Patent Publication Number: US-2022239570-A1

Title: Phase-shifted sampling module and method for determining filter coefficients

Description:
FIELD OF THE DISCLOSURE 
     Embodiments of the present disclosure generally relate to phase-shifted sampling circuits. Embodiments of the present disclosure further relate to measurement instruments and to methods of determining filter coefficients of an equalization circuit of a phase-shifted sampling circuit. 
     BACKGROUND 
     In order to achieve high sampling rates that exceed the sampling rate of a single analog-to-digital converter (ADC), it is known to use two or more ADCs that sample the corresponding signal in a phase-shifted manner. 
     The partial signals sampled by the individual ADCs are recombined into a single digitized signal, such that an integer multiple of the sampling rate of the individual ADCs is achieved. 
     If signals with a large bandwidth are to be digitized, such phase-shifted sampling circuits usually use ADCs that each have a large bandwidth. However, large-bandwidth ADCs are rather costly. 
     Thus, there is a need for a phase-shifted sampling circuit that can be produced at reduced costs while maintaining the quality of the digitized signal. 
     SUMMARY 
     Embodiments of the present disclosure provide phase-shifted sampling circuits. In an embodiment, a phase-shifted sampling circuit comprises a primary sampler circuit, an ADC circuit, and an equalization circuit. The primary sampler circuit comprises an analog signal input, a first signal path, and a second signal path. 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. 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. 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. 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. The equalization circuit comprises a primary sampler equalizer sub-circuit. The primary sampler equalizer sub-circuit is configured to compensate a mismatch between a transfer function associated with the first signal path and a transfer function associated with the second signal path based on the digital output signals. 
     In general, the primary sampler circuit may have M signal paths, wherein M is an integer bigger than or equal to 2. Accordingly, the ADC circuit may comprise M analog-to-digital converters, wherein each of the analog-to-digital converters is associated with one of the M signal paths. 
     Without restriction of generality, the case M=2 is described in the following. 
     The first analog sampling unit and the second analog sampling unit sample the analog input signal in a phase shifted manner with respect to each other. In other words, the analog sampling units sample the same analog input signal, but at different sampling times. Thus, the phase-shifted sampling takes place. 
     The resulting pre-sampled analog input signals are converted by the analog-to-digital converters (ADCs), and the digital output signals of the ADCs may be recombined downstream of the ADCs, thereby obtaining a digital version of the analog input signal that is sampled at a higher sampling rate than achievable with only a single ADC. 
     According to an embodiment of the present disclosure, the primary sampler circuit is provided that reduces the bandwidth of the analog input signal. For example, the first analog sampling unit and the second analog sampling unit reduce the bandwidth of the analog input signal. In other words, the first pre-sampled analog input signal and the second pre-sampled analog input signal each have a lower bandwidth than the analog input signal. 
     Thus, the ADCs each can have a lower bandwidth. Accordingly, cheaper ADCs can be used. The reduced costs of the ADCs overweigh the additional costs of the primary sampler circuit (which can be produced cost-efficiently), such that the overall production costs of the phase-shifted sampling circuit are reduced. 
     Due to (unavoidable) mismatches of real systems, the first signal path and the second signal path of the primary sampler circuit do not behave identically. This can cause unwanted disturbances in the digital output signals. 
     The present disclosure is further based on the idea to compensate the transfer function mismatches between the first signal path and the second signal path by filtering the digital output signals of the analog-to-digital converters by the primary equalizer sub-circuit. 
     For example, the primary sampler equalizer sub-circuit is configured to compensate transfer function mismatches between the analog signal paths within the primary sampler circuit. 
     In other words, the primary sampler equalizer sub-circuit is configured to remove unwanted signal perturbations caused by the transfer function mismatches between the signal paths from the digital output signals. This way, digital output signals with reduced perturbations are obtained. 
     According to an aspect of the present disclosure, the primary sampler equalizer sub-circuit is established as a linear periodic time variant (LPTV) filter. Thus, time variant perturbations in the digital output signals caused by time variant transfer function mismatches between the first signal path and the second signal path can be compensated by the primary sampler equalizer sub-circuit. In other words, perturbations that vary over time can be compensated by the primary sampler equalizer sub-circuit. 
     According to another aspect of the present disclosure, 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. In some embodiments, the filter units are established as FIR filter units. In general, a first one of the two inputs is associated with the first signal path, while a second one of the two inputs is associated with the second signal path. 
     A first one of the two outputs may be connected to the first input by a first filter unit. Moreover, the first output may be connected to the second input by a second filter unit. Thus, signals received by the first input and the second input are filtered by the first filter unit and the second filter unit, respectively, and the corresponding filtered signals are forwarded to the first output. In some embodiments, the corresponding filtered signals are superposed. 
     A second one of the two outputs may be connected to the first input by a third filter unit. Moreover, the second output may be connected to the second input by a fourth filter unit. Thus, signals received by the first input and the second input are filtered by the third filter unit and the fourth filter unit, respectively, and the corresponding filtered signals are forwarded to the second output. In some embodiments, the corresponding filtered signals are superposed. 
     In a further embodiment of the present disclosure, the equalization circuit comprises an ADC equalizer sub-circuit and/or an overall equalizer sub-circuit. The ADC equalizer sub-circuit is configured to compensate transfer function mismatches within the ADC circuit and/or transfer function mismatches in a connection path between the primary sampler circuit and the ADC circuit. The overall equalizer sub-circuit is configured to compensate transfer function mismatches originating upstream of the primary sampler circuit. 
     Therein, the “connection path between the primary sampler circuit and the ADC circuit” is understood to comprise all components downstream of the primary sampler circuit and upstream of the ADC circuit, e.g., connecting wires. These components may have a different (electrical) length, such that transfer function mismatches may be introduced by them. The ADC equalizer sub-circuit may be configured to compensate these mismatches. 
     The ADC equalizer sub-circuit may comprise a first ADC filter unit associated with the first signal path, and a second ADC filter unit associated with the second signal path. Accordingly, the first ADC filter unit may filter the first digital output signal (only), and the second ADC filter unit may filter the second digital output signal (only). Put differently, each digital output signal is filtered by a dedicated ADC filter unit. 
     In some embodiments, the first ADC filter unit and the second ADC filter unit may each be established as a linear time invariant filter, for example as a finite impulse response filter (FIR filter). While the individual ADC filter units may be established as time invariant filters, the ADC equalizer sub-circuit as a whole may be configured as a time variant filter, such that time variant perturbations can be removed by the ADC equalizer sub-circuit. 
     The overall equalizer sub-circuit may be established as a linear time invariant filter, for example as an FIR filter. 
     According to an aspect of the present disclosure, the ADC equalizer sub-circuit is provided upstream of the primary sampler equalizer sub-circuit, for example immediately upstream of the primary sampler equalizer sub-circuit. 
     According to a further aspect of the present disclosure, the overall equalizer sub-circuit is provided downstream of the primary sampler equalizer sub-circuit. 
     In some embodiments, the individual sub-circuits of the equalization circuit may be arranged in inverse order with respect to the sources of transfer function mismatches. Accordingly, the ADC equalizer sub-circuit may be arranged immediately downstream of the ADC circuit. The primary sampler equalizer sub-circuit may be arranged (immediately) downstream of the ADC equalizer sub-circuit. The overall equalizer sub-circuit may be arranged downstream of the primary sampler equalizer sub-circuit. 
     In a further embodiment of the present disclosure, 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. In general, the variation of the global cost functional corresponds to a non-linear optimization, as there may be no linear correlation between the filter coefficients of the equalizer sub-circuits and the input signal. The global cost functional may depend on a reference signal, namely an ideal reconstructed signal. The cost functional may further depend on the input signal and an equalized output signal, i.e., on a signal processed by the equalization circuit. 
     In some embodiments, the global cost functional corresponds to an error measure between an ideal output signal, i.e. the reference signal, and the equalized output signal, i.e., the signal processed by the equalization circuit. 
     According to another aspect of the present disclosure, 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. In other words, a first optimization is performed for the equalizer sub-circuits individually. The results of this first optimization, i.e., preliminary filter coefficients for the equalizer sub-circuits, may then be used for a second optimization, namely the variation of the global cost functional. 
     The individual local cost functionals may depend on filter coefficients of the respective equalizer sub-circuit, on a reference signal (i.e., an ideal output signal of the respective equalizer sub-circuit), and on an actual output signal of the respective equalizer sub-circuit. 
     In some embodiments, the local cost functionals correspond to an error measure between an ideal output signal, i.e., the reference signal, and the actual output signal of the respective equalizer sub-circuit. 
     The variation may be performed by a least squares technique. However, it is to be understood that any other suitable optimization technique may be employed. 
     In a further embodiment of the present disclosure, 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, for example in an alternating manner. In general, the output selector circuit is configured to recombine the digital output signals of the ADCs after the digital output signals have been filtered by the ADC equalizer sub-circuit and the primary sampler equalizer sub-circuit. The resulting digital output signal is sampled at a higher sampling rate than achievable by a single ADC. 
     The output selector circuit may be arranged downstream of the primary sampler equalizer sub-circuit, for example immediately downstream of the primary sampler equalizer sub-circuit. In other words, the output selector circuit may receive the output signals generated by the primary sampler equalizer sub-circuit, i.e., filtered digital output signals that have already been corrected for the transfer function mismatches between the first signal path and the second signal path. 
     Alternatively or additionally, the output selector circuit is arranged upstream of the overall equalizer sub-circuit, for example immediately upstream of the overall equalizer sub-circuit. 
     According to an aspect of the present disclosure, the primary sampler equalizer sub-circuit has a structure resembling the structure of the primary sampler circuit, for example a tree-like structure. For example, the structure of the primary sampler equalizer sub-circuit may be a mirror image of the structure of the primary sampler circuit. It has turned out that a particularly resource-efficient compensation of the transfer function mismatches between the signal paths of the primary sampler circuit is obtained in this way. 
     In some embodiments, individual filter units of the primary sampler equalizer sub-circuit may be arranged in a tree-like structure. For example, the individual filter units may be established as linear 2-periodic time variant (2-LPTV) filters. 
     The primary sampler equalizer sub-circuit may have a single filter structure being associated with all signal paths, for example a single linear n-periodic time variant filter. It has turned out that transfer function mismatches occurring between individual stages of the primary sampler circuit can be compensated in this way, as the single filter structure is linked to all signal paths. 
     Therein, n is an integer bigger than or equal to 2. In some embodiments, n may be equal to M, i.e., to the number of signal paths of the primary sampler circuit. 
     In some embodiments, the analog sampling units are established as sample &amp; hold units (S&amp;H units), as track &amp; hold units (T&amp;H units), or as return-to-zero sampling units (RZ sampling units). The difference between a sample &amp; hold unit and a return-to-zero sampling unit lies in that the sample &amp; hold unit keeps the signal at the sampled signal level for the whole sampling period, while the return-to-zero sampling unit returns the signal level to zero within the sampling period. It has turned out that return-to-zero sampling units are particularly suitable for primary samplers operating at high sampling rates, i.e. above 10 MS/s, for example above 100 MS/s, in particular above 1 GS/s. 
     Embodiments of the present disclosure further provide a measurement instrument that comprises the phase-shifted sampling circuit described above. In some embodiments, the measurement instrument is established as an oscilloscope. For example, the measurement instrument may be established as a digital oscilloscope. 
     Regarding the advantages and further properties of the measurement instrument, reference is made to the explanations given above with respect to the phase-shifted sampling circuit, which also hold for the measurement instrument and vice versa. 
     Embodiments of the present disclosure further provide methods of determining filter coefficients of an equalization circuit of a phase-shifted sampling circuit described above. In an embodiment, the equalization circuit comprises an ADC equalizer sub-circuit configured to compensate transfer function mismatches within the ADC circuit and/or an overall equalizer sub-circuit configured to compensate transfer function mismatches originating upstream of the primary sampler circuit. In an embodiment, the method comprises the following steps: receiving an input signal; processing the input signal by means of 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. 
     In general, the variation of the global cost functional corresponds to a non-linear optimization, as there may be no linear correlation between the filter coefficients of the equalizer sub-circuits and the input signal. The global cost functional may depend on a reference signal, namely an ideal reconstructed signal. The cost functional may further depend on the input signal and an equalized output signal, i.e., on a signal processed by the equalization circuit. 
     In some embodiments, the global cost functional corresponds to an error measure between an ideal output signal, i.e., the reference signal, and the equalized output signal, i.e., the signal processed by the equalization circuit. 
     Regarding the further advantages and properties of the method, reference is made to the explanations given above with respect to the phase-shifted sampling circuit, which also hold for the method and vice versa. 
     According to an aspect of the present disclosure, starting coefficients for the variation of the global cost functional are obtained by means of local variations of local cost functionals associated with the individual equalizer sub-circuits. In other words, a first optimization is performed for the equalizer sub-circuits individually. The results of this first optimization, i.e., preliminary filter coefficients for the equalizer sub-circuits, may then be used for a second optimization, namely the variation of the global cost functional. 
     The individual local cost functionals may depend on filter coefficients of the respective equalizer sub-circuit, on a reference signal (i.e., an ideal output signal of the respective equalizer sub-circuit), and on an actual output signal of the respective equalizer sub-circuit. 
     In some embodiments, the local cost functionals correspond to an error measure between an ideal output signal, i.e., the reference signal, and the actual output signal of the respective equalizer sub-circuit. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       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: 
         FIG. 1  schematically shows a phase-shifted sampling circuit according to an embodiment of the present disclosure; 
         FIG. 2  shows a representative flow chart of a phase-shifted sampling method according to an embodiment of the present disclosure; 
         FIG. 3  shows an example of equalization circuit of the phase-shifted sampling circuit of  FIG. 1  according to an embodiment of the present disclosure; 
         FIG. 4  shows another example of an equalization circuit of the phase-shifted sampling circuit of  FIG. 1  according to an embodiment of the present disclosure; and 
         FIG. 5  shows a representative flow chart of a method of determining filter coefficients according to an embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     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. 
       FIG. 1  schematically shows a representative measurement instrument  8  that comprises an example of a phase-shifted sampling circuit  10 . 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 circuit  10  described below may also be used in another type of electronic device. 
     As shown in the embodiment of  FIG. 1 , the phase-shifted sampling circuit  10  comprises a primary sampler circuit  12 , an ADC circuit  14 , an equalization circuit  16 , and an output selector circuit  17 . In the shown embodiment, the primary sampler circuit  12  comprises an analog signal input  18 , a first signal path  20 , and a second signal path  22 . In general, the primary sampler circuit  12  may have M signal paths, wherein M is an integer bigger than or equal to 2. 
     The first signal path  20  comprises a first analog sampling unit  24  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  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 &amp; hold unit, as a track &amp; hold unit, or as a return-to-zero sampling unit. 
     In some embodiments, the term “module,” “unit,” etc., refers to or includes, inter alia, a combination of hardware (e.g. a processor such as an integrated circuit, digital circuits 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. In some embodiments, hardware may include analog and/or digital circuitry. 
     In the shown embodiment, the ADC circuit  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 circuit  12 . In general, the ADC circuit  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 circuit  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. 
     In some embodiments, the equalization circuit  16  comprises an ADC equalizer sub-circuit  32 , a primary sampler equalizer sub-circuit  34 , and an overall equalizer sub-circuit  36 . The ADC equalizer sub-circuit  32  is arranged immediately downstream of the ADC circuit  14 . 
     The ADC equalizer sub-circuit  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 linear time invariant filter, for example as an FIR filter. While the individual ADC filter units  38 ,  40  may be established as time invariant filters, the ADC equalizer sub-circuit  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-circuit  32 . 
     The primary sampler equalizer sub-circuit  34  is arranged immediately downstream of the ADC equalizer sub-circuit  32 . In some embodiments, the primary sampler equalizer sub-circuit  34  comprises a first input  42 , a second input  44 , a first output  46 , and a second output  48 . 
     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 . 
     In general, the primary sampler equalizer sub-circuit  34  may be established as a linear periodic time variant (LPTV) filter. In some embodiments, the individual filter units  50 ,  52 ,  54 ,  56  may be established as FIR filter units. 
     The output selector circuit  17  is arranged immediately downstream of the primary sampler equalizer sub-circuit  34 . The overall equalizer sub-circuit  36  is arranged immediately downstream of the output selector circuit  17 . The overall equalizer sub-circuit  36  may be established as a linear time invariant filter, for example as an FIR filter. 
     The phase-shifted sampling circuit  10  is configured to perform a phase-shifted sampling method, such as one described in the following with reference to the example shown in  FIG. 2 . 
     As shown in  FIG. 2 , an analog input signal x(t) is received by the analog signal input  18  (step S 1 ). 
     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. Usually, the two signal paths  20 ,  22  are not completely identical, and thus have transfer functions that are not completely identical. 
     Thus, the first sampling unit  24  receives an analog input signal x 0 (t) due to the transfer function P 0 (j2πƒ) of the first signal path  20 . The second sampling unit  26  receives an analog input signal x 1 (t) due to the transfer function P 1 (j2πƒ) of the second signal path  22 , wherein in general x 1 (t)≠x 0 (t) due to P 1 (j2πƒ)≠P 0 (j2πƒ). Accordingly, there may be a transfer function mismatch between the first signal path  20  and the second signal path  22  within the primary sampler circuit  12 . 
     The analog input signals x 0 (t), x 1 (t) are sampled by the sampling units  24 ,  26 , thereby generating a first pre-sampled analog input signal x *,0 (t) and a second pre-sampled analog input signal (t), respectively (step S 2 ). The sampling units  24 ,  26  each operate with a sampling frequency ƒ 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 . 
     In some embodiments, the first sampling unit  24  samples the analog input signal x 0 (t) at sample times t=2 kT, while the second sampling unit  26  samples the analog input signal x 1 (t) at sample times t=(2k+1)T, wherein, k is an integer. In other words, the two sampling units  24  sample the input signal x(t) alternately. 
     The first pre-sampled analog input signal x *,0 (t) and the second pre-sampled analog input signal x *,1 (t) are forwarded to the first ADC  28  and to the second ADC  30 , respectively. 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 . Accordingly, the two connection paths may have different transfer functions. 
     Therein, the term “connection path” is understood to comprise all components downstream of the primary sampler circuit  12  and upstream of the ADC circuit  14 , e.g., connecting wires. 
     Moreover, signal paths within the ADC circuit  14  may also have transfer functions that are different from each other. 
     In  FIG. 1 , the overall effect of the connection paths and the signal paths within the ADC circuit  14  is summarized in the transfer functions Q 0 (j2πƒ) and Q 1 (j2πƒ). As a result of the transfer function Q 0 (j2πƒ), the first ADC  28  receives a modified version v 0 (t) of the first pre-sampled input signal (t). Similarly, the second ADC  30  receives a modified version v 1 (t) of the second pre-sampled input signal x *,1 (t) due to the transfer function Q 1 (j2πƒ). 
     The ADCs  28 ,  30  digitize the pre-sampled analog input signals v 0 (t), v 1 (t), thereby obtaining a first digital output signal y 0 (k) and a second digital output signal y 1 (k), respectively (step S 3 ). The ADCs  28 ,  30  each operate with a sampling frequency ƒ s =1/(2T), for example wherein the sampling frequency of the ADCs  28 ,  30  matches the sampling frequency of the sampling units  24 ,  26 . 
     However, the first ADC  28  samples the first pre-sampled analog input signal v 0 (t) in a phase-shifted manner compared to the second ADC  30  sampling the second pre-sampled analog input signal v 1 (t). For example, first ADC  28  samples the first pre-sampled analog input signal v 0 (t) at sample times t=2kT+ΔT, while the second ADC  30  samples the second pre-sampled analog input signal v 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 . 
     The digital output signals y 0 (k), y 1 (k) are forwarded to the equalization circuit  16 , or to the ADC equalizer sub-circuit  32  in some embodiments. Mismatches between the transfer functions Q 0 (j2πƒ), Q 1 (j2πƒ) are compensated by the ADC equalizer sub-circuit  32  (step S 4 ). Thus, the ADC equalizer sub-circuit  32  compensates transfer function mismatches originating within the ADC circuit  14  and in a connection path between the primary sampler circuit  12  and the ADC circuit  14 . 
     For example, the first ADC filter unit  38  filters the first digital output signal y 0 (k), thereby generating a first filtered digital output signal {tilde over (v)} 0 (k). The second ADC filter unit  40  filters the second ADC filter unit  40  filters the second digital output signal y 1 (k), thereby generating a second filtered digital output signal {tilde over (v)} 1 (k). 
     Therein, transfer functions G 0 (z) and G 1 (z) of the ADC filter units  38 ,  40  are chosen such that the mismatches between the transfer functions Q 0 (j2πƒ), Q 1 (j2πƒ) are compensated. In other words, filter coefficients of the ADC equalizer sub-circuit  32  are chosen such that the mismatches between the transfer functions Q 0 (j2πƒ), Q 1 (j2πƒ) are compensated. 
     The sampling times of the first ADC  28  and/or of the second ADC  30 , i.e., the times at which samples are taken by the first ADC  28  and/or the second ADC  30 ), may deviate from respective ideal sampling times. These deviations from the ideal sampling times are reflected in deviations of phase responses associated with the transfer functions Q 0 (j2πƒ), Q 1 (j2πƒ), as the transfer functions Q 0 (j2πƒ), Q 1 (j2πƒ) also comprise information on the respective phase response. 
     The transfer functions G 0 (z) and G 1 (z) of the ADC filter units  38 ,  40  may be chosen such that the deviations of the sampling times of the ADCs  28 ,  30  from the respective ideal sampling times are compensated. In other words, filter coefficients of the ADC equalizer sub-circuit  32  may be chosen such that the deviations of the sampling times of the ADCs  28 ,  30  from the respective ideal sampling times are compensated. 
     Accordingly, in the case of perfect reconstruction, it holds {tilde over (v)} 0 (k)=v 0 (t) for t=2kT+ΔT and {tilde over (v)} 1 (k)=v 1 (t) for t=(2k+1)T+ΔT. This will be explained in more detail below. 
     The filtered digital output signals {tilde over (v)} 0 (k), {tilde over (v)} 1 (k) are forwarded to the first input  42  and to the second input  44  of the primary sampler equalizer sub-circuit  34 , respectively. 
     Mismatches between the transfer functions P 0 (j2πƒ), (j2πƒ) are compensated by the primary sampler equalizer sub-circuit  34  (step S 5 ). Thus, the primary sampler equalizer sub-circuit  34  compensates transfer function mismatches originating within the primary sampler circuit  12 . 
     Therein, transfer functions H 00 (z), H 01 (z), H 10 (z), and H 11 (z) of the filter units  50 ,  52 ,  55 ,  56  are chosen such that the mismatches between the transfer functions P 0 (j2πƒ), P 1 (j2πƒ) are compensated. In other words, filter coefficients of the primary sampler equalizer sub-circuit  34  are chosen such that the mismatches between the transfer functions P 0 (j2πƒ), P 1 (j2πƒ) are compensated. 
     The sampling times of the first sampling unit  24  and/or of the second sampling unit  26 , i.e., the times at which samples are taken by the first sampling unit  24  and/or the second sampling unit  26 , may deviate from respective ideal sampling times. These deviations from the ideal sampling times are reflected in deviations of phase responses associated with the transfer functions P 0 (j2πƒ), P 1 (j2πƒ), as the transfer functions P 0 (j2πƒ), P 1 (j2πƒ) also comprise information on the respective phase response. 
     The transfer functions H 00 (z), H 01 (z), H 10 (z), and H 11 (z) of the filter units  50 ,  52 ,  55 ,  56  may be chosen such that the deviations of the sampling times of the sampling units  24 ,  26  from the respective ideal sampling times are compensated. In other words, filter coefficients of the primary sampler equalizer circuit  34  may be chosen such that the deviations of the sampling times of the sampling units  24 ,  26  from the respective ideal sampling times are compensated. 
     For example, the first filter unit  50  filters the first filtered digital output signal {tilde over (v)} 0 (k), thereby generating a first intermediate signal v 00 (k). The second filter unit  52  filters the first filtered digital output signal {tilde over (v)} 0 (k), thereby generating a second intermediate signal v 01 (k). The third filter unit  54  filters the second filtered digital output signal {tilde over (v)} 1 (k), thereby generating a third intermediate signal v 10 (k). The fourth filter unit  56  filters the second filtered digital output signal {tilde over (v)} 1 (k), thereby generating a fourth intermediate signal v 11 (k). 
     The first intermediate signal v 00 (k) and the third intermediate signal v 10 (k) are superposed, thereby generating a first compensated digital output signal {tilde over (x)} 0 (k) being associated with the first output  46 . The second intermediate signal v 01 (k) and the fourth intermediate signal v 11 (k) are superposed, thereby generating a second compensated digital output signal {tilde over (x)} 1 (k) being associated with the second output  48 . 
     Accordingly, in the case of perfect reconstruction, it holds {tilde over (x)} 0 (k)=x 0 (t) for t=2kT and {tilde over (x)} 1 (k)=x 1 (t) for t=(2k+1)T. This will be explained in more detail below. 
     The compensated digital output signals {tilde over (x)} 0 (k), {tilde over (x)} 1 (k) are forwarded to the output selector circuit  17 . The compensated digital output signals {tilde over (x)} 0 (k), {tilde over (x)} 1 (k) are recombined by the output selector circuit  17  (step S 6 ). 
     For example, the output selector circuit  17  selectively outputs a signal associated with the first signal path  20 , i.e., the first compensated digital output signal {tilde over (x)} 0 (k), and a signal associated with the second signal path  22 , i.e., the second compensated digital output signal {tilde over (x)} 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. 
     The digital output signal x(k) is forwarded to the overall equalizer sub-circuit  36 . The digital output signal x(k) is filtered by the overall equalizer sub-circuit  36 , thereby compensating transfer function mismatches originating upstream of the primary sampler circuit  12  (step S 7 ). As a result, a compensated digital output signal {tilde over (x)}(k) with reduced perturbations due to transfer function mismatches is obtained. 
     In some embodiments, time variant perturbations due to time variant transfer function 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 transfer function mismatches. 
       FIG. 3  shows another embodiment of the equalization circuit  16 , or more precisely of the ADC equalizer sub-circuit  32 , the primary sampler equalizer sub-circuit  34 , and the output selector circuit  17 . It is noted that the overall equalizer sub-circuit  36  is not shown in  FIG. 3 , but would be provided downstream of the primary sampler equalizer sub-circuit  34  and the output selector circuit  17 . 
     In the certain example of  FIG. 3 , the equalization circuit  16  is configured for a primary sampler circuit  12  having eight signal paths. Accordingly, the ADC equalizer sub-circuit  32  comprises eight ADC filter units (denoted as “ADCC” in  FIG. 3 ). Optionally, the ADC equalizer sub-circuit  32  may comprise an adder unit (not shown in  FIG. 3 ) upstream of each ADC filter unit, wherein the adder units are each configured to compensate offset errors. 
     The overall equalizer sub-circuit  36  and the output selector circuit  17  are established, for example, by several linear 2-periodic time variant filters  58 . In general, the 2-periodic time variant filters  58  are arranged in a structure resembling the internal structure of the primary sampler circuit  12 . For example, the 2-periodic time variant filters  58  are arranged in a tree-like structure, as the primary sampler circuit  12  typically also has a tree-like structure. 
     In some embodiments, the structure of the 2-periodic time variant filters  58  is a mirrored version of the internal structure of the primary sampler circuit  12 . 
     It has turned out that a particularly resource-efficient compensation of the transfer function mismatches between the signal paths  20 ,  22  of the primary sampler circuit  12  is obtained in this way. 
       FIG. 4  shows another embodiment of the equalization circuit  16 , or more precisely of the ADC equalizer sub-circuit  32 , the primary sampler equalizer sub-circuit  34 , and the output selector circuit  17 . It is noted that the overall equalizer sub-circuit  36  is not shown in  FIG. 4 , but would be provided downstream of the primary sampler equalizer sub-circuit  34  and the output selector circuit  17 . 
     In the certain example of  FIG. 4 , the equalization circuit  16  is configured for a primary sampler circuit  12  having eight signal paths. Accordingly, the ADC equalizer sub-circuit  32  comprises eight ADC filter units (denoted as “ADCC in  FIG. 4 ). Optionally, the ADC equalizer sub-circuit  32  may comprise an adder unit (not shown in  FIG. 4 ) upstream of each ADC filter unit, wherein the adder units are each configured to compensate offset errors. 
     The overall equalizer sub-circuit  36  and the output selector circuit  17  are established, for example, by a single 8-periodic time variant filter  60 . It is noted that this structure can be generalized to an arbitrary number of n signal paths. In this case, the equalizer sub-circuit  36  and the output selector circuit  17  are established by a single a-periodic time variant filter. 
     It has turned out that transfer function mismatches occurring between individual stages of the primary sampler circuit can be compensated in this way, as the single filter structure is linked to all signal paths. 
     In the following, a representative method of determining filter coefficients of the equalization circuit  16  of the phase-shifted sampling circuit  10  is described with respect to  FIG. 5 . 
     As shown in  FIG. 5 , the input signal x(t) is received by the analog signal input  18  (step F 1 ). 
     The input signal x(t) is processed by the primary sampler circuit  12  and the ADC circuit  14  (cf. steps S 2  and S 3  described above), thereby obtaining at least a first digital output signal y 0 (k) and a second digital output signal y 1 (k) (step F 2 ). 
     The first digital output signal y 0 (k) and the second digital output signal y 1 (k) are processed by the equalization circuit  16  (cf. steps S 4  to S 7  described above), thereby obtaining an equalized output signal {tilde over (x)}(k) (step F 3 ). 
     A global cost functional is provided, wherein the global cost functional depends on filter coefficients of the individual equalizer sub-circuits  32 ,  34 ,  36 , the input signal y(n)=(y 0 (n), y 1 (n)) of the equalization circuit  16 , and the equalized output signal {tilde over (x)}(k) (step F 4 ). 
     In general, the global cost functional has the following form: 
     
       
         
           
             Q 
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 
                   N 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   
                     ( 
                     
                       
                         
                           x 
                           ˜ 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       - 
                       
                         
                           
                             x 
                             ˜ 
                           
                           ideal 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 . 
               
             
           
         
       
     
     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 ges (b ges ,y,n) of a filter coefficient vector b 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. 
     {tilde over (x)} ideal  is an ideal reference signal, i.e., an ideal reconstructed signal. 
     Filter coefficients of the equalizer sub-circuits  32 ,  34 ,  36  are determined by a global variation of the global cost functional (step F 5 ). 
     For example, the global cost functional is minimized in order to find the filter coefficients of the individual equalizer sub-circuits  32 ,  34 ,  36 , i.e., 
     
       
         
           
             
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               . 
             
           
         
       
     
     Alternatively, the maximal error may be minimized, i.e., 
     
       
         
           
             
               min 
               b 
             
             ⁢ 
             
               
                 { 
                 
                   
                     max 
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               . 
             
           
         
       
     
     Therein, the starting coefficients for the variation of the global cost functional Q are obtained by means of local variations of local cost functionals being associated with the individual equalizer sub-circuits  32 ,  34 ,  36 . 
     Accordingly, three different local cost functionals Q ADCE , Q PSE , and Q OSE , and provided for the ADC equalizer sub-circuit  32 , the primary sampler equalizer sub-circuit  34 , and the overall equalizer sub-circuit  36 , respectively. 
     The local cost functionals have the following general form: 
     
       
         
           
             
               
                 Q 
                 X 
               
               = 
               
                 
                   ∑ 
                   
                     n 
                     = 
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                     N 
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                             , 
                             ref 
                           
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
             , 
           
         
       
     
     wherein X can be ACDE, PSE, or OSE. Moreover, x X (n) is the actual output signal of the respective equalizer sub-circuit  32 ,  34 ,  36 , and x X,ref (n) is the corresponding ideal output signal of the respective equalizer sub-circuit  32 ,  34 ,  36 , i.e., the corresponding reference signal. 
     In order to obtain the starting coefficients for the global optimization, the local cost functionals Q X  are minimized, i.e., 
     
       
         
           
             
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               . 
             
           
         
       
     
     Alternatively, the maximal error may be minimized, i.e., 
     
       
         
           
             
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               . 
             
           
         
       
     
     In other words, a first optimization is performed for the equalizer sub-circuits  32 ,  34 ,  36  individually. 
     The results of this first optimization, i.e., preliminary filter coefficients for the equalizer sub-circuits  32 ,  34 ,  36 , may then be used as starting conditions for a second optimization, namely the variation of the global cost functional Q. 
     It is noted that for the variations or rather optimizations described above, any suitable mathematical technique may be used, e.g. GauB-Newton or Nelder-Meat. 
     Certain embodiments disclosed herein utilize circuitry (e.g., one or more circuits) in order to implement 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. 
     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). In an embodiment, circuitry includes combinations of hardware 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, such as, for example, microprocessors, or portions thereof and accompanying software, firmware, hardware, and the like. 
     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. Each of these special purpose hardware-based computer systems or circuits, etc., or combinations of special purpose hardware circuits and computer instructions form specifically configured circuits, devices, etc., capable of implemented the functionality described herein. 
     Various embodiments are described above with reference to block diagrams and/or flowchart illustrations of apparatuses, methods, systems, and/or computer program instructions or program products. It should be understood that each block of any of the block diagrams and/or flowchart illustrations, respectively, of portions thereof, may be implemented in part by computer program instructions, e.g., as logical steps or operations executing on one or more computing devices. These computer program instructions may be loaded onto one or more computer or computing devices, such as special purpose computer(s) or computing device(s) or other programmable data processing apparatus(es) to produce a specifically-configured machine, such that the instructions which execute on one or more computer or computing devices or other programmable data processing apparatus implement the functions specified in the flowchart block or blocks and/or carry out the methods described herein. 
     These computer program instructions may also be stored in one or more computer-readable memory or portions thereof, that can direct one or more computers or computing devices or other programmable data processing apparatus(es) to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including computer-readable instructions for implementing the functionality specified in the flowchart block or blocks. 
     It will be appreciated that the term computer or computing device can include, for example, any computing device or processing structure, including but not limited to 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. 
     Accordingly, blocks of the block diagrams and/or flowchart illustrations support various combinations for performing the specified functions, combinations of operations for performing the specified functions and program instructions for performing the specified functions. Again, it should also be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and/or flowchart illustrations, or portions thereof, could be implemented by special purpose hardware-based computer systems or circuits, etc., that perform the specified functions or operations, or combinations of special purpose hardware and computer instructions. 
     Of course, in some embodiments, two or more of these components, 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 where the components are distributed, the components are accessible to each other via communication links. 
     In the foregoing description, specific details are set forth to provide a thorough understanding of representative embodiments of the present disclosure. It will be apparent to one skilled in the art, however, that the embodiments disclosed herein may be practiced without embodying all of the specific details. In some instances, well-known process steps have not been described in detail in order not to unnecessarily obscure various aspects of the present disclosure. Further, it will be appreciated that embodiments of the present disclosure may employ any combination of features described herein. 
     Throughout this specification, terms of art may be used. These terms are to take on their ordinary meaning in the art from which they come, unless specifically defined herein or the context of their use would clearly suggest otherwise. 
     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. 
     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.