Phase-shifted sampling module and method for determining filter coefficients

Abstract

A phase-shifted sampling module for sampling a signal is described. The phase-shifted sampling module includes a primary sampler module, an ADC module, and an equalization module. The primary sampler module includes an analog signal input, a first signal path, and a second signal path. The equalization module includes a primary sampler equalizer sub-module. The primary sampler equalizer sub-module is configured to compensate low-frequency mismatches between the first signal path and the second signal path. Further, a method for determining filter coefficients of an equalization module of a phase-shifted sampling module is described.

Claims

1. A phase-shifted sampling module for sampling a signal, the phase-shifted sampling module comprising: a primary sampler circuit, an ADC circuit, and an equalization circuit, the primary sampler circuit comprising an analog signal input, a first signal path, and a second signal path, wherein the first signal path comprises a first analog sampling unit configured to sample an analog input signal received by the analog signal input, thereby generating a first pre-sampled analog input signal, wherein the second signal path comprises a second analog sampling unit configured to sample the analog input signal in a phase-shifted manner compared to the first analog sampling unit, thereby generating a second pre-sampled analog input signal, wherein the ADC circuit comprises a first analog-to-digital converter associated with the first signal path, wherein the first analog-to-digital converter is configured to sample the first pre-sampled analog input signal, thereby generating a first digital output signal, wherein the ADC circuit comprises a second analog-to-digital converter associated with the second signal path, wherein the second analog-to-digital converter is configured to sample the second pre-sampled analog input signal in a phase-shifted manner compared to the first analog-to-digital converter, thereby generating a second digital output signal, and wherein the equalization circuit comprises a primary sampler equalizer sub-circuit, wherein the primary sampler equalizer sub-circuit is configured to compensate low-frequency mismatches between the first signal path and the second signal path based on the digital output signals.

2. The phase-shifted sampling module according to claim 1, wherein the primary sampler equalizer sub-circuit is established as a time variant filter.

3. The phase-shifted sampling module according to claim 2, wherein the primary sampler equalizer sub-circuit comprises at least two inputs and at least two outputs, wherein the primary sampler equalizer sub-circuit comprises filter units connecting each input with each output.

4. The phase-shifted sampling module according to claim 3, wherein the filter units each are established as a multi-rate filter circuit.

5. The phase-shifted sampling module according to claim 4, wherein the multi-rate filter circuit comprises at least one of a delay element, a decimator, a digital correction filter, and an interpolator.

6. The phase-shifted sampling module according to claim 1, wherein the equalization circuit comprises at least one of an ADC equalizer sub-circuit and an overall equalizer subcircuit, wherein the ADC equalizer sub-circuit is configured to compensate at least one of low-frequency mismatches within the ADC circuit and low-frequency mismatches in a connection path between the primary sampler circuit and the ADC circuit, and wherein the overall equalizer sub-circuit is configured to compensate low-frequency mismatches originating upstream of the primary sampler circuit.

7. The phase-shifted sampling module according to claim 6, wherein the ADC equalizer sub-circuit is provided upstream of the primary sampler equalizer sub-circuit.

8. The phase-shifted sampling module according to claim 7, wherein the ADC equalizer sub-circuit is provided immediately upstream of the primary sampler equalizer sub-circuit.

9. The phase-shifted sampling module according to claim 6, wherein the overall equalizer sub-circuit is provided downstream of the primary sampler equalizer sub-circuit.

10. The phase-shifted sampling module according to claim 6, wherein filter coefficients of the equalizer sub-circuits are obtained by a variation of a global cost functional, wherein the global cost functional depends on the filter coefficients of the individual equalizer sub-circuits.

11. The phase-shifted sampling module according to claim 10, wherein starting coefficients for the variation of the global cost functional are obtained by local variations of local cost functionals being associated with the individual equalizer sub-circuits.

12. The phase-shifted sampling module according to claim 10, wherein the variation is performed by a least squares technique.

13. The phase-shifted sampling module according to claim 1, wherein the phase-shifted sampling circuit comprises an output selector circuit configured to selectively output a signal associated with the first signal path or with the second signal path.

14. The phase-shifted sampling module according to claim 13, wherein the output selector circuit is arranged downstream of the primary sampler equalizer sub-circuit.

15. The phase-shifted sampling module according to claim 14, wherein the output selector circuit is arranged immediately downstream of the primary sampler equalizer sub-circuit.

16. The phase-shifted sampling module according to claim 1, wherein the analog sampling units are established as sample & hold units, as track & hold units, or as return-to-zero sampling units.

17. A measurement instrument, the measurement instrument comprising a phase-shifted sampling module according to claim 1.

18. The measurement instrument of claim 17, wherein the measurement instrument is established as an oscilloscope.

19. A method for determining filter coefficients of an equalization circuit of a phase-shifted sampling module according to claim 1, wherein the equalization circuit comprises at least one of an ADC equalizer sub-circuit configured to compensate low-frequency mismatches within the ADC circuit, and an overall equalizer sub-circuit configured to compensate low-frequency mismatches originating upstream of the primary sampler circuit, the method comprising: receiving an input signal; processing the input signal by the primary sampler circuit and the ADC circuit, thereby obtaining at least a first digital output signal and a second digital output signal; processing the first digital output signal and the second digital output signal by the equalization circuit, thereby obtaining an equalized output signal; providing a global cost functional that depends on filter coefficients of the individual equalizer sub-circuits, the input signal, and the equalized output signal; and determining the filter coefficients of the equalizer sub-circuits by a global variation of the global cost functional.

20. The method according to claim 19, wherein starting coefficients for the variation of the global cost functional are obtained by local variations of local cost functionals associated with the individual equalizer sub-circuits.

21. The method according to claim 19, wherein the variation is performed by a least squares technique.

22. The method according to claim 19, wherein the global cost functional corresponds to an error measure between an ideal output signal of the equalizer sub-circuits and an actual output signal of the equalizer sub-circuits.

Description

DESCRIPTION OF THE DRAWINGS

(1) The foregoing aspects and many of the attendant advantages of the claimed subject matter will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:

(2) FIG. 1 schematically shows a phase-shifted sampling module according to an embodiment of the present disclosure;

(3) FIG. 2 schematically shows a multi-rate filter circuit of the phase-shifted sampling module of FIG. 1;

(4) FIG. 3 shows a flow chart of an embodiment of a phase-shifted sampling method according to the present disclosure; and

(5) FIG. 4 shows a flow chart of a method for determining filter coefficients according to an embodiment of the present disclosure.

DETAILED DESCRIPTION

(6) The detailed description set forth below in connection with the appended drawings, where like numerals reference like elements, is intended as a description of various embodiments of the disclosed subject matter and is not intended to represent the only embodiments. Each embodiment described in this disclosure is provided merely as an example or illustration and should not be construed as preferred or advantageous over other embodiments. The illustrative examples provided herein are not intended to be exhaustive or to limit the claimed subject matter to the precise forms disclosed.

(7) FIG. 1 schematically shows a measurement instrument 8 comprising a phase-shifted sampling module 10. In some embodiments, the measurement instrument 8 may be established as an oscilloscope. For example, the measurement instrument 8 may be established as a digital oscilloscope. However, it is to be understood that the phase-shifted sampling module 10 described below may also be used in another type of electronic device.

(8) The phase-shifted sampling module 10 comprises a primary sampler module 12, an ADC module 14, an equalization module 16, and an output selector module 17.

(9) In some embodiments, the term “module” or “unit” refers to a combination of hardware (e.g. a processor such as an integrated circuit or other circuitry) and software (e.g. machine- or processor-executable instructions, commands, or code such as firmware, programming, or object code). Furthermore, a combination of hardware and software may include hardware only (i.e. a hardware element with no software elements), software hosted at hardware (e.g. software that is stored at a memory and executed or interpreted at a processor), or hardware with the software hosted thereon. In some embodiments, the hardware may, inter alia, comprise a CPU, a GPU, an FPGA, an ASIC, or other types of electronic circuitry.

(10) In the shown embodiment, the primary sampler module 12 comprises an analog signal input 18, a first signal path 20, and a second signal path 22. In general, the primary sampler module 12 may have M signal paths, wherein M is an integer bigger than or equal to 2.

(11) The first signal path 20 comprises a first analog sampling unit 24 including circuitry configured to sample an analog input signal received by the analog signal input 18. The second signal path 22 comprises a second analog sampling unit 26 including circuitry configured to sample the analog input signal in a phase-shifted manner compared to the first analog sampling unit 24. The analog sampling units 24, 26 are each established as a sample & hold circuit, as a track & hold circuit, or as a return-to-zero sampling circuit.

(12) In the shown embodiment, the ADC module 14 comprises a first analog-to-digital converter 28 associated with the first signal path 20 and a second analog-to-digital converter 30 associated with the second signal path 22 of the primary sampler module 12. In general, the ADC module 14 may comprise M analog-to-digital converters, wherein each of the analog-to-digital converters is associated with one of the M signal paths of the primary sampler module 12. Hence, each signal path may be associated with a dedicated analog-to-digital converter. Without restriction of generality, the case M=2 is described in the following.

(13) The equalization module 16 comprises an ADC equalizer sub-module 32, a primary sampler equalizer sub-module 34, and an overall equalizer sub-module 36. Each of these sub-modules, or other sub-modules disclosed herein, may include one or more circuits.

(14) The ADC equalizer sub-module 32 is arranged immediately downstream of the ADC module 14. The ADC equalizer sub-module 32 comprises a first ADC filter unit 38 being associated with the first signal path 20, and a second ADC filter unit 40 being associated with the second signal path 22. The first ADC filter unit 38 and the second ADC filter unit 40 may each be established as a time invariant filter, for example as a multi-rate filter circuit. While the individual ADC filter units 38, 40 may be established as time invariant filters, the ADC equalizer sub-module 32 as a whole may be configured as a time variant filter, such that time variant perturbations can be removed by the ADC equalizer sub-module 32.

(15) The primary sampler equalizer sub-module 34 is arranged immediately downstream of the ADC equalizer sub-module 32. The primary sampler equalizer sub-module 34 comprises a first input 42, a second input 44, a first output 46, and a second output 48.

(16) The first input 42 is connected with the first output 46 via a first filter unit 50. Moreover, the first input 42 is connected with the second output 48 via a second filter unit 52. The second input 44 is connected with the first output 46 via a third filter unit 54. Moreover, the second input 44 is connected with the second output 48 via a fourth filter unit 56.

(17) In general, the primary sampler equalizer sub-module 34 may be established as a time variant filter. The individual filter units 50, 52, 54, 56 may be established as multi-rate filter circuits.

(18) The output selector module 17 is arranged immediately downstream of the primary sampler equalizer sub-module 34. The overall equalizer sub-module 36 is arranged immediately downstream of the output selector module 17. The overall equalizer sub-module 36 may be established as a time invariant filter, for example as a multi-rate filter circuit.

(19) FIG. 2 schematically shows a multi-rate filter circuit 58 that may be used in the ADC equalizer sub-module 32, in the primary sampler equalizer sub-module 34 and/or in the overall equalizer sub-module 36. As can be seen in the block diagram of FIG. 2, the multi-rate filter circuit 58 is established as a low frequency, LF, correction circuit (LFCCI) that can be used for the correction of low frequency disturbances within a digital signal applied to a signal input terminal 60 of the multi-rate filter circuit 58.

(20) In the embodiment shown, the multi-rate filter circuit 58 comprises a main signal path 62 and at least one correction signal path 64. The main signal path 62 and the correction signal path 64 are implemented in parallel within the multi-rate filter circuit 58. The main signal path 62 of the multi-rate filter circuit 58 is provided to delay the received digital input signal with a predetermined delay time. In a possible embodiment, the main signal path 62 comprises a delay unit 66 adapted to delay the received digital input signal with a predetermined delay time. The correction signal path 64 comprises at least one digital correction filter 68 adapted to filter the received digital input signal. In an embodiment, the main signal path 62 may comprise a long signal line having a length adapted to the required delay time.

(21) The multi-rate filter circuit 58 further comprises an adder 70 adapted to add the digital signal delayed by the main signal path 62 and the digital signal corrected by the correction signal path 64 to generate a digital output signal output by a signal output terminal 72 of the multi-rate filter circuit 58.

(22) More details on the structure of examples of the multi-rate filter circuit 58 are given in co-pending US patent application US 2020 0 295 850 A1, which is hereby incorporated by reference in its entirety.

(23) Embodiments of the phase-shifted sampling module 10 are configured to perform a phase-shifted sampling method that is described in the following with reference to FIG. 3. In some embodiments, the phase-shifted sampling module 10 includes program code, instructions, scripts, etc. (“a program element”), which is configured and arranged when executed on a computer or suitable circuitry (e.g., microprocessor, DSP, ASIC, SoC, etc.,) to perform one or more steps of the method set forth in FIG. 3 or in the subsequent claims.

(24) As shown in FIG. 3, an analog input signal x(t) is received by the analog signal input 18 (step S1). The analog input signal is forwarded to the first sampling unit 24 and to the second sampling unit 26 via the first signal path 20 and the second signal path 22, respectively.

(25) Due to thermal effects, the two signal paths 20, 22 may not be completely identical, and thus may have transfer functions that are not completely identical in a low-frequency range, i.e. at frequencies below 200 MHz, for example below 150 MHz, in particular below 100 MHz. This thermal effect is also known as “droop-effect”.

(26) Thus, the first sampling unit 24 receives an analog input signal x.sub.0(t) due to the transfer function P.sub.0(j2πƒ) of the first signal path 20. The second sampling unit 26 receives an analog input signal x.sub.1(t) due to the transfer function P.sub.1(j2πƒ) of the second signal path 22, wherein in general x.sub.1(t)≠x.sub.0(t) due to P.sub.1(j2πƒ)≠P.sub.0(j2πƒ).

(27) Accordingly, there may be a low-frequency mismatch between the first signal path 20 and the second signal path 22 within the primary sampler module 12.

(28) The analog input signals x.sub.0(t), x.sub.1(t) are sampled by the sampling units 24, 26, thereby generating a first pre-sampled analog input signal x.sub.*,0(t) and a second pre-sampled analog input signal x.sub.*,1(t), respectively (step S2).

(29) The sampling units 24, 26 each operate with a sampling frequency ƒ.sub.S=1/(2T). However, the first sampling unit 24 samples the analog input signal x(t) in a phase-shifted manner compared to the second sampling unit 26.

(30) In some embodiments, the first sampling unit 24 samples the analog input signal x.sub.0(t) at sample times t=2kT, while the second sampling unit 26 samples the analog input signal x.sub.1(t) at sample times t=(2k+1)T, wherein, k is an integer.

(31) In other words, the two sampling units 24 sample the input signal x(t) alternately.

(32) The first pre-sampled analog input signal x.sub.*,0(t) and the second pre-sampled analog input signal x.sub.*,1(t) are forwarded to the first ADC 28 and to the second ADC 30, respectively.

(33) In general, a connection path between the first sampling unit 24 and the first ADC 28 may be different from the connection path between the second sampling unit 26 and the second ADC 30. In some embodiments, the droop-effect explained above may also occur between the connection paths. Accordingly, the two connection paths may have different transfer functions.

(34) Therein, the term “connection path” is understood to comprise all components downstream of the primary sampler module 12 and upstream of the ADC module 14, e.g. connecting wires.

(35) Moreover, signal paths within the ADC module 14 may also have transfer functions that are different from each other due to the droop-effect explained above. Thus, additional low-frequency mismatches may occur.

(36) In FIG. 1, the overall effect of the connection paths and the signal paths within the ADC module 14 is summarized in the transfer functions Q.sub.0(j2πƒ) and Q.sub.1(j2πƒ). As a result of the transfer function Q.sub.0(j2πƒ), the first ADC 28 receives a modified version v.sub.0(t) of the first pre-sampled input signal x.sub.*,0(t). Similarly, the second ADC 30 receives a modified version v.sub.1(t) of the second pre-sampled input signal x.sub.*,1(t) due to the transfer function Q.sub.1(j2πƒ).

(37) The ADCs 28, 30 digitize the pre-sampled analog input signals v.sub.0(t), v.sub.1(t), thereby obtaining a first digital output signal y.sub.0(k) and a second digital output signal y.sub.1(k), respectively (step S3). The ADCs 28, 30 each operate with a sampling frequency ƒ.sub.S=1/(2T), for example wherein the sampling frequency of the ADCs 28, 30 matches the sampling frequency of the sampling units 24, 26.

(38) However, the first ADC 28 samples the first pre-sampled analog input signal v.sub.0(t) in a phase-shifted manner compared to the second ADC 30 sampling the second pre-sampled analog input signal v.sub.1(t). In some embodiments, first ADC 28 samples the first pre-sampled analog input signal v.sub.0(t) at sample times t=2kT+ΔT, while the second ADC 30 samples the second pre-sampled analog input signal v.sub.1(t) at sample times t=(2k+1)T+ΔT, wherein, k is an integer and ΔT is a time shift accounting for the propagation time of the signals from the sampling units 24, 26 to the ADCs 28, 30.

(39) The digital output signals y.sub.0(k), y.sub.1(k) are forwarded to the equalization module 16, for example to the ADC equalizer sub-module 32.

(40) Low-frequency mismatches, i.e. mismatches between the transfer functions Q.sub.0(j2πƒ), Q.sub.1(j2πƒ) in the low-frequency range are compensated by the ADC equalizer sub-module 32 (step S4). Thus, the ADC equalizer sub-module 32 compensates low-frequency mismatches originating within the ADC module 14 and in a connection path between the primary sampler module 12 and the ADC module 14.

(41) For example, the first ADC filter unit 38 filters the first digital output signal y.sub.0(k), thereby generating a first filtered digital output signal {tilde over (v)}.sub.0(k). The second ADC filter unit 40 filters the second ADC filter unit 40 filters the second digital output signal y.sub.1(k), thereby generating a second filtered digital output signal {tilde over (v)}.sub.1(k).

(42) Therein, transfer functions MRF.sub.0(z) and MRF.sub.1(z) of the ADC filter units 38, 40 are chosen such that the low-frequency mismatches between the transfer functions Q.sub.0(j2πƒ), Q.sub.1(j2πƒ) are compensated. In other words, filter coefficients of the ADC equalizer sub-module 32 are chosen such that the low-frequency mismatches between the transfer functions Q.sub.0(j2πƒ), Q.sub.1(j2πƒ) are compensated.

(43) Accordingly, in the case of perfect reconstruction, it holds {tilde over (v)}.sub.0(k)=v.sub.0(t) for t=2kT+ΔT and {tilde over (v)}.sub.1(k)=v.sub.1(t) for t=(2k+1)T+ΔT. This will be explained in more detail below.

(44) The filtered digital output signals {tilde over (v)}.sub.0(k), {tilde over (v)}.sub.1(k) are forwarded to the first input 42 and to the second input 44 of the primary sampler equalizer sub-module 34, respectively.

(45) Low-frequency mismatches between the transfer functions P.sub.0(j2πƒ), P.sub.1(j2πƒ) are compensated by the primary sampler equalizer sub-module 34 (step S5). Thus, the primary sampler equalizer sub-module 34 compensates low-frequency mismatches originating within the primary sampler module 12.

(46) Therein, transfer functions MRF.sub.00(z), MRF.sub.01(z), MRF.sub.10(z), and MRF.sub.11(z) of the filter units 50, 52, 55, 56 are chosen such that the low-frequency mismatches between the transfer functions P.sub.0(j2πƒ), P.sub.1(j2πƒ) are compensated. In other words, filter coefficients of the primary sampler equalizer sub-module 34 are chosen such that the low-frequency mismatches between the transfer functions P.sub.0(j2πƒ), P.sub.1(j2πƒ) are compensated.

(47) For example, the first filter unit 50 filters the first filtered digital output signal {tilde over (v)}.sub.0(k), thereby generating a first intermediate signal v.sub.00(k). The second filter unit 52 filters the first filtered digital output signal {tilde over (v)}.sub.0(k), thereby generating a second intermediate signal v.sub.01(k). The third filter unit 54 filters the second filtered digital output signal {tilde over (v)}.sub.1(k), thereby generating a third intermediate signal v.sub.10(k). The fourth filter unit 56 filters the second filtered digital output signal {tilde over (v)}.sub.1(k), thereby generating a fourth intermediate signal v.sub.11(k).

(48) The first intermediate signal v.sub.00(k) and the third intermediate signal v.sub.10(k) are superposed, thereby generating a first compensated digital output signal {tilde over (v)}.sub.0(k) being associated with the first output 46. The second intermediate signal v.sub.01(k) and the fourth intermediate signal v.sub.11(k) are superposed, thereby generating a second compensated digital output signal {tilde over (x)}.sub.1(k) being associated with the second output 48.

(49) Accordingly, in the case of perfect reconstruction, it holds {tilde over (x)}.sub.0(k)=x.sub.0(t) for t=2k T and {tilde over (x)}.sub.1(k)=x.sub.1(t) for t=(2k+1)T. This will be explained in more detail below.

(50) The compensated digital output signals {tilde over (x)}.sub.0(k), {tilde over (x)}.sub.1(k) are forwarded to the output selector module 17. The compensated digital output signals {tilde over (x)}.sub.0(k), {tilde over (x)}.sub.1(k) are recombined by the output selector module 17 (step S6). For example, the output selector module 17 selectively outputs a signal associated with the first signal path 20, i.e. the first compensated digital output signal {tilde over (x)}.sub.0(k), and a signal associated with the second signal path 22, i.e. the second compensated digital output signal {tilde over (x)}.sub.1(k), in an alternating manner. The resulting digital output signal x(k) is sampled at a higher sampling rate than achievable by a single ADC.

(51) The digital output signal x(k) is forwarded to the overall equalizer sub-module 36. The digital output signal x(k) is filtered by the overall equalizer sub-module 36, thereby compensating low-frequency mismatches originating upstream of the primary sampler module 12 (step S7).

(52) As a result, a compensated digital output signal {tilde over (x)}(k) with reduced perturbations due to low-frequency mismatches is obtained.

(53) In some embodiments, time variant perturbations due to time variant low-frequency mismatches can be reduced or even removed, such that the compensated digital output signal {tilde over (x)}(k) bears reduced or no time variant perturbations due to low-frequency mismatches.

(54) In the following, a method for determining filter coefficients of the equalization module 16 of the phase-shifted sampling module 10 is described with respect to FIG. 4. In some embodiments, one or more components described herein includes a program element (e.g., program code, instructions, scripts, etc., which is configured and arranged when executed on a computer or suitable circuitry (e.g., microprocessor, DSP, ASIC, SoC, etc.,) to perform one or more steps of the method set forth in FIG. 4 or in the subsequent claims.

(55) The input signal x(t) is received by the analog signal input 18 (step F1).

(56) The input signal x(t) is processed by the primary sampler module 12 and the ADC module 14 (cf. steps S2 and S3 described above), thereby obtaining at least a first digital output signal y.sub.0(k) and a second digital output signal y.sub.1(k) (step F2).

(57) The first digital output signal y.sub.0(k) and the second digital output signal y.sub.1(k) are processed by the equalization module 16 (cf. steps S4 to S7 described above), thereby obtaining an equalized output signal {tilde over (x)}(k) (step F3).

(58) A global cost functional is provided, wherein the global cost functional depends on filter coefficients of the individual equalizer sub-modules 32, 34, 36, the input signal y(n)=(y.sub.0(n),y.sub.1(n)) of the equalization module 16, and the equalized output signal {tilde over (x)}(k) (step F4).

(59) In general, the global cost functional has the following form:

(60) $Q=\underset{n=0}{\overset{N-1}{.Math.}}{\left(\tilde{x}\left(n\right)-{\tilde{x}}_{\mathrm{ideal}}\left(n\right)\right)}^2.$

(61) Therein, the dependence of the global cost functional on the filter coefficients is implicit, as the equalized output signal is a function {tilde over (x)}(n)=ƒ.sub.ges(b.sub.ges,y,n) of a filter coefficient vector b.sub.ges and of the input signal vector y. Moreover, N is the signal length (i.e. the number of samples) over which the cost functional is evaluated.

(62) {tilde over (x)}.sub.ideal is an ideal reference signal, i.e. an ideal reconstructed signal.

(63) Filter coefficients of the equalizer sub-modules 32, 34, 36 are determined by a global variation of the global cost functional (step F5).

(64) In some embodiments, the global cost functional is minimized in order to find the filter coefficients of the individual equalizer sub-modules 32, 34, 36, i.e.

(65) $\underset{b}{\min }\left\{\underset{n=0}{\overset{N-1}{.Math.}}{\left(f_{ges}\left(b_{\mathrm{ges}},y,n\right)-{\tilde{x}}_{\mathrm{ideal}}\left(n\right)\right)}^2\right\}.$

(66) Alternatively, the maximal error may be minimized, i.e.

(67) $\underset{b}{\min }\left\{\underset{n}{\max }\left\{f_{ges}\left(b_{\mathrm{ges}},y,n\right)-{\tilde{x}}_{\mathrm{ideal}}\left(n\right)\right\}\right\}.$

(68) Therein, the starting coefficients for the variation of the global cost functional Q are obtained by local variations of local cost functionals being associated with the individual equalizer sub-modules 32, 34, 36.

(69) Accordingly, three different local cost functionals Q.sub.ADCE, Q.sub.PSE, and Q.sub.OSE are provided for the ADC equalizer sub-module 32, the primary sampler equalizer sub-module 34, and the overall equalizer sub-module 36, respectively.

(70) The local cost functionals have the following general form:

(71) $Q_X=\underset{n=0}{\overset{N-1}{.Math.}}{\left(x_X\left(n\right)-x_{X,\mathrm{ref}}\left(n\right)\right)}^2,$

(72) wherein X can be ACDE, PSE, or OSE. Moreover, x.sub.X(n) is the actual output signal of the respective equalizer sub-module 32, 34, 36, and x.sub.X,ref(n) is the corresponding ideal output signal of the respective equalizer sub-module 32, 34, 36, i.e. the corresponding reference signal.

(73) In order to obtain the starting coefficients for the global optimization, the local cost functionals Q.sub.X are minimized, i.e.

(74) $\underset{b}{\min }\left\{\underset{n=0}{\overset{N-1}{.Math.}}{\left(x_X\left(n\right)-x_{X,\mathrm{ref}}\left(n\right)\right)}^2\right\}.$

(75) Alternatively, the maximal error may be minimized, i.e.

(76) $\underset{b}{\min }\left\{\underset{n}{\max }\left\{x_X\left(n\right)-x_{X,\mathrm{ref}}\left(n\right)\right\}\right\}.$

(77) In other words, a first optimization is performed for the equalizer sub-modules 32, 34, 36 individually.

(78) The results of this first optimization, i.e. preliminary filter coefficients for the equalizer sub-modules 32, 34, 36, may then be used as starting conditions for a second optimization, namely the variation of the global cost functional Q.

(79) It is noted that for the variations or rather optimizations described above, any suitable mathematical technique may be used, e.g. Gauß-Newton or Nelder-Meat.

(80) Certain embodiments disclosed herein, for example the respective module(s), unit(s), etc., utilize circuitry (e.g., one or more circuits) in order to implement standards, protocols, methodologies or technologies disclosed herein, operably couple two or more components, generate information, process information, analyze information, generate signals, encode/decode signals, convert signals, transmit and/or receive signals, control other devices, etc. Circuitry of any type can be used. It will be appreciated that the term “information” can be use synonymously with the term “signals” in this paragraph. It will be further appreciated that the terms “circuitry,” “circuit,” “one or more circuits,” etc., can be used synonymously herein.

(81) In an embodiment, circuitry includes, among other things, one or more computing devices such as a processor (e.g., a microprocessor), a central processing unit (CPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), a system on a chip (SoC), or the like, or any combinations thereof, and can include discrete digital or analog circuit elements or electronics, or combinations thereof. In an embodiment, circuitry includes hardware circuit implementations (e.g., implementations in analog circuitry, implementations in digital circuitry, and the like, and combinations thereof).

(82) In an embodiment, circuitry includes combinations of circuits and computer program products having software or firmware instructions stored on one or more computer readable memories that work together to cause a device to perform one or more protocols, methodologies or technologies described herein. In an embodiment, circuitry includes circuits, such as, for example, microprocessors or portions of microprocessor, that require software, firmware, and the like for operation. In an embodiment, circuitry includes one or more processors or portions thereof and accompanying software, firmware, hardware, and the like.

(83) In some examples, the functionality described herein can be implemented by special purpose hardware-based computer systems or circuits, etc., or combinations of special purpose hardware and computer instructions.

(84) Of course, in some embodiments, two or more of the components described herein, or parts thereof, can be integrated or share hardware and/or software, circuitry, etc. In some embodiments, these components, or parts thereof, may be grouped in a single location or distributed over a wide area. In circumstances were the components are distributed, the components are accessible to each other via communication links.

(85) The present application may reference quantities and numbers. Unless specifically stated, such quantities and numbers are not to be considered restrictive, but exemplary of the possible quantities or numbers associated with the present application. Also in this regard, the present application may use the term “plurality” to reference a quantity or number. In this regard, the term “plurality” is meant to be any number that is more than one, for example, two, three, four, five, etc. The terms “about,” “approximately,” “near,” etc., mean plus or minus 5% of the stated value. For the purposes of the present disclosure, the phrase “at least one of A and B” is equivalent to “A and/or B” or vice versa, namely “A” alone, “B” alone or “A and B.”. Similarly, the phrase “at least one of A, B, and C,” for example, means (A), (B), (C), (A and B), (A and C), (B and C), or (A, B, and C), including all further possible permutations when greater than three elements are listed.

(86) The principles, representative embodiments, and modes of operation of the present disclosure have been described in the foregoing description. However, aspects of the present disclosure which are intended to be protected are not to be construed as limited to the particular embodiments disclosed. Further, the embodiments described herein are to be regarded as illustrative rather than restrictive. It will be appreciated that variations and changes may be made by others, and equivalents employed, without departing from the spirit of the present disclosure. Accordingly, it is expressly intended that all such variations, changes, and equivalents fall within the spirit and scope of the present disclosure, as claimed.