Design of pre-filter and post-filter of switched capacitor filter

Abstract: How to reasonably configure the pre-filter and post-filter for the switched capacitor filter (SCF) has always lacked systematic analysis and explanation. Based on the study of the working characteristics of SCF, the design method of the pre-filter and post-filter of SCF is proposed. The main design parameters of the SFC pre-filter and post-filter are the corner frequency and attenuation. When designing, the attenuation is first determined according to the system requirements, and then the corner frequency is determined according to the attenuation coefficient of the selected filter structure. Using this method, combined with the needs of the 12-bit data acquisition system, a programmable low-pass filter is designed with MAX295 as the core. The results show that the pre-filter and post-filter of MAX295 can cover its 10 Hz to 50 kHz corner frequency range with 4 steps of corner frequency, and the minimum attenuation of the pre-filter and post-filter are -74 dB and -10 dB respectively.
Keywords: switched capacitor filter; pre-filter; post-filter; anti-aliasing; reconstruction; MAX295

The switched capacitor filter is a filter element based on the charge transfer principle. Compared with the traditional analog filter, it has the advantages of easy production and ease of use, and is increasingly widely used in various test systems.
On the other hand, the switched capacitor filter itself is also a sampling system. The maximum frequency of its input signal is subject to the limitation of the sampling theorem, and its output is a step-shaped discrete time signal. It is usually required to configure appropriate pre- and post-filters for the switched capacitor filter to solve the problems of anti-aliasing and reconstruction, but there is a lack of systematic analysis and explanation of the design criteria and process of the pre- and post-filters.
For this reason, the author takes the anti-aliasing filter in the data acquisition system as an example to discuss the characteristics of the switched capacitor filter and its pre- and post-filtering problems.

1 Characteristics of the switched capacitor filter
Figure 1 is a typical structure of a switched capacitor filter. In this switched capacitor filter, in addition to the input and output terminals, there is also a clock input terminal to provide a sampling pulse with a frequency of fclk. This clock frequency fclk has a certain correspondence with the center frequency or corner frequency fc of the switched capacitor filter, that is, fclk=Nfc, and N is generally 100 or 50. In this way, the analog input signal with a frequency of fin becomes a discrete sampling sequence after being processed by the switched capacitor filter.

Obviously, there is also an anti-aliasing problem for the switched capacitor filter itself, so the input signal must be pre-filtered before the switched capacitor filter to limit the highest frequency of the input signal.
[page] The output of the switched capacitor filter is a series of steps, which is not only discontinuous in the time domain, but also adds new high-frequency components in the frequency domain. Therefore, the required waveform must be reconstructed through appropriate post-filtering.
It is worth pointing out that for the convenience of practical application, the pre- and post-filters of the switched capacitor filter should be borne by analog filters to avoid repeating the previous problems. Otherwise, some unnecessary troubles may occur in the application.
Taking the above factors into consideration, the general application mode of the switched capacitor filter can be summarized as Figure 2. The following is a further discussion of the relevant issues.

1.1 Pre-filtering
1.1.1 Design indicators
The main function of pre-filtering is to limit the maximum frequency of the input signal of the switched capacitor filter. Ideally, the design parameter of this pre-low-pass filter is only one, that is, the corner frequency fpre of the low-pass filter. However, before determining fpre, the frequency components to be removed should be clearly defined.
For a general sampling system, if its sampling frequency is fs and the input signal frequency is fin, when fin≥1/2fs, aliasing will occur. The frequency fs after aliasing can be calculated by formula (1), where N is a natural number greater than 0.
fa=|Nfs-fin| (1)
The situation in which the switched capacitor filter generates aliasing is different from formula (1). Not all aliasing generated by input frequencies higher than 1/2fclk can form an effective output. When the aliased frequency is outside the passband of the switched capacitor filter, it is attenuated by the switched capacitor filter and its influence can be ignored. Therefore, when designing the pre-filter of the switched capacitor filter, only those frequency components that will fall into the passband of the switched capacitor filter after aliasing need to be considered.
To simplify the discussion, only the situation within the clock base frequency is considered here. Taking a low-pass switched capacitor filter with fclk=Nfc as an example, it is not difficult to know through simple analysis that: in an ideal state, only those aliases generated by input frequencies higher than fST=(N-1)fc will fall within the passband of the switched capacitor filter to form a valid output (as shown in Figure 3). At this time, it is only necessary to use a pre-filter to filter out all components in the input signal that are higher than the starting frequency fst.

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Considering the limited attenuation coefficient of the actual switched capacitor filter, the starting frequency fST can be selected as follows:
fST=(ND)fc (2)
Where D is the coefficient reflecting the attenuation characteristics of the switched capacitor filter, which is generally 2 to 4.
In this way, under ideal conditions, the corner frequency fpre of the pre-filter only needs to meet the following conditions:
fc Where fc is the corner frequency set by the switched capacitor filter, and fST is the starting frequency of the switched capacitor filter to generate aliasing output. In physical terms, the first part of the above formula ensures that no loss is caused to the useful signal, and the second part prevents any components that may cause aliasing from passing through.
For actual filters, due to the limited attenuation coefficient, two indicators need to be considered when designing the actual pre-low-pass filter: corner frequency fpre and attenuation A. The former determines the frequency range of the blocked signal, and the latter determines the degree of attenuation of these frequency components.
In actual design, the attenuation A is determined first, and then the corner frequency fpre is determined.

It is worth noting that the above attenuation is equivalent to the case when the signal-to-noise ratio is 1. If the signal-to-noise ratio of the input signal is known in advance, the attenuation can be reduced accordingly.
1.1.3 Determination of corner frequency
After determining the attenuation of the prefilter, the required corner frequency fpre can be determined according to the attenuation coefficient of the selected prefilter, that is, to ensure that:
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Taking the fourth-order Butterworth low-pass filter as an example, its attenuation coefficient Ca is 80 dB per 10 octave. For a 12-bit data acquisition system and a switched capacitor filter with fclk=50fc, the total attenuation A is 74 dB, and D is 4. Then, the range of the corner frequency fpre of the prefilter can be calculated from formula (6):
fc 1.2 Post-filtering
1.2.1 Design indicators
The function of post-filtering is to eliminate the sampling steps and clock interference of the switched capacitor filter and reconstruct the analog waveform, so a continuous analog filter is still used.
Ideally, there is only one design parameter for the post-pre low-pass filter, which is the corner frequency fpost of the low-pass filter. At this time, the selection of fpost only needs to meet the following conditions:
fc where fc is the corner frequency set by the switched capacitor filter and fclk is the clock frequency of the switched capacitor filter. In physical terms, the first part of equation (9) ensures that no loss is caused to the useful signal, while the second part prevents any component higher than the clock frequency from passing through.
Since the attenuation coefficient of the actual filter is limited, the attenuation A and the corner frequency fpost should also be considered when designing the post-filter.
In actual design, the attenuation A is determined first, and then the corner frequency fpost is determined.
1.2.2 Determination of attenuation
The post-filter is required to have sufficient attenuation capability to attenuate any frequency component ≥fclk to a level acceptable to the subsequent data acquisition system (within the input range corresponding to 1 LSB). On the other hand, considering the actual level of high-frequency noise output by the switched capacitor filter, the attenuation can be taken as about 30dB.
1.2.3 Determination of corner frequency
After determining the attenuation of the post filter, the required corner frequency fpost is determined according to the attenuation coefficient of the selected post filter, so that

Taking the second-order Butterworth low-pass filter as an example, its attenuation coefficient CA is 40 dB per 10 octave. For a switched capacitor filter with fclk=50fc, the total attenuation A is taken as 30 dB, then the range of the post filter corner frequency fpost can be calculated from formula (12):
fc [page]2 Follow-up tracking of pre- and post-filters
2.1 Follow-up method
In actual data acquisition systems, the programmable performance of the switched capacitor filter can be used to specify the clock frequency fclk of the switched capacitor filter at any time within a wide frequency range, thereby changing the filter's corner frequency fc. At this time, the frequencies of the pre- and post-filters of the switched capacitor filter should also change accordingly, that is, to achieve follow-up tracking of pre- and post-filters.
When implementing follow-up tracking filtering, in order to simplify the hardware design, the corner frequency of the switched capacitor filter can be changed by several steps under the premise of meeting the minimum attenuation, and then the corner frequency of the pre- and post-filters will change by one step, that is, the several steps of the corner frequency of the pre- and post-filters are combined into one step. The specific judgment method is as follows:
According to formula (7), for the corner frequency fc specified by the switched capacitor filter, the corner frequency fpre of its pre-filter can be taken within a certain frequency range, and its minimum value is recorded as min(fpre) and the maximum value is recorded as max(fpre). For fci min(fprej)≤fpre≤max(fprei) (14)
The situation of the post filter is similar. If min(fprej)

2.2 Example
In a data acquisition system, the MAX295 switched capacitor filter is used as the anti-aliasing filter of the A/D converter.
MAX295 is a clock-programmable eighth-order Butterworth low-pass filter. The corner frequency can be set arbitrarily in the range of 0.1 Hz to 50 kHz. The ratio of its clock frequency fclk to the corner frequency fc is N=50. According to the system requirements, the corner frequency of the anti-aliasing filter should be in the range of 10 Hz to 50 kHz, divided into 12 levels according to the 1-2-5 scale.
The resolution of the A/D converter of this data acquisition system is 12 bits. According to Table 1, it can be determined that the attenuation A of the pre-filter should be greater than 74 dB. Considering the actual possibility, two LM318s are used here to form a fourth-order Chebyshev filter with a passband ripple of 0.5 dB as a pre-filter, and the four corner frequencies of 53 Hz, 530 Hz, 5.3 kHz and 53 kHz are set to cover the entire frequency range.
The MAX295 output noise is very low (THD+N typical value -70 dB), so the attenuation of the post-filter is greater than 10 dB to meet the requirements. For this purpose, a second-order Butterworth low-pass filter is formed with its own operational amplifier, and four corner frequencies of 53Hz, 530Hz, 5.3kHz and 53kHz are set to cover the entire frequency range.

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Figure 5 shows the structure of the programmable anti-aliasing filter. The frequency settings and minimum attenuation of the three-stage filter are shown in Table 2. Among them, one corner frequency of the pre-filter and the post-filter can cover the three frequencies of the switched capacitor filter. For example, the corner frequency of the pre-filter and the post-filter set at 53 Hz can correspond to the three frequencies of 10 Hz, 20 Hz and 50 Hz of the switched capacitor filter, and so on.

The parameter calculation and verification of all filter components are completed by the "FiherPro low-pass filter design program" of TI Company in the United States. The theoretical calculation and actual measurement results show that the filter can fully meet the requirements of the data acquisition system.

3 Conclusion
The author proposed the design parameters and design steps of the pre-filter (anti-aliasing) and post-filter (reconstruction) of the switched capacitor filter, and combined with the engineering example of the data acquisition system, discussed the design problems of the pre-filter and post-filter of the tracking. The experiment shows that the design method proposed by the author is concise, reliable and effective, and can be used as a reference for the application of switched capacitor filters.