DE102014200914B4 - Method and system for compensating mismatches between analog-to-digital converters that scan at different times - Google Patents

Abstract

Method for compensating mismatches in the transfer function between several analog-to-digital converters (20, 21, 22, 23, ...), which are sampled at different times from one another, by converting the signals from the respective analog-to-digital converter (20, 21, 22, 23 , ...) generated sampled values are convoluted with an impulse response of a compensation filter (40, 41, 42, 43, ...) belonging to the respective analog-digital converter (20, 21, 22, 23, ...), where for at least two compensation filters (40, 41, 42, 43, ...) at least one filter coefficient is determined in advance by minimizing a cost function in order to minimize the difference in amount and / or phase between those with the transfer function of the associated compensation filter (40, 41, 42, 43, ...) in each case to achieve folded transfer functions in at least two analog-to-digital converters (20, 21, 22, 23, ...), in the case of more than two staggered in time analog-digital converters (20, 21, 22, 23, ...) scanning each other, the filter coefficients of the compensation filter (40, 41 , 42, 43, ...) result from the convolution of the filter coefficients of several partial compensation filters (40, 41, 42, 43, ...), with a pair of first partial compensation filters (40, 41, 42 , 43, ...) compensates for the mismatch of the transfer functions of two respective analog-to-digital converters (20, 21, 22, 23, ...) to one another, and each additional pair of partial compensation filters (40, 41 , 42,43, ...) the mismatching of the transfer function of a respective first subgroup of analog-to-digital converters (20, 21, 22, 23, ...), which by means of already determined partial compensation filters (40, 41, 42 , 43, ...) are already compensated for each other, with regard to the transfer function of a respective second subgroup with the same number of analog-digital walls learn (20, 21, 22, 23, ...), which are already compensated for each other by means of already determined partial compensation filters (40, 41, 42, 43, ...), compensated for each other.

Description

The invention relates to a method and a system for compensating for mismatches in the transfer function between analog-to-digital converters that are sampled at different times.

The acquisition of increasingly broadband signals with the aid of digital oscilloscopes requires signal sampling that cannot be achieved with conventional analog-to-digital converters. To increase the sampling rate, several analog-to-digital converters are connected in parallel and the same input signal to be sampled is applied. The analog-digital converters connected in parallel sample the input signal with the same sampling rate, which is reduced by a factor corresponding to the number of analog-digital converters connected in parallel compared to the desired sampling rate, and by a time offset corresponding to the desired sampling time interval offset from (so-called "time-interleaved" A / D converter or time-shifted A / D converter).

• • unterschiedliche Impulsantworten h_i(t) und h_j(t) sowie unterschiedliche Übertragungsfunktionen H_i(f) bzw. H_j(f) zwischen unterschiedlichen Analog-Digital-Wandlern i und j,
• • unterschiedliche Gleichanteile o_i und o_j zwischen unterschiedlichen Analog-Digital-Wandlern i und j und
• • normierte Zeitversätze ε_i und ε_j zu den einzelnen normierten Abtastzeitpunkten M · v + i bzw. M · v + j (M : Anzahl der parallel geschalteten Analog-Digital-Wandler).

Such A / D converter structures require completely identically functioning analog-digital converters. Due to mismatches in real A / D converters, there is no completely identical operating behavior of the individual A / D converters. Real analog-to-digital converters typically have the following errors compared to ideal analog-to-digital converters:

• • different impulse responses h _i (t) and h _j (t) as well as different transfer functions H _i (f) or H _j (f) between different analog-to-digital converters i and j,
• • different DC components o _i and o _j between different analog-to-digital converters i and j and
• • normalized time offsets ε _i and ε _j to the individual normalized sampling times M · v + i or M · v + j (M: number of analog-digital converters connected in parallel).

DE 10 2011 011 711 A1 discloses a method and a system for compensating for such mismatches in real A / D converters. The impulse response of the individual compensation filters is determined in real time from the impulse responses of the associated analog-digital converters and the input signal using auto- and cross-correlation functions. The method has the disadvantage that the input signal is not known in advance and is only approximated by a sync function.

No. 7,782,235 B1 describes a device and a method for compensating for mismatching in an analog-digital converter with analog-digital converters arranged in parallel. The input signal is distributed evenly to the inputs of the analog-digital converter using a splitter. For time-interleaved sampling of the input signal, the sampling clock of each analog-digital converter can be phase-shifted with respect to one another. Each signal path of the parallel array of analog-to-digital converters can each have an associated mismatch. An adaptive mismatch estimation unit determines the mismatch of the individual signal paths by analyzing the digitized signals and forwards the adaptation parameters calculated therefrom to an equalization unit. The adjustment parameters are representative of the DC offset, amount and phase of the mismatch between each of the digital signals converted from the analog signal. The adjustment parameters can include a variety of amplitude and phase levels at different frequencies so that a digital filter design algorithm, such as least squares optimization, can calculate a digital finite impulse response (FIR) filter that approximates the desired profile . The equalization unit compensates for the mismatches in the output values of the analog-digital converter based on the calculated filters and outputs digital values of the signals that have been adjusted with regard to amplification, phase and DC offset.

A similar prior art can be found in US 2011/0260898 A1.

DE 10 2005 039 684 A1 describes a system that includes delay elements, at least two analog-to-digital converters (ADCs) and finite impulse response (FIR) filters. An analog input signal is fed to a first ADC. The same input signal is z. B. in a system that comprises three ADCs, with a delay of a third of the sampling time to a second ADC and with a delay of a further third of the sampling time to a third ADC. To correct a mismatch, the output values of the three ADCs in this case are fed into an FIR filter with a 3 × 3 matrix. The adjusted measured values are available for further processing at the output of the FIR filter.

US Pat. No. 7,541,958 B2 discloses a technique for reducing errors in a parallel, time-interleaved analog-to-digital converter (PTIC). 3 Figure 12 shows a calibration process for acquiring values for H-matrices. There is one for each of the parallel analog-to-digital converters separate calibration carried out. Each determined calibration value is entered in an H matrix. With the help of the H matrix, the measured values can be output corrected.

US 2011/0006933 A1 describes an analog-digital converter system consisting of several time-interleaved analog-digital converters with adjustable delay elements and an estimation unit. The estimation unit is coupled to each of the adjustable delay elements and to each of the ADCs in order to adapt the adjustable delay elements in order to compensate for differences in the delay of the sampling times between the ADCs.

The object of the invention is therefore to create a method and a system for the exact compensation of mismatches in the transfer function between a plurality of analog-to-digital converters, each of which is sampled at different times.

The object is achieved by a method according to the invention for compensating mismatches in the transfer function between several analog-to-digital converters, each scanning at different times, with the features of claim 1 and by a system according to the invention of several analog-to-digital converters and downstream compensation filters with the features of claim 9 solved. Advantageous technical extensions are listed in the respective dependent claims.

According to the invention, the parameters of the individual compensation filters are determined in advance by minimizing a cost function on the basis of the least squares criterion. The minimization of the cost function creates an identity between the transfer functions of at least two analog-to-digital converters.

The transfer function of each individual analog-to-digital converter is preferably modeled by a digital filter of first order with an infinite length of the impulse response and measured for several frequencies with a view to minimizing the cost function.

In the first two embodiments of a parameter optimization according to the invention, the compensation filter belonging to the respective analog-digital converter is preferably a digital filter of the first order with a finite length of the impulse response and is ideally inverse to the digital filter of the associated analog-digital converter. The parameter to be determined for the respective compensation filter is the weighting factor of the digital filter. While in a first exemplary embodiment of the parameter optimization according to the invention by minimizing the cost function, the difference between the transfer functions of two analog-digital converter channels is minimized, in a second exemplary embodiment of the parameter optimization according to the invention by minimizing the cost function, the difference between the amount-squared transfer functions minimized by two analog-digital converter channels each.

In a third exemplary embodiment of a parameter optimization, the difference between the transfer functions of the compensation filters belonging to two analog-digital converters is achieved by minimizing the cost function, each of the two compensation filters each representing a digital filter of the order L, each with a finite length of the impulse response . The parameters of the respective compensation filter to be optimized are thus a total of L filter coefficients.

In the case of broadband signals, more than two analog-digital converter channels are preferably connected in parallel. The number of analog-to-digital converters that are sampled at a time offset from one another is preferably an integer power of two.

In this case, in a first optimizing iteration step, the mismatches in the transfer function between two analog-digital converters are compensated by adding the filter coefficients of the partial compensation filters belonging to two analog-digital converters, for example according to one of the three embodiments of a parameter optimization according to the invention be optimized.

In further optimizing iteration steps, mismatches in the transfer function between the next higher power of two of analog-digital converters are compensated by preferably two subgroups of analog-digital converters whose mismatches in the transfer function to each other already by means of two partial compensation filters optimized for this are compensated, are fed to a common compensation.

For this purpose, for each subgroup of analog-digital converters, a sum of summands belonging to each individual analog-digital converter of the subgroup is preferably formed. Each summand contains the multiplication of the transfer function of the individual analog-digital converter channel with the transfer functions of all partial compensation filters for several frequencies that have been determined in the previous optimizing iteration steps and belong to the individual analog-digital converter. In the respective optimizing iteration step, the filter coefficients of the associated partial compensation filter are determined for each of the two subgroups of analog-digital converters using one of the three embodiments of a parameter optimization according to the invention.

The filter coefficients of the compensation filter belonging to the respective analog-digital converter are preferably obtained by convolution of the filter coefficients of the partial compensation filters determined in the individual optimizing iteration steps for the respective analog-digital converter.

Since the filter length of the determined compensation filter is too long due to the convolution, an associated compensation filter with an optimized filter length is determined by minimizing a cost function using the least squares criterion. For this purpose, the cost function contains differences between the determined Fourier-transformed filter coefficients of the unabridged compensation filter and a reference transfer function of the compensation filter with an optimized filter length for several frequencies. The filter coefficients of a compensation filter with an optimized filter length can be determined from the minimization of this cost function.

• 1A,1B ein Blockdiagramm einer ersten und zweiten Ausführungsform des erfindungsgemäßen Systems von mehreren Analog-Digital-Wandlern und nachgeschalteten Kompensationsfiltern,
• 2A,2B,2C,2D ein Flussdiagramm des erfindungsgemäßen Verfahrens zur Kompensation von Fehlanpassungen in der Übertragungsfunktion zwischen mehreren jeweils zeitlich versetzt zueinander abtastenden Analog-Digital-Wandlern unter Benutzung einer ersten, zweiten, dritten und vierten Ausführungsform einer Parameteroptimierung,
• 3A,3B,3C,3D eine spektrale Darstellung des Betrags und der Phase der Übertragungsfunktion eines ersten und zweiten Analog-Digital-Wandlers und des Betrags und der Phase des Verhältnisses der Übertragungsfunktionen zwischen ersten und zweiten Analog-Digital-Wandler,
• 4A,4B,4C,4D eine spektrale Darstellung des Betrags und der Phase der gestörten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals und des Betrags und der Phase des Verhältnisses zwischen den gestörten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals,
• 5A,5B,5C,5D eine spektrale Darstellung des Betrags und der Phase der kompensierten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals, und des Betrags und der Phase des Verhältnisses zwischen den kompensierten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals bei Anwendung der ersten Ausführungsform einer Parameteroptimierung,
• 6A,6B,6C,6D eine spektrale Darstellung des Betrags und der Phase der kompensierten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals, und des Betrags und der Phase des Verhältnisses zwischen den kompensierten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals bei Anwendung der zweiten Ausführungsform einer Parameteroptimierung,
• 7A,7B,7C,7D eine spektrale Darstellung des Betrags und der Phase der kompensierten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals, und des Betrags und der Phase des Verhältnisses zwischen den kompensierten Übertragungsfunktionen des ersten und zweiten Analog-Digital-Wandler-Kanals bei Anwendung der dritten Ausführungsform einer Parameteroptimierung,
• 8A,8B,8C,8D eine spektrale Darstellung des Betrags und der Phase der gestörten Übertragungsfunktionen eines ersten, zweiten, dritten und vierten Analog-Digital-Wandler-Kanals und des Betrags und der Phase des Verhältnisses zwischen den gestörten Übertragungsfunktionen des zweiten, dritten und vierten Analog-Digital-Wandler-Kanals zum ersten Analog-Digital-Wandler-Kanal und
• 9A,9B,9C,9D eine spektrale Darstellung des Betrags und der Phase des Verhältnisses zwischen den kompensierten Übertragungsfunktionen des zweiten, dritten und vierten Analog-Digital-Wandler-Kanals und dem ersten Analog-Digital-Wandler-Kanal bei Anwendung der vierten Ausführungsform einer Parameteroptimierung.

Embodiments of the method according to the invention for compensating mismatches in the transfer function between several analog-to-digital converters, each scanning at different times, and the system according to the invention of several analog-to-digital converters and downstream compensation filters is explained in detail below with reference to the drawing. The figures in the drawing show:

• 1A , 1B a block diagram of a first and second embodiment of the system according to the invention of several analog-digital converters and downstream compensation filters,
• 2A , 2 B , 2C , 2D a flowchart of the method according to the invention for compensating for mismatches in the transfer function between a plurality of analog-to-digital converters, each sampling with a time offset from one another, using a first, second, third and fourth embodiment of a parameter optimization,
• 3A , 3B , 3C , 3D a spectral representation of the amount and the phase of the transfer function of a first and second analog-to-digital converter and of the amount and the phase of the ratio of the transfer functions between the first and second analog-to-digital converter,
• 4A , 4B , 4C , 4D a spectral representation of the amount and the phase of the disturbed transfer functions of the first and second analog-to-digital converter channels and of the amount and the phase of the ratio between the disturbed transfer functions of the first and second analog-to-digital converter channels,
• 5A , 5B , 5C , 5D a spectral representation of the magnitude and phase of the compensated transfer functions of the first and second analog-to-digital converter channels, and of the magnitude and phase of the ratio between the compensated transfer functions of the first and second analog-to-digital converter channels when using the first Embodiment of a parameter optimization,
• 6A , 6B , 6C , 6D a spectral representation of the magnitude and phase of the compensated transfer functions of the first and second analog-to-digital converter channels, and of the magnitude and phase of the ratio between the compensated transfer functions of the first and second analog-to-digital converter channels when the second is used Embodiment of a parameter optimization,
• 7A , 7B , 7C , 7D a spectral representation of the magnitude and phase of the compensated transfer functions of the first and second analog-to-digital converter channels, and of the magnitude and phase of the ratio between the compensated transfer functions of the first and second analog-to-digital converter channels when using the third Embodiment of a parameter optimization,
• 8A , 8B , 8C , 8D a spectral representation of the amount and the phase of the disturbed transfer functions of a first, second, third and fourth analog-to-digital converter channel and of the amount and the phase of the ratio between the disturbed transfer functions of the second, third and fourth analog-to-digital converter Channel to the first analog-to-digital converter channel and
• 9A , 9B , 9C , 9D a spectral representation of the magnitude and the phase of the relationship between the compensated transfer functions of the second, third and fourth analog-digital converter channels and the first analog-digital converter channel when using the fourth embodiment of a parameter optimization.

In the following, the mathematical fundamentals required to understand the invention are first explained.

An analog-to-digital converter essentially consists of a sample and hold element and a downstream level quantizer. If there is a mismatch between the input impedance of the level quantizer and the impedance of the connecting line and if there is a mismatch between the output impedance of the sample and hold element and the impedance of the connection line, reflections occur at the level quantizer or the sample and hold element.

If the transit time T _{transit time} according to equation (1) is an integer multiple of the sampling time T _{a, ADC '} , the reflected signal enters the sample-and-hold element phase-synchronously with the signal to be sampled and is sampled phase-synchronously with the signal to be sampled. $$T_{\mathrm{A.}btast}=N\cdot T_{a,\mathrm{A.}\mathrm{D.}\mathrm{C.}}$$

Under this condition, the transmission behavior of the analog-to-digital converter in channel i can be determined by a first-order delay element with the transfer function H _i (f / f _a ) according to equation (2) with the delay D and the gain factor g _i - corresponding to a digital filter with infinite pulse length (English: Infinite Impulse Response (IIR) filter). The frequency f is normalized to the sampling frequency f _a. $$H_i\left(f/f_a\right)=\frac{1}{1-G_i\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/fa\cdot \mathrm{D.}}}$$

_{The associated compensation filter thus has a transfer function H equi} (f / f _a ) inverted to the transfer function H _i (f / f _a ) of the analog-to-digital converter according to equation (3), that of the transfer function of a digital filter of the first order with finite Pulse length (English: Finite Impulse Response (FIR) filter) corresponds. $$H_{\mathrm{E.}qui}\left(f/f_a\right)=1-{\widehat{G}}_i\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/fa\cdot \mathrm{D.}}$$

If two analog-digital converters are connected in parallel, each of which has different transfer functions H _i (f / f _a ), the transfer functions H _equi (f / f _a ) of the associated compensation filter must be determined for each analog-digital converter channel become. To simplify the optimization problem, the delay D is assumed to be known and the two gain factors ĝ ₀ and ĝ _{1 are determined} by minimizing a cost function K.

$$H\left(f/f_a\right)=\frac{{\tilde{H}}_0\left(f/f_a\right)}{{\tilde{H}}_1\left(f/f_a\right)}=\frac{H_a\left(f/f_a\right)\cdot H_0\left(f/f_a\right)}{H_a\left(f/f_a\right)\cdot H_1\left(f/f_a\right)}=\frac{1-{\widehat{g}}_1{e}^{j2\pi f/f_a\cdot D}}{1-{\widehat{g}}_0{e}^{j2\pi f/f_a\cdot D}}$$

To determine the cost function K, in a first embodiment of a parameter optimization according to the invention, the transfer functions H̃ _i (f / f _a ) belonging to the two analog-digital converter channels between the input of the front end and the output of the analog-digital converter are used. Converter measured according to equation (4) for different frequencies f / f _a. Then the ratio H (f / f _a ) between the measured transfer function H̃ (f / f _a ) of the first analog-digital converter channel and the measured transfer function H̃ ₁ (f / f _a ) of the second analog-digital converter Channel determined for different frequencies f / f _a according to equation (5). Equation (5) takes into account that the transfer function of the compensation filter belonging to an analog-digital converter channel is ideally inverted to the transfer function of the associated analog-digital converter channel. $${\tilde{H}}_i\left(f/f_a\right)=H_a\left(f/f_a\right)\cdot H_i\left(f/f_a\right)$$

$$H\left(f/f_a\right)=\frac{{\tilde{H}}_0\left(f/f_a\right)}{{\tilde{H}}_1\left(f/f_a\right)}=\frac{H_a\left(f/f_a\right)\cdot H_0\left(f/f_a\right)}{H_a\left(f/f_a\right)\cdot H_1\left(f/f_a\right)}=\frac{1-{\widehat{G}}_1{\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">e</figure-callout>}}^j}{1-{\widehat{G}}_0{\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">e</figure-callout>}}^j}$$

An ideal identity between the transfer functions H̃ ₀ (f / f _a ) and H̃ ₁ (f / f _a ) leads to a ratio H (f / f _a ) of one. On the basis of this condition, starting from equation (5) and using the least squares criterion, a relationship for the cost function K according to equation (6) is obtained. $$K={\displaystyle \sum _{f/f_a}{\left|{\widehat{G}}_0\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mathrm{D.}}\cdot H\left(f/f_a\right)-{\widehat{G}}_1\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mathrm{D.}}-\left[H\left(f/f_a\right)-1\right]\right|}^2}$$

$${\underset{\_}{\underset{\_}{A}}}_{f/f_a,links}=e^{-j2\pi f/f_a\cdot D}\cdot H\left(f/f_a\right)$$

$${\underset{\_}{\underset{\_}{A}}}_{f/f_a,rechts}=e^{-j2\pi f/f_a\cdot D}$$

$$\underset{\_}{b}=H\left(f/f_a\right)-1$$

$$\underset{\_}{\widehat{g}}={\left[\begin{array}{cc}{\widehat{g}}_0& {\widehat{g}}_1\end{array}\right]}^T$$

The cost function K can be represented on the basis of equation (6) in general notation according to equation (7), the coefficients in the left-hand column of matrix A according to equation (8), the coefficients in the right-hand column of matrix A according to equation ( 9), the coefficients of the vector b according to equation (10) and the vector ĝ of the gain factors according to equation (11) result in: $$K={\left[\underset{\_}{\underset{\_}{\mathrm{A.}}}\cdot \underset{\_}{\widehat{G}}-\underset{\_}{b}\right]}^H\cdot \left[\underset{\_}{\underset{\_}{\mathrm{A.}}}\cdot \underset{\_}{\widehat{G}}-\underset{\_}{b}\right]$$

$${\underset{\_}{\underset{\_}{\mathrm{A.}}}}_{f/f_a,links}=e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mathrm{D.}}\cdot H\left(f/f_a\right)$$

$${\underset{\_}{\underset{\_}{\mathrm{A.}}}}_{f/f_a,recHts}=e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mathrm{D.}}$$

$$\underset{\_}{b}=H\left(f/f_a\right)-1$$

$$\underset{\_}{\widehat{G}}={\left[\begin{array}{cc}{\widehat{G}}_0& {\widehat{G}}_1\end{array}\right]}^T$$

$$\underset{\_}{\underset{\_}{\tilde{A}}}=Real\left\{\underset{\_}{\underset{\_}{A^H}}\cdot \underset{\_}{\underset{\_}{A}}\right\}$$

$$\underset{\_}{\tilde{b}}=-2\cdot Real\left\{{\underset{\_}{\underset{\_}{A}}}^H\cdot \underset{\_}{b}\right\}$$

The vector ĝ of the gain factors can be determined by differentiating the cost function K according to the vector ĝ and then setting it to zero and results from equation (12) with the matrix à according to equation (13) and the vector b̃ according to equation (14). $$\underset{\_}{\widehat{G}}=-\frac{1}{2}\cdot {\underset{\_}{\underset{\_}{\widetilde{\mathrm{A.}}}}}^{-1}\cdot \underset{\_}{\underset{\_}{\tilde{b}}}$$

$$\underset{\_}{\underset{\_}{\widetilde{\mathrm{A.}}}}=\mathrm{R.}eal\left\{\underset{\_}{\underset{\_}{{\mathrm{A.}}^H}}\cdot \underset{\_}{\underset{\_}{\mathrm{A.}}}\right\}$$

$$\underset{\_}{\tilde{b}}=-2\cdot \mathrm{R.}eal\left\{{\underset{\_}{\underset{\_}{\mathrm{A.}}}}^H\cdot \underset{\_}{b}\right\}$$

In a second embodiment of a parameter optimization according to the invention, the amount-squared ratio | H (f / f _a ) | ² between the amount-squared measured transfer function H̃ ₁ (f / f _a ) of the first analog-digital converter channel and the amount-squared measured transfer function H̃ ₁ (f / f _a ) of the second analog-digital converter channel according to equation (15) determined. $${\left|H\left(f/f_a\right)\right|}^2=\frac{{\left|{\tilde{H}}_0\left(f/f_a\right)\right|}^2}{{\left|{\tilde{H}}_1\left(f/f_a\right)\right|}^2}=\frac{{\left|H_a\left(f/f_a\right)\cdot H_0\left(f/f_a\right)\right|}^2}{{\left|H_a\left(f/f_a\right)\cdot H_1\left(f/f_a\right)\right|}^2}=\frac{1+{\widehat{G}}_1{}^2-2\cdot {\widehat{G}}_1\cdot \text{cos}\left(2\pi f/f_a\cdot \mathrm{D.}\right)}{1+{\widehat{G}}_0{}^2-2\cdot {\widehat{G}}_0\cdot \text{cos}\left(2\pi f/f_a\cdot \mathrm{D.}\right)}$$

An ideally existing identity between the transfer functions | H̃ ₀ (f / f _a | ² and | H ₁ (f / f _a ) | ² , squared in amount, leads to a ratio | H (f / f _a ) | ² of one. On the basis This condition results in a relationship for the cost function K in the second embodiment of the invention according to equation (16) on the basis of equation (15) using the least squares criterion. $$K={\displaystyle \sum _{f/f_a}{\left[\frac{1+{\widehat{G}}_1{}^2-2\cdot {\widehat{G}}_1\cdot \text{cos}\left(2\pi f/f_a\cdot \mathrm{D.}\right)}{1+{\widehat{G}}_0{}^2-2\cdot {\widehat{G}}_0\cdot \text{cos}\left(2\pi f/f_a\cdot \mathrm{D.}\right)}-{\left|H\left(f/f_a\right)\right|}^2\right]}^2}$$

The cost function K according to equation (16) represents a non-linear optimization problem. This non-linear optimization problem can be solved numerically, for example, with the Coleman “confidence range approach”.

In a third embodiment of a parameter optimization according to the invention, a compensation filter is determined for each analog-digital converter channel, which is implemented as a digital filter of order L with a finite impulse response length and a transfer function H inverted _{to the measured transfer function H̃ i} (f / f _{a) equi} (f / f _a . The ratio H (f / f _a ) between the measured transfer function H̃ ₁ (f / f _a ) of the first analog-digital converter channel and the measured transfer function H̃ ₁ (f / f _a ) of the second analog-to-digital converter channel is thus obtained according to equation (17). $$H\left(f/f_a\right)=\frac{{\tilde{H}}_0\left(f/f_a\right)}{{\tilde{H}}_1{\left(f/f_a\right)}^2}=\frac{H_a\left(f/f_a\right)\cdot H_0\left(f/f_a\right)}{H_a\left(f/f_a\right)\cdot H_1\left(f/f_a\right)}=\frac{{\displaystyle \sum _{\xi =0}^{{\mathrm{L.}}_1-1}{H}_1\left(\xi \right)\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a<figure-callout\; id="1"\; label="\cdot \; \xi "\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0052.png"\; state="\{\{state\}\}">\cdot </figure-callout>\xi }}}{1+{\displaystyle \sum _{\mu =1}^{{\mathrm{L.}}_0-1}{H}_0\left(\xi \right)\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mu }}}$$

An ideal identity between the measured transfer functions H̃ ₁ (f / f _a ) of the first analog-digital converter channel and H̃ ₁ (f / f _a ) of the second analog-digital converter channel leads to a ratio H (f / f _a ) from one. With this condition, the cost function K results using the least squares criterion according to equation (18). $$K={{\displaystyle \sum _{f/f_a}\left|-H\left(f/f_a\right)\cdot {\displaystyle \sum _{\mu =1}^{{\mathrm{L.}}_{\mu }-1}{\widehat{H}}_0\left(\mu \right)\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mu }}+{\displaystyle \sum _{\xi =0}^{{\mathrm{L.}}_{\xi }-1}{\widehat{H}}_1\left(\xi \right)\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/{\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">f</figure-callout>}}_0\cdot \xi }-H\left(f/f_a\right)}\right|}}^2$$

$${\underset{\_}{\underset{\_}{A}}}_{f/f_a,\mu }=-H\left(f/f_a\right)\cdot e^{-j2\pi f/f_a\cdot \mu }\text{ f}\ddot{\text{u}}\text{r }\mu \text{=1}\text{,2}\text{,}\dots {\text{L}}_0-1$$

$${\underset{\_}{\underset{\_}{A}}}_{f/f_a,\xi }=e^{-j2\pi f/f_a\cdot \xi }\text{ f}\ddot{\text{u}}\text{r }\xi \text{=0}\text{,1}\text{,}\dots {\text{L}}_1-1$$

$${\underset{\_}{b}}_{f/f_a}=H\left(f/f_a\right)$$

$$\underset{\_}{\widehat{h}}=\left[\begin{array}{cccccccc}{\widehat{h}}_0\left(1\right)& {\widehat{h}}_0\left(2\right)& \dots & {\widehat{h}}_0\left(L_0-1\right)& {\widehat{h}}_0\left(0\right)& {\widehat{h}}_1\left(1\right)& \dots & {\widehat{h}}_1\left(L_1-1\right)\end{array}\right]$$

The cost function K according to equation (18) also results in a more general notation according to equation (19) with the coefficients in the left-hand columns of matrix A according to equation (20), the coefficients in the right-hand columns of matrix A according to equation (21) , the coefficient of the vector b according to equation (22) and the vector ĥ of the filter coefficients to be determined according to equation (23). $$K={\left[\underset{\_}{\underset{\_}{\mathrm{A.}}}\cdot \underset{\_}{\widehat{H}}-\underset{\_}{b}\right]}^H\cdot \left[\underset{\_}{\underset{\_}{\mathrm{A.}}}\cdot \underset{\_}{\widehat{H}}-\underset{\_}{b}\right]$$

$${\underset{\_}{\underset{\_}{\mathrm{A.}}}}_{f/f_a,\mu }=-H\left(f/f_a\right)\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \mu }\text{f}\ddot{\text{u}}\text{r}\mu \text{= 1}\text{, 2}\text{,}...{\text{L.}}_0-1$$

$${\underset{\_}{\underset{\_}{\mathrm{A.}}}}_{f/f_a,\xi }=e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \xi }\text{f}\ddot{\text{u}}\text{r}\xi \text{= 0}\text{,1}\text{,}...{\text{L.}}_1-1$$

$${\underset{\_}{b}}_{f/f_a}=H\left(f/f_a\right)$$

$$\underset{\_}{\widehat{H}}=\left[\begin{array}{cccccccc}{\widehat{H}}_0\left(1\right)& {\widehat{H}}_0\left(2\right)& ...& {\widehat{H}}_0\left({\mathrm{L.}}_0-1\right)& {\widehat{H}}_0\left(0\right)& {\widehat{H}}_1\left(1\right)& ...& {\widehat{H}}_1\left({\mathrm{L.}}_1-1\right)\end{array}\right]$$

$$\underset{\_}{\underset{\_}{\tilde{A}}}=Real\left\{{\underset{\_}{\underset{\_}{A}}}^H\cdot \underset{\_}{\underset{\_}{A}}\right\}$$

$$\underset{\_}{\tilde{b}}=-2\cdot Real\left\{{\underset{\_}{\underset{\_}{A}}}^H\cdot \underset{\_}{b}\right\}$$

The vector ĥ of the filter coefficients can be determined by differentiating the cost function K according to the vector ĥ and then setting it to zero and results from equation (24) with the matrix à according to equation (25) and the vector b̃ according to equation (26). $$\underset{\_}{\widehat{H}}=-\frac{1}{2}\cdot {\underset{\_}{\underset{\_}{\widetilde{\mathrm{A.}}}}}^{-1}\cdot \underset{\_}{\tilde{b}}$$

$$\underset{\_}{\underset{\_}{\widetilde{\mathrm{A.}}}}=\mathrm{R.}eal\left\{{\underset{\_}{\underset{\_}{\mathrm{A.}}}}^H\cdot \underset{\_}{\underset{\_}{\mathrm{A.}}}\right\}$$

$$\underset{\_}{\tilde{b}}=-2\cdot \mathrm{R.}eal\left\{{\underset{\_}{\underset{\_}{\mathrm{A.}}}}^H\cdot \underset{\_}{b}\right\}$$

If more than two analog-digital converters are connected in parallel, it makes sense to connect an integer power of two of analog-digital converters in parallel. In this case, in a first step, the partial compensation filters for two analog-digital converters are determined with their filter coefficients with regard to a compensation of mismatches between their transfer functions. One of the three above-described embodiments of a parameter optimization according to the invention can be used for this purpose.

In the next step, further partial compensation filters are determined to compensate for mismatches in the transfer function between two pairs of two analog-digital converter channels, whose mismatches in the transfer function can already be compensated for by means of a partial compensation filter optimized in the first iteration step.

_{For this purpose, the transfer functions H Equ1} (f / f _a ) and H _Equ2 (f / f _a ) of the associated partial compensation filter with the measured transfer functions already determined in the first step for the pair of analog-digital converter channels 1 and 2 H̃ ₁ (f / f _a ) and H̃ ₁ (f / f _a ) of the associated analog-to-digital converter channels are multiplied and added to each other and which are also used in the first step for the pair of analog-to-digital converter channels 3 and 4 determined transfer functions H _Equ3 (f / f _a ) and H _Equ4 (f / f _a ) of the associated partial compensation _{filter with the measured transfer functions H̃ 3} (f / f _a ) and H̃ ₄ (f / f _a ) of the associated analog -Digital converter channels are multiplied and added together and the ratio H (f / f _a ) between the two sums is calculated according to equation (27). $$H\left(f\text{/}{f}_a\right)=\frac{{\tilde{H}}_0\left(f\text{/}fa\right)\cdot H_{\mathrm{E.}qu0}\left(f\text{/}{f}_a\right)+{\tilde{H}}_1\left(f\text{/}fa\right)\cdot H_{\mathrm{E.}qu1}\left(f\text{/}{f}_a\right)}{{\tilde{H}}_2\left(f\text{/}fa\right)\cdot H_{\mathrm{E.}qu2}\left(f\text{/}{f}_a\right)+{\tilde{H}}_3\left(f\text{/}fa\right)\cdot H_{\mathrm{E.}qu3}\left(f\text{/}{f}_a\right)}$$

_{Using one of the three embodiments of a parameter optimization according} to the invention presented above, the filter coefficients of the partial compensation filters H Equ10 (f / f _a ) and H _Equ11 (f / f _a ) belonging to the two pairs of two analog-to-digital converters are determined.

$${\widehat{h}}_{Equ1}\left(k\right)={\widehat{h}}_{Equ01}\left(k\right)\ast {\widehat{h}}_{Equ10}\left(k\right)$$

$${\widehat{h}}_{Equ2}\left(k\right)={\widehat{h}}_{Equ02}\left(k\right)\ast {\widehat{h}}_{Equ11}\left(k\right)$$

$${\widehat{h}}_{Equ3}\left(k\right)={\widehat{h}}_{Equ03}\left(k\right)\ast {\widehat{h}}_{Equ11}\left(k\right)$$

By convolution of the filter coefficients ĥ _Equ00 (k), ĥ _Equ01 (k), ĥ _Equ02 (k), ĥ _Equ03 (k), ĥ _Equ10 (k) and ĥ _Equ11 (k) determined in the first and second iteration steps according to equation (28A) ) to (28D), the filter coefficients of the compensation filters associated with each of the four analog-digital converter channels can be determined. $${\widehat{H}}_{\mathrm{E.}qu0}\left(k\right)={\widehat{H}}_{\mathrm{E.}qu00}\left(k\right)\ast {\widehat{H}}_{\mathrm{E.}qu10}\left(k\right)$$

$${\widehat{H}}_{\mathrm{E.}qu1}\left(k\right)={\widehat{H}}_{\mathrm{E.}qu01}\left(k\right)\ast {\widehat{H}}_{\mathrm{E.}qu10}\left(k\right)$$

$${\widehat{H}}_{\mathrm{E.}qu2}\left(k\right)={\widehat{H}}_{\mathrm{E.}qu02}\left(k\right)\ast {\widehat{H}}_{\mathrm{E.}qu11}\left(k\right)$$

$${\widehat{H}}_{\mathrm{E.}qu3}\left(k\right)={\widehat{H}}_{\mathrm{E.}qu03}\left(k\right)\ast {\widehat{H}}_{\mathrm{E.}qu11}\left(k\right)$$

If more than four analog-to-digital converters are connected in parallel, then in the following iteration steps for two sub-groups of analog-to-digital converters each, which are compensated for mismatches in the transfer function by two partial compensation filters determined in the previous iteration step , the associated partial compensation filter determines the mismatches in the transfer function between the analog-to-digital converters of the two sub-groups.

For this purpose, a ratio between two sums is formed for the two sub-groups of analog-digital converters based on equation (27), each sum being formed for a sub-group of analog-digital converters. Each sum results from a summand for each individual analog-digital converter from the respective sub-group of analog-digital converters. Each summand is in turn determined from the weighting of the transfer function of the respective analog-digital converter channel with the transfer functions of all partial compensation filters determined in the previous iteration steps for the respective analog-digital converter.

On the basis of this ratio, the filter coefficients of two partial compensation filters are determined for each group of analog-digital converters using, for example, one of the three embodiments of a parameter optimization according to the invention presented above.

Since the number of filter coefficients - ie the filter length - of the individual compensation filters is increased due to the convolution compared to the filter length of the partial compensation filters, the filter length must be shortened. For this purpose, a cost function K is minimized by calculating the difference between the determined and Fourier-transformed filter coefficients ĥ _Equi (k) of a compensation filter i with a filter length that is too long and the transfer function H _Equi ^(Ref) (f / f _a ) of an associated compensation filter i with an optimal filter length according to equation (29) is minimized. $$K={\displaystyle \sum _{f\text{/}{f}_a}{\left|{\displaystyle \sum _{\xi =0}^{\mathrm{L.}-1}{\widehat{H}}_{\mathrm{E.}qui}\left(\xi \right)\cdot e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f\text{/}{f}_a\cdot \xi }}-H_{\mathrm{E.}qui}{}^{\left(\text{re}f\right)}\left(f\text{/}{f}_a\right)\right|}^2}$$

The cost function K according to equation (29) can in turn in a more general notation according to equation (30) with the matrix A _{f / f a , ξ} according to equation (30), with the vector

$${\underset{\_}{\underset{\_}{A}}}_{f/f_a,\xi }=e^{-j2\pi f/f_a\cdot \xi }$$

$${\underset{\_}{b}}_{f/f_a}=H_{Equi}{}^{(j)(Ref)}\left(f/f_a\right)$$

b _{f / fa} according to equation (31) and the determined vector ĥ _Equi with the filter coefficients of the compensation filter i. $$k={\left[\underset{\_}{\underset{\_}{\mathrm{A.}}}\cdot {\underset{\_}{\widehat{H}}}_{\mathrm{E.}qui}-\underset{\_}{b}\right]}^H\cdot \left[\underset{\_}{\underset{\_}{\mathrm{A.}}}\cdot {\underset{\_}{\widehat{H}}}_{\mathrm{E.}qui}-\underset{\_}{b}\right]$$

$${\underset{\_}{\underset{\_}{\mathrm{A.}}}}_{f/f_a,\xi }=e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="DE102014200914B4\_0049.png,DE102014200914B4\_0056.png"\; state="\{\{state\}\}">j</figure-callout>}2\pi f/f_a\cdot \xi }$$

$${\underset{\_}{b}}_{f/f_a}=H_{\mathrm{E.}qui}{}^{(j)(\mathrm{R.}ef)}\left(f/f_a\right)$$

$$\underset{\_}{\underset{\_}{\tilde{A}}}=Real\left\{{\underset{\_}{\underset{\_}{A}}}^H\cdot \underset{\_}{\underset{\_}{A}}\right\}$$

$$\widetilde{\underset{\_}{b}}=-2\cdot Real\left\{{\underset{\_}{\underset{\_}{A}}}^H\cdot \underset{\_}{b}\right\}$$

The vector ĥ _{Equi of} the filter coefficients can be determined by differentiating the cost function K according to the vector ĥ _Equi and then setting it to zero and results from equation (33) with the matrix à according to equation (34) and the vector b̃ according to equation (35). $${\widehat{\underset{\_}{H}}}_{\mathrm{E.}qui}{}^{\left(j\right)}=-\frac{1}{2}\cdot {\underset{\_}{\underset{\_}{\widetilde{\mathrm{A.}}}}}^{-1}\cdot \widetilde{\underset{\_}{b}}$$

$$\underset{\_}{\underset{\_}{\widetilde{\mathrm{A.}}}}=\mathrm{R.}eal\left\{{\underset{\_}{\underset{\_}{\mathrm{A.}}}}^H\cdot \underset{\_}{\underset{\_}{\mathrm{A.}}}\right\}$$

$$\widetilde{\underset{\_}{b}}=-2\cdot \mathrm{R.}eal\left\{{\underset{\_}{\underset{\_}{\mathrm{A.}}}}^H\cdot \underset{\_}{b}\right\}$$

In the following, the method according to the invention for compensating for mismatches in the transfer function between a plurality of analog-to-digital converters, each sampling with a time offset from one another, is described in detail, taking into account a first embodiment of a parameter optimization according to the invention, using the flowchart in FIG 2A and the block diagram in 1A presented.

The in the flowchart of the 2A The method according to the invention presented relates to the compensation of mismatches in the transfer function between two analog-to-digital converters that are sampled at a time offset from one another 1A .

In such a system according to the invention of several analog-to-digital converters and downstream compensation filters, an input signal x (t) to be measured in a digital oscilloscope is fed to _{an analog front end (input area) 1 with the transfer function H a} (f / f _a). Input amplification and filtering of the input signal x (t) to be measured typically take place in this front end 1.

The signal at the output of the front end 1 is fed in parallel to _{two analog-to-digital converters 2 0} and 2 _{1 which are sampled at different times.} In order to implement the temporally offset sampling between the two analog-digital converters 2 ₀ and 2 ₁ , the signal at the output of the front end 1 is fed to a delay element 6 _{before it is applied to the analog-digital converter 2 1,} which realizes a delay of the signal by half a sampling period T _a / 2.

Each analog-to-digital converter 2 ₀ and 2 ₁ consists of an in 1A sample and hold element, not shown, and a downstream level quantizer. The transmission behavior of the sample and hold element, the level quantizer and the connecting line connected in between can be approximated by a first-order delay element. The delaying transfer behavior of the first analog-digital converter 2 ₀ is described by the transfer function H ₀ (f / f _a ), while the delaying transfer behavior of the second analog-digital converter 2 _{1 is described} by the transfer function H ₁ (f / f _a ) according to 1A is modeled.

The stochastic transmission behavior in the entire analog-digital converter channel - for example the thermal amplifier noise in the front end 1 or the quantization noise in the level quantizer of the analog-digital converter 2 ₀ and 2 ₁ - is determined by the noise signal n ₀ (t / T _a ) in the case of the analog-to-digital converter channel 2 ₀ and by the noise signal n ₁ (t / T _a ) in the case of the analog-to-digital converter channel 2 ₁ , each by a summation element 3 ₀ and 3 ₁ are superimposed on the analog-digital converted signal.

In a subsequent compensation _{filter 4 0} or 4 ₁ with the transfer function H Equ0 (f / f _a ) or H _Equ1 (f / f _a ), there are mismatches between the transfer functions H ₀ (f / f _a ) and H ₁ (f / f _a ) of the first analog-digital converter channel 2 ₀ and the second analog-digital converter channel 2 ₁ compensated.

The output signals y ₀ (t / T _a ) and y ₁ (t / T _a ) of the two compensation filters 4 ₀ and 4 ₁ thus each have a sequence of sample values of the original input signal x (t) to be measured, the sample values of the output signal y ₁ {t / T _a ) the sample values of the input signal to be measured x (t) delayed by half a sampling period T _{a / 2 and the sample values of the output signal y 0} (t / T _a ) the non-delayed sample values of the to measuring input signal x (t) included. Since the sample and hold elements of the two analog-to-digital converters 2 ₀ and 2 ₁ each sample with the sampling period T _a , a sequence of Generated sampling values which follow one another in half the sampling period T _a / 2 and thus realize a doubling of the sampling frequency.

In the first method step S10 of the method according to the invention 2A the transfer functions H̃ ₀ (f / f _a ) and H̃ ₁ (f / f _a ) of the two analog-digital converter channels, which result from the transfer function H _a (f / f _a ) of the front end 1 and the transfer function H ₀ (f / f _a ) of the first analog-digital converter 2 ₀ or the transfer function H ₁ (f / f _a ) of the second analog-digital converter 2 ₁ , for several frequencies f / f _a with a suitable measuring instrument - for example a network analyzer - measured. The consideration of the transfer function of the front end 1 in the measurement of the transfer functions is based on the fact that uniform and accessible input and output measuring points are used for both analog-digital converter channels.

In addition, in the first method step S10, the ratio H (f / f _a ) between the measured transfer function H̃ ₀ (f / f _a ) of the first analog-to-digital converter channel and the measured transfer function H̃ ₁ (f / f _a ) of the second Analog-digital converter channel determined at the individual measurement _{frequencies f / f a.}

In the next method step S20, the left-hand and right-hand column of the matrix A is determined according to equations (8) and (9) from the _{ratios H (f / f a ) determined for the individual measurement frequencies f / f a} and the ratios H (f / f a) for the individual measurement frequencies f / f _a ^{complex variables e -j2πf / f} determined in each case ^{a · D} determined.

In addition, in the same method step S20, the elements of the vector b are determined in accordance with equation (10) from the ratios H (f / f _a ) determined in _{each case for the individual measurement frequencies f / f a.}

In the same method step S20, the elements of the matrix à according to equation (12) are determined from the elements of the matrix A and the elements of the vector b̃ according to equation (13) are determined from the elements of the matrix A and the vector b .

Using the matrix à determined in method step S20 and the vector b̃ determined in method step S20, the weighting factors ĝ _{for the two compensation filters 4 0} and 4 ₁ implemented as first-order digital filters with finite impulse response length are determined in accordance with equation (12). Conventional numerical methods for matrix inversion can be used for the necessary inversion of the matrix Ã.

In the final method step S40, the two analog-digital-converted signals in the individual analog-to-digital converters 2 ₁ and 2 ₂ are compensated for with respect to the mismatches in the transfer function in the respectively assigned compensation filter 4 ₁ and 4 _2. The compensated sample values of the two analog-digital converter channels are finally combined in a subsequent summation element 5.

While method steps S10 to S30 are carried out in advance in a calibration phase, method step S40 is carried out continuously while the signal x (t) is being measured with the aid of the digital oscilloscope.

In the 3A to 3D the transfer functions of analog-digital converter channels are shown, which have to be compensated for each other. The 3A shows the amount history and the 3B the phase curve of the transfer functions of the two analog-digital converter channels over the frequency for a delay D = 1, a weighting factor g ₀ = 0.01 of the first-order IIR filter characterizing the first analog-digital converter 2 _{0, a weighting factor} g ₁ = 0.014 of the first-order IIR filter characterizing the second analog-digital converter 2 ₁ and an ideally assumed signal-to-noise ratio of infinity.

From the 3C and 3D the absolute value curve and the phase curve of the relationship between the transfer functions of the first and second analog-to-digital converter channels result. It can be seen that with a given ratio of the transfer functions in the measurement frequency range shown, a signal-to-noise ratio of the order of magnitude of 50 dB can be achieved and compensation is therefore required.

In the 4A to 4D Disturbed transfer functions of analog-digital converter channels are shown, which have to be compensated. The 4A shows the amount history and the 4B the phase curve of the disturbed transfer functions of the two analog-digital converter channels and the associated undisturbed transfer functions of the two analog-digital converter channels with the same parameters for the delay D and the weighting factors g ₀ and g ₁ of the two analog-digital Converter 2 ₀ or 2 ₁ characterizing first-order IIIR filter as in the case of the 3A to 3D . Due to the disturbance of the transfer functions, a signal-to-noise ratio of 50 dB is assumed.

From the 4C and 4D the absolute value curve and the phase curve of the ratio between the disturbed transfer functions of the first and second analog-digital converter channels and the ratio between the associated undisturbed transfer functions of the first and second analog-digital converter channels result.

In the 5A and 5B the magnitude curves or the phase curves of the compensated transfer functions of the two analog-digital converter channels are shown, which were determined according to the first embodiment of the parameter optimization according to the invention. From the 5C and 5D the absolute value curve or the phase curve of the ratio between the compensated transfer functions of the two analog-digital converter channels emerge. It can be seen that the difference between the compensated transfer functions of the two analog-digital converter channels is less than 4 · 10 ^-4 dB.

In the following, the method according to the invention for compensating for mismatches in the transfer function between a plurality of analog-to-digital converters, each sampling with a time offset from one another, is described in detail, taking into account a second embodiment of a parameter optimization according to the invention, using the flow chart in 2 B and the block diagram in 1A presented.

In the first method step S100, the two weighting factors ĝ ₀ and ĝ _{1 of the compensation filters 4 0} and 4 ₁ implemented as FIR filters are determined by numerically minimizing the cost function K according to equation (16) with the aid of a numerical optimization method. The Coleman trust region method is the preferred minimization method. Alternatively, other numerical minimization methods can also be used and are also covered by the invention.

In the next method step S110, the two _{analog-digital-converted signals in the individual analog-digital converters 2 0} and 2 ₁ are compensated for with respect to the mismatches in the transfer function in the respectively assigned compensation filters 4 ₀ and 4 _1. The compensated sample values of the two analog-digital converter channels are finally combined in a subsequent summation element 5. While method step S100 is carried out in advance in a calibration phase, method step S110 is carried out continuously while measuring the signal x (t) with the aid of the digital oscilloscope.

In the 6A and 6B the magnitude curves or the phase curves of the compensated transfer functions of the two analog-digital converter channels are shown, which were determined according to the second embodiment of the parameter optimization according to the invention. The parameters used in the first embodiment of the parameter optimization according to the invention for the delay D, for the weighting factors ĝ ₀ and ĝ _{1 are} the first-order IIR filters characterizing the two analog-digital converters 2 ₀ and 2 _{1, and for the signal} - Use noise ratio.

From the 6C and 6D the absolute value curve and the phase curve of the relationship between the compensated transfer functions of the two analog-digital converter channels emerge. It can be seen that the difference in the compensated transfer functions of the analog-digital converter channels is less than 2 · 10 ^-4 dB. In the case of the second embodiment of the parameter optimization according to the invention, the compensation results are even improved compared to the first embodiment of the parameter optimization according to the invention if the magnitude and phase curves are included in the 5C and 5D with the amount and phase progression in the 6C and 6D compares.

In the following, the method according to the invention for compensating for mismatches in the transfer function between a plurality of analog-to-digital converters, each sampling with a time offset from one another, is described in detail, taking into account a third embodiment of a parameter optimization according to the invention, using the flowchart in FIG 2C and the block diagram in 1A presented.

In the first method step S200 of the method according to the invention in 2C the transfer functions H̃ ₀ (f / f _a ) or H _i (f / f _a ) of the two analog-digital converter channels, which result from the transfer function H _a (f / f _a ) of the front end 1 and the transfer function H ₀ (f / f _a ) of the first analog-digital converter 2 ₀ or the transfer function H ₁ (f / f _a ) of the second analog-digital converter 2 ₁ , for several frequencies f / f _a with a suitable measuring instrument - for example a network analyzer - measured.

In addition, in the first method step S200, the ratio H (f / f _a ) between the measured transfer function H _{des 0} (f / f _a ) of the first analog-digital converter channel and the measured transfer function H̃ ₁ (f / f _a ) of the second Analog-digital converter channel determined at the individual measurement _{frequencies f / f a.}

In the next method step S210, the left-hand and right-hand columns of the matrix A are determined according to equations (20) and (21) from the _{ratios H (f / f a ) determined for the individual measurement frequencies f / f a} and the ratios H (f / f a) for the individual measurement frequencies f / f _a ^{complex variables e -j2πf / f} determined in each case ^{a · Μ} or e ^{-j2πf / f a · Ξ} definitely.

In addition, in the same method step S210, the elements of the vector b are determined in accordance with equation (22) from the ratios H (f / f _a ) determined in _{each case for the individual measurement frequencies f / f a.}

In the same method step S210, the elements of the matrix à according to equation (25) are determined from the elements of the matrix A and the elements of the vector b̃ according to equation (26) from the elements of the matrix A and the vector b .

Using the matrix à determined in method step S210 and the vector b̃ determined in method step S210, the filter coefficients ĥ of the two compensation _{filters 4 0} and 4 ₁ , which are implemented as digital filters of order L ₀ and L ₁ with a finite impulse response length, are determined in the subsequent method step S220 , determined according to equation (24).

In the final step S220, the two analog-digital-converted signals in the individual analog-to-digital converters 2 ₀ and 2 ₁ are compensated for in the respective associated compensation filters 4 ₀ and 4 ₁ with regard to the mismatches in the transfer function. The compensated sample values of the two analog-digital converter channels are finally combined in a subsequent summation element 5.

While method steps S200 to S220 are carried out in advance in a calibration phase, method step S230 is carried out continuously while measuring the signal x (t) with the aid of a digital oscilloscope.

In the 7A and 7B the magnitude and phase curves of the compensated transfer functions of the two analog-digital converter channels for the third embodiment of the parameter optimization according to the invention are shown. The parameters used in the first embodiment of the parameter optimization according to the invention for the delay D, for the weighting factors g ₀ and g ₁ of the first-order IIR filters characterizing the two analog-to-digital converters 2 ₀ and 2 _{1, and for the signal} -Noise ratio used.

In addition, the filter lengths L ₀ and L _{1 of} the two compensation filters 4 ₀ and 4 ₁ are each set to two. From the 7C and 7D the absolute value and the phase profile of the relationship between the compensated transfer functions of the two analog-digital converter channels emerge. It can be seen that the difference in the compensated transfer functions of the analog-digital converter channels is less than 4 · 10 ^-4 dB. In the case of the third embodiment of the parameter optimization according to the invention, the compensation results are even better than the first and second embodiment of the parameter optimization according to the invention if the absolute value and the phase progression are in the 7C and 7D with the amount and the phase progression in the 6C and 6D or. 5C and 5D compares.

In the following, the method according to the invention for compensating for mismatches in the transfer function between a plurality of analog-to-digital converters, each sampling with a time offset from one another, taking into account a fourth embodiment of a parameter optimization according to the invention, is described in detail using the flowchart in FIG 2D and the block diagram in 1B presented.

The method according to the invention for compensating for mismatches in the transfer function between a plurality of analog-to-digital converters, each sampling offset in time, taking into account the fourth embodiment of a parameter optimization according to the invention, is the compensation of mismatches in the transfer function of more than two analogs sampling offset in time -Digital converters. In this case, an integer power of two is preferably always used by parallel analog-digital converters. Based on 1B the example of the case of four analog-to-digital converters that scan at different times is presented as an example.

The input signal x (t) to be measured with the digital oscilloscope is fed to an analog front end 1 for input amplification and filtering. The signal at the output of the analog front end 1 is immediately delayed to a first analog-digital converter 2 ₀ with the transfer function H ₀ (f / f _a ) by a quarter sampling period T _a / 4 to a second analog-to-digital converter 2 ₁ with the transfer function H ₁ (f / f _a ), delayed by half a sampling period T _a / 2 to a third analog-to-digital converter 2 ₂ with the transfer function H ₂ (f / f _a ) and delayed by three quarters of a sampling period 3 · T _a / 4 to a fourth analog-to-digital converter 2 ₃ with the transfer function H ₃ (f / f _a ).

The signal at the output of the front end 1 is delayed by a quarter sampling period T _a / 4 in a first delay unit 6 ₁ . The delay of the signal delayed by a quarter sampling period by a further quarter sampling period T _a / 4 takes place in a second delay unit 6 _{2 connected downstream of the first delay unit 6 1} , which delivers a signal delayed by half a sampling period T _{a / 2 at its output.} The delay of the signal delayed by half a sampling period by a further quarter sampling period T _a / 4 takes place in a third delay unit 6 _{3 connected downstream of the second delay unit 6 2} , which supplies a signal delayed by three quarters sampling period 3 · T _{a / 4 at its output} . In the individual analog-to-digital converters 2 ₀ , 2 ₁ , 2 ₂ and 2 ₃ , the signals supplied in each case are sampled _{with the sampling period T a.}

Stochastic transmission behavior in the individual analog-digital converter channels - thermal amplifier noise in the analog front end 1 or quantization noise in the level quantizers of the individual analog-digital converters 2 ₀ , 2 ₁ , 2 ₂ and 2 ₃ - are caused by the noise signal n ₀ (t / T _a ) in the case of the analog-to-digital converter channel 2 ₀ , through which the noise signal n ₁ (t / T _a ) in the case of the analog-to-digital converter channel 2 ₁ , through which the noise signal n ₂ (t / T _a ) in the case of the analog-to-digital converter channel 2 ₂ and by which the noise signal n ₃ (t / T _a ) in the case of the analog-to-digital converter channel 2 _{3 is} described, each by a Summation element 3 ₀ , 3 ₁ , 3 ₂ or 3 ₃ are superimposed on the analog-digital converted signal. In the subsequent compensation filter 4 ₀ , 4 ₁ , 4 ₂ and 4 ₃ with the respective transfer functions H _Equ0 (f / f _a ), H _Equ1 (f / f _a ), H _Equ1 (f / f _a ) and H _Equ3 (f / f _a ) mismatches in the transfer functions between the four analog-to-digital converters 2 ₀ , 2 ₁ , 2 ₂ and 2 _{3 are} compensated for.

The samples of the signals y ₀ (t / T _a ), y ₁ (t / T _a ), y ₂ (t / T _a ) and y ₃ (t / T _a ) at the outputs of the four compensation filters 4 ₀ , 4 ₁ , 4 ₂ and 4 ₃ are combined in a subsequent summation element 5 and result in a sequence of sample values y (t / (T _a / 4)) which, compared to the sequence of sample values at the outputs of the individual analog-to-digital converters 2 ₀ , 2 ₁ , 2 ₂ and 2 _{3 is} increased by the sampling rate of four.

In the first method step S300, the individual transfer functions H̃ ₀ (f / f _a ), H̃ ₁ (f / f _a ), ..., H̃ _{2 N -1} (f / f _a ) of the total of 2 ^N analog-digital converter channels, which result from the transfer function H _a (f / f _a ) of the front end 1 and the transfer function H ₀ (f / f _a ), H _i (f / f _a ), ..., H _{2 N -1} (f / f _a ) of the respective analog-digital converter 2 ₀ , 2 ₁ , ..., 2 _{N-1 ,} measured over several frequencies f / f a.

(für k = 1,...,N) im Fall der in den folgenden Verfahrensschritten jeweils angewendeten ersten, zweiten und dritten Ausführungsform einer erfindungsgemäßen Parameteroptimierung bestimmt.In addition, in method step S300, the ratio is determined for each pair of two analog-to-digital converters $H\left(f/f_a\right)=\frac{{\tilde{H}}_{2k}\left(f/f_a\right)}{{\tilde{H}}_{2k-1}\left(f/f_a\right)}$

(for k = 1, ..., N) in the case of the first, second and third embodiment of a parameter optimization according to the invention used in each case in the following method steps.

In the next method step S310, the elements in the left-hand column of the matrix A ⁽¹⁾ for the first iteration step using the ratio H (f / f _a ) and the complex variable e ^{-j2πf / f a · D} for all measurement frequencies f / f _a according to equation (8), the elements in the right-hand column of matrix A ⁽¹⁾ for the first iteration step based on the complex variables e ^{-j2πf / fa · D} for all measurement frequencies f / f _a accordingly Equation (9) and the elements of the vector b ⁽¹⁾ for the first iteration step are determined using the complex variables e ^{-j2πf / fa · D} for all measurement frequencies f / f _a according to equation (10).

Finally, in the case of the first embodiment of the parameter optimization according to the invention, the matrix à ^{(1) (1)} for the first iteration step using the calculated matrix A ⁽¹⁾ for the first iteration step according to equation ( 13) and the vector b̃ ⁽¹⁾ for the first iteration step based on the calculated matrix A ⁽¹⁾ for the first iteration step and the calculated vector b ⁽¹⁾ for the first iteration step according to equation (14).

When using the third embodiment of the parameter optimization according to the invention, the elements in the left-hand side become equivalent for each pair of two analog-digital converters Column matrix A ⁽¹⁾ for the first iteration step based on the ratio H (f / f _a ) and the complex variable e ^{-j2πf / f a · Μ} for all measurement frequencies f / f _a according to equation (20), the elements in the right-hand columns of matrix A ⁽¹⁾ for the first iteration step using the complex variables e ^{-j2πf / f a · Ξ} for all measurement frequencies f / f _a according to equation (21) and the elements of the vector b ⁽¹⁾ for the first iteration step based on the complex variable e ^{-j2πf / f a · D determined} for all measurement frequencies f / f _a according to equation (22).

Finally, in the case of the third embodiment of the parameter optimization according to the invention, the matrix à ⁽¹⁾ for the first iteration step based on the matrix A ⁽¹⁾ calculated for the first iteration step according to equation (25) and the vector b̃ ⁽¹⁾ for the first iteration step based on the matrix A ⁽¹⁾ calculated for the first iteration step and the vector b ⁽¹⁾ calculated for the first iteration step according to equation (26).

In the subsequent method step S320, the parameters ĝ _2k and ĝ _2k-1 for the two partial compensation filters determined in the first iteration step of each pair of two analog-to-digital converters when using the first embodiment of a parameter optimization according to the invention according to equation (12) using the each determined matrix à ⁽¹⁾ in the previous method step S310 and the vector b̃ ^{(1) also} determined in each case in method step S310.

When using the second embodiment of a parameter optimization according to the invention, the parameters ĝ _2k and ĝ _2k-1 for the two partial compensation filters determined in the first iteration step of each pair of two analog-to-digital converters after numerical minimization of the cost function K according to equation (16) using Use of a suitable numerical minimization method, preferably the Coleman "confidence interval method".

If the third embodiment of a parameter optimization according to the invention is used, the parameters _{ĥ 2k} ⁽¹⁾ and ĥ _2k-1 ⁽¹⁾ for the two partial compensation filters determined in the first iteration step for each pair of two analog-to-digital converters are calculated using the equation (24) is determined using the matrix à ⁽¹⁾ determined in each case in the previous method step S310 and the vector b̃ ^{(1) also} determined in each case in method step S310.

In the subsequent method step S330, based on equation (27), a ratio H (f / f _a ) is formed between two sums that each belong to a subgroup of analog-digital converters and contain those analog-digital converters whose mismatches are in of the transfer function can already be compensated for one another by means of a partial compensation filter determined in each case in the previous iteration steps.

Each sum consists of a number of summands corresponding to the number of analog-digital converters within the subgroup, with the respective summand multiplying the transfer function of the respective analog-digital converter channel with the transfer functions of all in the previous iteration steps for the respective analog -Digital converter channel each contains determined partial compensation filters.

In the subsequent method step S340, the elements in the left-hand columns of the matrix A ⁽¹⁾ are used for each pair of two subgroups of analog-digital converters when using an FIR filter of order L as compensation filter and corresponding application of the third embodiment of the parameter optimization according to the invention. for the i-th iteration step based on equation (20) using the complex variables e ^{-j2πf / f a · Μ} and the ratio H (f / f _a ) for all measurement frequencies f / f _a and the elements in the right-hand columns of the matrix A ⁽¹⁾ for the i-th iteration step based on equation (21) using the complex variable e ^{-j2πf / f a · Ξ determined} for all measurement frequencies f / f _a and the elements of the vector b ⁽ⁱ⁾ for the i-th iteration step based on equation (22) using the ratios H (f / f _a ) for all measurement frequencies f / f _a .

Alternatively, when using a FIR filter of the first order as a compensation filter and corresponding application of the first embodiment of the parameter optimization according to the invention, the elements in the left-hand and right-hand column of the matrix A ⁽¹⁾ based on equations (8) and (9) and the Elements of the vector b ⁽ⁱ ) can be determined based on equation (10).

Finally, in this method step S340, when using the third embodiment of the parameter optimization according to the invention, for each pair of two subgroups of analog-digital Converter determines the matrix à ⁽ⁱ⁾ based on equation (25) and the vector b̃ ⁽ⁱ⁾ based on equation (26).

In this method step S340, if the first embodiment of the parameter optimization according to the invention is used accordingly, the matrix à ⁽ⁱ⁾ based on equation (13) and the vector b̃ ⁽ⁱ⁾ based on equation (14) for the case of using an FIR Filters of the first order can be determined as a compensation filter.

In the next method step S350, when using an L-order FIR filter as a compensation filter and applying the third embodiment of the inventive parameter optimization _{accordingly, the parameters ĥ 2k} ⁽ⁱ⁾ and ĥ _2k-1 ⁽ⁱ⁾ for the part to be determined in the i-th iteration step -Compensation filter of the respective pair of subgroups of analog-to-digital converters determined on the basis of equation (24).

For the case of using a first-order FIR filter as a compensation filter and corresponding application of the first embodiment of the parameter optimization ^{according to the invention, the parameters i (i)} for the partial compensation filter to be determined in the i-th iteration step of the respective pair of partial groups of analog-digital -Converters determined on the basis of equation (12).

If a first-order FIR filter is used as a compensation filter and the second embodiment of the parameter optimization ^{according to the invention is used, the parameters ĝ (i)} for the partial compensation filters to be determined in the i-th iteration step of the respective pair of subgroups of analog-digital Converters determined by numerically minimizing the cost function K according to equation (16).

In the next method step S360, the parameters of the individual partial compensation filters determined in all previous iteration steps for the respective subgroup of the respective analog-digital converter are convolved with one another based on equations (28A) to (28D), for each analog-digital converter, in order to calculate the compensation filter of the respective analog-digital converter determined up to and including the i-th iteration step.

In the subsequent method step S370, it is determined whether all the analog-digital converters connected in parallel are compensated for with respect to mismatches in the transfer function.

If this is the case, then the compensation filter associated with each analog-digital converter is completely determined with regard to its parameters.

In this case, in a subsequent method step S390, the filter length of the compensation filter determined in each case for the individual analog-digital converter, the filter length of which is not optimized due to the convolution (s) in method step S360, while minimizing a cost function K according to equation (29) reduced. _{For this purpose, a reference transfer function H Equi} ^(Ref) (f / f _a ) with an optimized filter length must be specified for the compensation filter i to be determined for the respective analog-digital converter.

The parameters of the compensation filter to be determined for the individual analog-digital converter result from the elements of the matrix à , which can be determined according to equation (34) from the matrix à , and the vector b̃ der according to equation (35) from the matrix A and the vector b is to be calculated.

To determine the elements of the matrix A ^{, the complex variables e -j2πf / f} are given in equation (31) ^{a ξ to be determined} for all measurement frequencies f / f _a and all values ξ from 1 to the filter length L. To determine the elements of the vector b , the reference transfer function H _Equi ^(Rrf) (f / f _a ) with an optimized filter length for the compensation filter i and all measurement frequencies f / f _{a is} to be applied according to equation (32).

The parameters ĥ _Equi of the compensation filter i belonging to the respective analog-digital converter with an optimized filter length results from the elements of the matrix A and the elements of the vector b̃ according to equation (34).

In the final method step S400, an input signal that has been converted from analog to digital in the respective analog-to-digital converter 2 ₀ , 2 ₁ , ..., 2 _N-1 is continuously transmitted to the associated compensation filter 4 ₀ , 4 ₁ , ..., 4 _{N- 1 is} compensated for its mismatches in the transfer function to the transfer functions of the other analog-digital converters during the ongoing measurement with the digital oscilloscope.

If it is determined in step S370 that not all analog-to-digital converters have yet to be compensated for the mismatches in the transfer function, then in the subsequent step S380 the analog-to-digital converters that were previously compensated for mismatches in the transfer function become a new part -Group of analog-to-digital converters combined.

Subsequently, in method step S330, resumed iteratively for the next iteration step, a ratio H (f / f _a ) based on equation (27) is formed for two of the newly formed sub-groups of analog-to-digital converters.

From the 8A and 8B the magnitude and phase progression of the disturbed transfer function determined by measurement of four analog-to-digital converters connected in parallel can be seen. A delay D = 1, a weighting factor g ₀ = 0.01 of the first-order IIR filter characterizing the analog-digital converter 2 _{0 , a weighting factor g 1} = 0.0113 of the first-order filter characterizing the analog-digital converter 2 ₁ First order IIR filter, a weighting factor g ₂ = 0.0127 of the IIR filter characterizing the analog-digital converter 2 ₂ , a weighting factor g ₃ = 0.014 of the IIR filter characterizing the analog-digital converter 2 _{3 and a signal} -Noise ratio set at 50.

zwischen der gestörten Übertragungsfunktion bzw. der zugehörigen ungestörten Übertragungsfunktion des zweiten und ersten Analog-Digital-Wandlers, das Verhältnis $\frac{H_2}{H_0}$

zwischen der gestörten Ubertragungsfunktion bzw. der zugehörigen ungestörten Übertragungsfunktion des dritten und ersten Analog-Digital-Wandlers und das Verhältnisses $\frac{H_3}{H_0}$

zwischen der gestörten Übertragungsfunktion bzw. der zugehörigen ungestörten Übertragungsfunktion des vierten und ersten Analog-Digital-Wandlers dargestellt.In the 8C and 8D the relationship becomes to the same parameters $\frac{H_1}{{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_0}$

between the disturbed transfer function or the associated undisturbed transfer function of the second and first analog-to-digital converter, the ratio $\frac{H_2}{{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_0}$

between the disturbed transfer function or the associated undisturbed transfer function of the third and first analog-digital converter and the ratio $\frac{H_3}{{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_0}$

shown between the disturbed transfer function or the associated undisturbed transfer function of the fourth and first analog-to-digital converter.

der kompensierten Übertragungsfunktion zwischen dem zweiten und dem ersten Analog-Digital-Wandler-Kanal, des Verhältnisses $\frac{H_{c2}}{H_{c0}}$

der kompensierten Übertragungsfunktion zwischen dem dritten und dem ersten Analog-Digital-Wandler-Kanal und des Verhältnisses $\frac{H_{c3}}{H_{c0}}$

der kompensierten Übertragungsfunktion zwischen dem vierten und dem ersten Analog-Digital-Wandler-Kanal. Für alle Kompensationsfilter wird jeweils eine nicht-verkürzte Filterlänge von drei angenommen. Zu erkennen ist, dass der Betrags-Unterschied zwischen den kompensierten Übertragungsfunktionen zweier Analog-Digital-Wandler-Kanäle in der Größenordnung von -13 · 10^-4 dB deutlich geringer als der Betrags-Unterschied zwischen den nicht-kompensierten Übertragungsfunktionen zweier Analog-Digital-Wandler-Kanäle in der Größenordnung von -3 · 10^-2 dB ist.The 9A and 9B contain the absolute and phase progression of the relationship $\frac{H_{c1}}{{\mathrm{<figure-callout\; id="0"\; label="H\; c"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_c}$

the compensated transfer function between the second and the first analog-digital converter channel, the ratio $\frac{H_{c2}}{{\mathrm{<figure-callout\; id="0"\; label="H\; c"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_c}$

the compensated transfer function between the third and the first analog-digital converter channel and the ratio $\frac{H_{c3}}{{\mathrm{<figure-callout\; id="0"\; label="H\; c"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_c}$

the compensated transfer function between the fourth and the first analog-digital converter channel. A non-shortened filter length of three is assumed for all compensation filters. It can be seen that the difference in amount between the compensated transfer functions of two analog-digital converter channels in the order of magnitude of -13 · 10 ^-4 dB is significantly smaller than the difference in amount between the uncompensated transfer functions of two analog-digital Transducer channels is on the order of -3 · 10 ^-2 dB.

der kompensierten Übertragungsfunktion zwischen dem zweiten und dem ersten Analog-Digital-Wandler-Kanal, des Verhältnisses $\frac{H_{c2}}{H_{c0}}$

der kompensierten Übertragungsfunktion zwischen dem dritten und dem ersten Analog-Digital-Wandler-Kanal und des Verhältnisses $\frac{H_{c3}}{H_{c0}}$

der kompensierten Übertragungsfunktion zwischen dem vierten und dem ersten Analog-Digital-Wandler-Kanal. Zu erkennen ist, dass der Betrags-Unterschied von maximal -2 · 10^-3 im Fall der verkürzten Filterlänge von zwei deutlich höher ausfällt als der Betrags-Unterschied von maximal -13·10^-4 im Fall der nicht-verkürzten Filterlänge von drei.Assuming a shortened filter length of two for all compensation filters, the results in the 9C and 9D depicted amount and phase curves of the relationship $\frac{H_{c1}}{{\mathrm{<figure-callout\; id="0"\; label="H\; c"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_c}$

the compensated transfer function between the second and the first analog-digital converter channel, the ratio $\frac{H_{c2}}{{\mathrm{<figure-callout\; id="0"\; label="H\; c"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_c}$

the compensated transfer function between the third and the first analog-digital converter channel and the ratio $\frac{H_{c3}}{{\mathrm{<figure-callout\; id="0"\; label="H\; c"\; filenames="DE102014200914B4\_0054.png,DE102014200914B4\_0055.png"\; state="\{\{state\}\}">H</figure-callout>}}_c}$

the compensated transfer function between the fourth and the first analog-digital converter channel. It can be seen that the difference in amount of a maximum of -2 · 10 ^-3 in the case of the shortened filter length of two turns out to be significantly higher than the difference in amount of a maximum of -13 · 10 ^-4 in the case of the non-shortened filter length of three.

The invention is not limited to the illustrated embodiments. The invention also covers all combinations of all of the features claimed in the individual patent claims, of all of the features disclosed in the description and of all of the features shown in the figures of the drawing.

Claims (10)

Method to compensate for mismatches in the transfer function between several analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), which are sampled at different times from one another, by converting the signals from the respective analog-to-digital converter (2 ₀ , 2 _1, 2 _2, 2 _3, ...) samples generated with an impulse response for each of a respective analog-to-digital converter (2 _0, 2 _1, 2 _2, 2 _3, ...) corresponding compensation filter (4 _0, 4 ₁ , 4 ₂ , 4 ₃ , ...), with at least one filter coefficient being determined in advance by minimizing a cost function for at least two compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 _{3, ...)} In order to achieve a minimized difference in magnitude and / or phase between the transfer functions _{folded with the transfer function of the associated compensation filter (4 0} , 4 ₁ , 4 ₂ , 4 ₃ , ...) in each case in at least two analog-digital -Converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) to achieve where in the case of more than two analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) that are sampled at different times, the filter coefficients of the respective analog-to-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) associated compensation _{filter (4 0} , 4 ₁ , 4 ₂ , 4 ₃ , ...) from the convolution of the filter coefficients of several partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...), with a pair of first partial compensation filters to be determined (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) the mismatching of the transfer functions of two respective analog-to-digital converters ( 2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) are compensated for each other, and each additional pair of partial compensation filters to be determined (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) the mismatch of the Transfer function of a respective first subgroup of analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), which by means of already determined partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 _3,. ..) already knocked out each other are compensated, with regard to the transfer function of a respective second subgroup, with the same number of analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), which by means of already determined partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) are already compensated for each other, compensated for each other. Procedure according to Claim 1 , characterized in that, in order to minimize the cost function, the transfer function _{is measured in the respective two analog-to-digital converters (2 0} , 2 ₁ , 2 ₂ , 2 ₃ , ...) for several frequencies in each case. Procedure according to Claim 1 or 2 , characterized in that the transfer function of each analog-to-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) corresponds to the transfer function of a digital filter of first order with an infinite length of the impulse response. Method according to one of the Claims 1 to 3 , characterized in that, by minimizing the cost function, a difference, which is only minimized in terms of amount, between the transfer functions of at least two analogue filters, each folded _{with the transfer function of the associated compensation filter (4 0} , 4 ₁ , 4 ₂ , 4 _{3, ...)} Digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) is achieved. Method according to one of the Claims 1 to 3 , characterized in that the transfer function of the compensation filter (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) belonging to the respective analog-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 _{3, ...)} ) is the transfer function of a digital filter with a finite number of several filter coefficients. Method according to one of the Claims 1 to 5 , characterized in that the respective first and respective second subgroups each contain an integer power of two of analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...). Method according to one of the Claims 1 to 6th , characterized in that for the determination of each pair of further partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) a cost function is minimized in order to minimize the difference between the amount and / or the phase one for the respective first subgroup of analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) and one for the respective second subgroup of Analog-to-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) to achieve the corresponding sum, with each sum for each analog-to-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) of the respective first or respective second subgroup has an addend, each addend being the multiplication of the transfer function of the respective analog-to-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) with the transfer functions of all partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) already determined for the respective analog-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 _{3, ...)} ) contains. Method according to one of the Claims 1 to 7th , characterized in that the filter length of the respective compensation filter (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) is shortened by minimizing a cost function that is the difference between the determined and Fourier-transformed filter coefficients of the respective compensation filter (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) and a reference transfer function of the respective compensation _{filter (4 0} , 4 ₁ , 4 ₂ , 4 ₃ , ...) with a shortened filter length for several frequencies. System of several analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) and subsequent compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) to compensate for mismatches in the transfer function between the analog-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), the system being designed so that a pre-determination of at least one filter coefficient for at least two compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) by minimizing at least one cost function in order to achieve a difference, which is minimized in terms of amount and / or phase, between those with the transfer function of the associated compensation filter (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) in each case folded transfer functions takes place in at least two analog-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), the system being designed in this way is that in the case of more than two, each time staggered to one another of the scanning analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), the filter coefficients of the respective analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , .. .) compensation filter associated (4 _0, 4 _1, 4 _2, 4 _3, ...), ...) resulting from the convolution of the filter coefficients of plural sub-cancellation filters (4 _0, 4 _1, 4 _2, 4 _3, where in the system the mismatching of the transfer functions of two respective analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) by a pair of first partial compensation filters to be determined (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) are compensated to each other, and each additional pair of partial compensation filters to be determined (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) in the system the mismatch of the transfer function of a respective first subgroup of analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), which by means of already determined partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) are already compensated for each other, with respect to the transfer function of a respective second subgroup with the same number of analog-to-digital converters (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...), which by means of already determined partial compensation filters (4 ₀ , 4 ₁ , 4 ₂ , 4 ₃ , ...) are already compensated for each other, compensated for each other. System according to Claim 9 , characterized in that the transfer function of each analog-to-digital converter (2 ₀ , 2 ₁ , 2 ₂ , 2 ₃ , ...) corresponds to the transfer function of a digital filter of first order with an infinite length of the impulse response.

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