help, complex mathematical calculations, such as, for example, FFT calculations, can also be made available on resource-limited platforms. It provides an improvement in terms of energy consumption and system complexity. This enables, for example, signal processing, for example signals from one or more sensors, on platforms close to the hardware / sensors. In addition, the possibility of an improved calculation speed also offers the advantage that larger data volumes can be processed at a faster rate, so that, for example, the extraction of frequency components can be made possible in real time. This also enables transformation of signals, as well as multiplication with large numbers and polynomials. Therefore, video and / or audio signal analysis is now also made possible on such platforms.

The invention makes it possible to calculate sine and / or cosine function values even for 8-bit microcontrollers without a floating point calculation option and to make them available for further processing.

The invention is based on the knowledge of recursive schemes.

In particular, recursive methods are described below in which the argument increases linearly. These allow simple implementation despite a possible accumulation of rounding errors.

Furthermore, it is assumed below that the increment ϑ of the argument (in the angular range) is constant.

die Werte eines Sinus bzw. Kosinus bestimmt werden können.For such values, an algorithm should be assumed in the following, in which in a simple manner for successive increments, ie $$\text{A * cos}\left({\mathrm{\gamma }}_0\right)\text{A * cos}\left({\mathrm{\gamma }}_0+\mathrm{\vartheta }\right)\mathrm{,,}\text{A * cos}\left({\mathrm{\gamma }}_0+2\mathrm{\vartheta }\right)\mathrm{,\; ....}$$

the values of a sine or cosine can be determined.

This should also be able to run on hardware platforms with a limited memory space (e.g. a few hundred bytes) and with a small number of registers and a low clock rate (e.g. a few megahertz). Therefore, the goal is to keep both the size of the programming code required and the size of the data memory used low, while at the same time the goal is to keep the accuracy high.

Diese Gleichung kann umgestellt werden: $$\text{cos}\left(\mathrm{\omega -\vartheta }\right)=2\text{cos}\left(\mathrm{\omega }\right)*\text{cos}\left(\mathrm{\vartheta }\right)-\text{cos}\left(\mathrm{\omega -\vartheta }\right)$$

The invention makes use of the mathematical relationship between the product of two cosine values: $$\text{cos}\left(\mathrm{\omega }\right)*\text{cos}\left(\mathrm{\vartheta }\right)=\frac{\text{cos}\left(\mathrm{\omega \; -\; \vartheta }\right)+\text{cos}\left(\mathrm{\omega \; -\; \vartheta }\right)}{2}$$

This equation can be rearranged: $$\text{cos}\left(\mathrm{\omega \; -\; \vartheta }\right)=2\text{cos}\left(\mathrm{\omega }\right)*\text{cos}\left(\mathrm{\vartheta }\right)-\text{cos}\left(\mathrm{\omega \; -\; \vartheta }\right)$$

• • Y₀ = A
• • Y₁ = Acos (ϑ)
• • Y_n = 2 cos (ϑ) Y_n-1 - Y_n-2

From this, however, a procedure can be derived how the subsequent value (cos (ω + ϑ)) can be determined from the two previous values (cos (ω) and cos (ω - ϑ)) with the same increment ϑ:

• • Y ₀ = A
• • Y ₁ = Acos (ϑ)
• • Y _n = 2 cos (ϑ) Y _n-1 - Y _n-2

• // Verfahren 1
• //Data: step angle ϑ, required samples number N, amplitude A
• //Result: N samples
• Y-₂ ← A;
• Y ← Acos (ϑ);

while less than N samples generated do $${\text{Y}}_{\text{tmp}}\leftarrow 2\text{cos}\left(\mathrm{\vartheta }\right)\text{Y}-{\text{Y}}_{-2};$$

$${\text{Y}}_{-2}\leftarrow \text{Y};$$

$$\text{Y}\leftarrow {\text{Y}}_{\text{tmp}};$$

endA simple calculation method can be developed from this:

• // Procedure 1
• // Data: step angle ϑ, required samples number N, amplitude A
• // Result: N samples
• Y- ₂ ← A;
• Y ← Acos (ϑ);

while less than N samples generated do $${\text{Y}}_{\text{tmp}}\leftarrow 2\text{cos}\left(\mathrm{\vartheta }\right)\text{Y}-{\text{Y}}_{-2};$$

$${\text{Y}}_{-2}\leftarrow \text{Y};$$

$$\text{Y}\leftarrow {\text{Y}}_{\text{tmp}};$$

end

Assuming that there were no rounding errors, this procedure would produce correct values. However, the use of a floating point library here would lead to a large increase in the programming code.

$$\text{Y}\left(\text{t}\right)={\text{K}}_2\text{dX}\left(\text{t}\right)/\text{dt},{\text{K}}_2<0$$

A harmonic oscillator can also be represented in the following differential equation system using Euler's integration technique. $$\text{X}\left(\text{t}\right)={\text{K}}_1\text{dY}\left(\text{t}\right)/\text{German},{\text{K}}_1>0$$

$$\text{Y}\left(\text{t}\right)={\text{K}}_2\text{dX}\left(\text{t}\right)/\text{German},{\text{K}}_2<0$$

• , wobei A, B und K Konstanten sind, die die Amplitude (A, B) der jeweiligen Schwingung bzw. deren Frequenz (K) bestimmen.

The solution to this is: $$\begin{array}{cc}\text{X}\left(\text{t}\right)=\text{A sin}\left(\text{Kt}\right),& \text{Y}\left(\text{t}\right)=\text{B cos}\left(\text{Kt}\right)\end{array}$$

• , where A, B and K are constants that determine the amplitude (A, B) of the respective oscillation or its frequency (K).

• • Y₀ = A
• • X₀ = 0
• • X_n = X_n -₁ + kY_n-1
• • Y_n = Y_n-1 - kX_n-1

The discretization is provided here continuously via the variable t (unlike in the method given above). The so-called Forward Euler can be developed from this:

• • Y ₀ = A
• • X ₀ = 0
• • X _n = X _n - ₁ + kY _n-1
• • Y _n = Y _n-1 - kX _n-1

• // Verfahren 2
• // Data: k = f(step angle θ), required samples number N, amplitude A
• // Result: N samples
• Y ← A;
• X <— 0;

while less than N samples generated do $${\text{X}}_{\text{tmp}}\leftarrow \text{X};$$

$$\text{X}\leftarrow \text{X}+\text{kY};$$

$$\text{Y}\leftarrow \text{Y}-{\text{kX}}_{\text{tmp}};$$

endThe parameter k can be understood as a function of the increment ϑ. From this a procedure can in turn be developed:

• // Procedure 2
• // Data: k = f (step angle θ), required samples number N, amplitude A
• // Result: N samples
• Y ← A;
• X <- 0;

while less than N samples generated do $${\text{X}}_{\text{tmp}}\leftarrow \text{X};$$

$$\text{X}\leftarrow \text{X}+\text{kY};$$

$$\text{Y}\leftarrow \text{Y}-{\text{kX}}_{\text{tmp}};$$

end

By a suitable choice of k as a power of two, the multiplications shown above can also be implemented as a shift operation.

However, since this approach tends to generate an increase in amplitude in practice, it has already been proposed in the past to use the value Y _n = Y _n-1 - kX _n instead of Y _n = Y _n-1 - kX _n-1 . This so-called intermediate Euler can be used in the practical implementation to ensure that method 2 only needs two variables instead of three variables. This leads to a reduction in the requirements for the hardware platform.

$${\text{X}}_{\text{n}-\text{1}}=\left({\text{Y}}_{\text{n}-2}-{\text{Y}}_{\text{n}-1}\right)/\text{k}$$

• // Verfahren 3
• //Data: k = f(step angle ϑ), required samples number N, amplitude A
• //Result: N samples
• Y ← A;
• X ← 0;

while less than N samples generated do $$\text{X}\leftarrow \text{X}+\text{kY};$$

$$\text{Y}\leftarrow \text{Y}-\text{kY};$$

endHowever, this can also be changed and Y _{n be} replaced by Y _n-1 and Y _n-2 : $${\text{X}}_{\text{n}}={\text{X}}_{\text{n}-\text{1}}+{\text{kY}}_{\text{n}-1}$$

$${\text{X}}_{\text{n}-\text{1}}=\left({\text{Y}}_{\text{n}-2}-{\text{Y}}_{\text{n}-1}\right)/\text{k}$$

• // Procedure 3
• // Data: k = f (step angle ϑ), required samples number N, amplitude A
• // Result: N samples
• Y ← A;
• X ← 0;

while less than N samples generated do $$\text{X}\leftarrow \text{X}+\text{kY};$$

$$\text{Y}\leftarrow \text{Y}-\text{kY};$$

end

• • Y₀ = A
• • X₀ = 0
• • X_n = X_{n -1} + kY_n-1
• • Y_n = (2 - k²)Y_{n - 1} - Y_n-2

From the conversion, however, Y _n ← (2 - k ² ) Y _n-1 - Y _n-2 can be deduced. Ie

• • Y ₀ = A
• • X ₀ = 0
• • X _n = X _{n -1} + kY _n-1
• • Y _n = (2 - k ² ) Y _{n - 1} - Y _n-2

• • Y₀ = A
• • Y₁ = Acos (ϑ)
• • Y_n = 2 cos (ϑ) Y_n-1 - Y_n-2

kann folgender Verglich gezogen werden:

• • $\text{k}=\sqrt{2-2cos\mathrm{\vartheta }}$
  
• • ${\text{X}}_0=-\text{A}\frac{\sqrt{2}}{2}\sqrt{1-cos\mathrm{\vartheta }}=-Asin\left(\frac{\mathrm{\vartheta }}{2}\right)$
  
• • Y₀ = A

By comparison with

• • Y ₀ = A
• • Y ₁ = Acos (ϑ)
• • Y _n = 2 cos (ϑ) Y _n-1 - Y _n-2

the following comparison can be drawn:

• • $\text{k}=\sqrt{2-2cOs\mathrm{\vartheta }}$
  
• • ${\text{X}}_0=-\text{A}\frac{\sqrt{2}}{2}\sqrt{1-cOs\mathrm{\vartheta }}=-Asin\left(\frac{\mathrm{\vartheta }}{2}\right)$
  
• • Y ₀ = A

• • ${\text{X}}_0=\text{A sin }\left(-\frac{\mathrm{\vartheta }}{2}\right)$
  
• • ${\text{X}}_1=\text{A sin }\left(\frac{\mathrm{\vartheta }}{2}\right)$
  
• • ${\text{X}}_{\text{n}}=\text{A sin}\left(-\frac{\mathrm{\vartheta }}{2}+\text{n}\mathrm{\vartheta }\right)$
  

However, this can also be changed as follows:

• • ${\text{X}}_0=\text{A sin}\left(-\frac{\mathrm{\vartheta }}{2}\right)$
  
• • ${\text{X}}_1=\text{A sin}\left(\frac{\mathrm{\vartheta }}{2}\right)$
  
• • ${\text{X}}_{\text{n}}=\text{A sin}\left(-\frac{\mathrm{\vartheta }}{2}+\text{n}\mathrm{\vartheta }\right)$
  

I.e. With the above considerations, it is possible to provide a sequence of exact cosine values or exact cosine values and sine values, based on a phase shift of half the step size el.

The actual implementation must, however, take into account rounding errors and error propagation, since with an increasing number of iterations the signal / values obtained deviate from the target variable actually to be generated. I.e. there is an increase in amplitude errors and / or offset of the signal curve produced from the target curve.

In the context of the invention, it is therefore proposed to make use of the symmetry properties of the sine / cosine function or the phase dependencies of the sine function on the cosine function (and vice versa).

This is in the 1 to 3 shown as an example. While in 1 a sine function is shown in 2 a cosine function is shown. In the 1 and 2 individual subdivisions are shown. The subdivisions shown there can be broken down into the proportions according to 3 can be summarized, the symmetries being more clearly visible.

An oscillator according to the invention, which can be implemented in software but also in an application-specific hardware layout (FGA, ASIC, etc.), can then be made available (purely) in integer logic. The oscillator continues to use recursive processing. However, the recursive processing is limited to certain sub-areas. A representation of a phase angle is used as the input variable x of the oscillator, a calculation of the oscillator value based on recursive sine processing being determined depending on the phase angle in a first phase angle range, and a calculation of the oscillator value in a second phase angle range, which is different from the first phase angle range is determined using recursive cosine processing.

In other words, we assume that the oscillator has a sinusoidal oscillation or values of a sinusoidal oscillation according to 1 should produce. In an input variable range that corresponds to a value from 0 to π / 4, X is calculated by means of recursive sine processing (for example according to method 3 in the forward direction). In an input variable range that corresponds to a value from π / 4 to π / 2, X is calculated by means of recursive cosine processing (for example according to method 3 in the backward direction). In an input variable range that corresponds to a value from π / 2 to 3π / 4, X is calculated by means of recursive cosine processing (for example according to method 3 in the forward direction). In an input variable range that corresponds to a value from 3π / 4 to π, X is calculated by means of recursive sine processing (for example according to method 3 in the reverse direction). A (positive) half-wave of a sinusoidal oscillation with small input variables could thus be represented for the respective recursive configuration. This allows rounding errors and propagation errors to be reduced, with the code size and data size remaining small.

Obviously, this positive half-wave can also serve as the basis for the calculation of the negative half-wave, since sin (α + π) = - sin (α).

In the same way, however, one can assume that the oscillator has a cosine oscillation or values according to a cosine oscillation 2 should produce. In an input variable range that corresponds to a value from 0 to π / 4, X is determined using a recursive cosine processing (eg according to method 3 in the forward direction). In an input variable range that corresponds to a value from π / 4 to π / 2, X is calculated by means of recursive sine processing (for example according to method 3 in the backward direction). In an input variable range that corresponds to a value from π / 2 to 3π / 4, X is calculated with a negative sign by means of recursive sine processing (eg according to method 3 in the forward direction). In an input variable range which corresponds to a value from 3π / 4 to π, X is calculated with a negative sign by means of recursive cosine processing (for example according to method 3 in the backward direction). A first partial wave of a cosine oscillation with small input variables could thus be represented for the respective recursive configuration. This allows rounding errors and propagation errors to be reduced, with the code size and data size remaining small.

Obviously, this first partial wave can also serve as a basis for the calculation of the second partial wave, since cos (α) = cos (-α) = cos (2π-α).

The invention can make use of the fact that the method 3 shown above is reversible.

That is, in the first quadrant of 1 X can initially be determined in the forward direction by means of the sine calculation, while Y is initially determined in the forward direction by means of the cosine calculation. As soon as the value of the X calculation is greater than or equal to the value of the Y calculation (ie sin (X) = cos (X)), the oscillator is no longer operated in a forward direction, but in a reverse direction. However, the output is then redirected. This means that the previous sine is ordered to the cosine and vice versa. Since the oscillator goes back to its previous forward steps, the rounding errors are caught up again backwards.

The invention makes use of two types of knowledge. On the one hand, the error increases with the number of samples. On the other hand, the error is more serious with small values.

The assignment can also be represented as follows, where HQ stands for half quadrant, starting with 0 corresponding to the 1st HQ, and where fwd. For forward direction and rev. Stands for reverse direction. HQ 1 2 3 4th 5 6 7th 8th Oscillator direction Fwd Rev. Fwd Rev. Fwd Rev. Fwd Rev. Sine X Y Y X -X -Y -Y -X cosine Y X -X -Y -Y -X X Y

This processing can be implemented with integer values of any format, i.e. 8-bit as well as 16-bit as well as 32-bit and 64-bit up to so-called bigintegers can be used.

The particular value of the invention is shown, however, on resource-limited platforms which have no floating point units for providing the desired values, or which are required for other calculations.

Without loss of generality, the digital oscillator can be configured on a wide variety of platforms. Due to the simplicity and robustness paired with low demands on memory and processing speed, the oscillator can be operated with a clock rate corresponding to frequencies 32 MHz or less, e.g. 16 MHz, 12 MHz, 10 MHz or less can be implemented. Of course, nothing stands in the way of using it on higher-performance platforms.

Even if a subdivision at π / 4 was carried out in the example described above and this proves to be advantageous in relation to the processing according to method 3, neither the use of another method nor the exact location of a subdivision is mandatory. Rather, using this example, the invention shows that there are possibilities of avoiding errors, error propagation, etc. through the clever use of recursion properties and symmetry properties of the sine / cosine functions.

Furthermore, the performance of the digital oscillator can also be made available on hardware platforms that have no or slowly functioning division functions. This can be implemented, for example, by cleverly choosing a division by 2 ⁿ , with n, n> 1, as a shift operation. It goes without saying that this would also be possible in the same way with a multiplication. In other words, the invention can also be implemented on hardware platforms, such as PIC controllers.

In particular, the digital oscillator according to the invention can be implemented in software on an 8-bit or 16-bit microcontroller. Such microcontrollers can be found e.g. also on chip cards.

The digital implementation as well as the hardware implementation allow the use of a digital oscillator in encryption and / or decryption operations.

Without loss of generality, both the digital implementation and the hardware implementation can be used in a route calculation, or in a frequency synthesizer, in the provision of FFT and / or inverse FFT operations or in wavelet operations.

Claims (13)

Digital oscillator, where the digital oscillator operates with integer values, with recursive processing being used, with a representation of a phase angle being used as the input variable, with a calculation of the oscillator value based on a first phase angle range to reduce rounding errors and error propagation depending on the phase angle in a first phase angle range recursive sine processing is determined, and in a second phase angle range, which is different from the first phase angle range, a calculation of the oscillator value is determined on the basis of recursive cosine processing. Digital oscillator according to Claim 1 , characterized in that integer values have 16 bits or 8 bits. Digital oscillator according to Claim 1 or 2 , characterized in that the clock rate at which the oscillator is clocked is less than or equal to 32 MHz. Digital oscillator according to one of the preceding claims, characterized in that a change from sine processing to cosine processing takes place with a representation of a phase angle of approximately pi / 4. Digital oscillator according to one of the preceding claims, characterized in that the input variable for recursive sine processing and / or recursive cosine processing originates in a phase angle range from 0 to pi / 4. Digital oscillator according to one of the preceding claims, characterized in that the output variable of the recursive sine processing and / or recursive cosine processing is available as a positive or negative output variable of the digital oscillator depending on the input variable of the digital oscillator. Digital oscillator according to one of the preceding claims, characterized in that the recursive sine processing or the recursive cosine processing have shift operations, so that the digital oscillator can be implemented on microcontrollers without a division operation. Digital oscillator according to one of the preceding claims, characterized in that the digital oscillator is implemented in software on an 8-bit or 16-bit microcontroller. Use of a digital oscillator according to one of the preceding claims in encryption and / or decryption operations. Use of a digital oscillator according to one of the preceding claims for route calculations. Use of a digital oscillator according to one of the preceding claims as a frequency synthesizer. Use of a digital oscillator according to one of the preceding claims for FFT and / or inverse FFT operations. Use of a digital oscillator according to one of the preceding claims in wavelet operations

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