Switched-Capacitor Filters:Sampled-Data Analog Filters

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Introduction

The accuracy of the absolute value of integrated passive devices (R and C) is very poor. As a consequence, the frequency response accuracy of integrated active-RC filters is poor and they are not feasible when high- accuracy performance is needed. A possible solution for the implementation of analog filters with accurate frequency response was given by the switched-capacitor (SC) technique since the late 1970s [1,2]. Their popularity has further increased since they can be realized with the same standard CMOS technology used for digital circuits. In this way, fully integrated, low-cost, high-flexibility, mixed-mode systems have become possible. The main reasons of the large popularity of SC networks can be summarized as follows:

1. The basic requirements of SC filters fit the popular MOS technology features. In fact, the infinite input impedance of the operational amplifier (opamp) is obtained using a MOS input device. MOS transconductance amplifiers can be used since only-capacitive load is present, precise switches are realized with MOS transistor, and capacitors are available in the MOS process.

2. SC filter performance accuracy is based on the matching of integrated capacitors (and not on their absolute values). In a standard CMOS process, the capacitor matching error can be less than 0.2%. As a consequence, SC systems guarantee very accurate frequency response without component trimming. For the same reason, temperature and aging coefficients track, reducing performance sensitivity to temperature and aging variations.

3. It is possible to realize SC filters with long time constants without using large capacitors and resistors. This means a chip area saving with respect to active-RC filter implementations.

4. SC systems operate with closed-loop structures; this allows to process large swing signals and to achieve large dynamic range.

In contrast the major drawbacks of SC technique are as follows:

1. To process a fully analog signal, an SC filter has to be preceded by an anti-aliasing (AA) filter and followed by a smoothing filter, which complicate the overall system and increase power and die size.

2. The opamps embedded in an SC filter have to perform a large DC gain and a large unity-gain bandwidth, much larger than the bandwidth of the signal to be processed. This limits the maximum signal bandwidth.

3. The power of noise of all the sources in the SC filter is folded in the band [0 - Fs/2]. Thus their noise power density is increased by factor (Fs/2)/Fb, where Fs is the sampling frequency and Fb the noise bandwidth at the source.

From its first proposals the SC technique has been highly developed. Many different circuit solutions have been realized with SC technique not only in analog filtering, but also in analog equalizers, analog- to-digital and digital-to-analog conversion (including in particular the oversampled l.: converters), Sample&Hold, Track&Hold, etc.

In this chapter an overview of the main aspects of the SC technique is given; the reader can refer to more details in the literature. Few advanced solutions feasible for future SC systems are given in the last section.

Sampled-Data Analog Filters

A SC filter is a continuous-amplitude, sampled-data system. This means that the amplitude of the signals can assume any value within the possible range in a continuous manner. In contrast, these values are assumed at certain time instants and then they are held for all the sampling period. Thus the resulting waveforms are not continuous in time but looks like a staircase.

In an SC filter, the continuous-time input signal is first sampled at sampling frequency Fs and then processed through the SC network. This sampling operation results in a particular feature of the frequency response of the SC system. In the following the aspects relative to the sampling action are illustrated from an intuitive point of view, while a more rigorous description can be found in Ref. [3].

The sampling operation extracts from the continuous-time waveform the values of the input signal at the instant kTs (k = 1, 2, 3, ¼), where Ts = 1/Fs is the sampling period. This is shown in Figure 62.1 for a single-input sine wave at fo = 0.16Fs (i.e., with fo < Fs/2).

If the input sine wave is at Fs + fo, the input sequence of analog samples is exactly equal to that previously obtained with fo as input frequency (see Figure 62.2). Both sequences should then be processed exactly in the same way by the SC network, and the overall filter output sequence should then result to be again identical. The two input sine waves result then to be indistinguishable after the sampling action. This effect is called aliasing. It can be demonstrated that a sine wave at frequency fo in the range [0-Fs/2] is aliased by the components at frequency f al given by

Switched-Capacitor Filters-0166

As a consequence, to avoid frequency aliasing (which means signal corruption), the input signal band of a sample data system must be limited in the [0-Fs/2] range. The range [0-Fs/2] is called baseband and the above limitation is an expression of the Nyquist theorem.

After the sampling, the SC network processes the sequence of samples, independently of how they have been produced. Since all the frequencies given in Eq. (62.1) produce the same sequence of samples, the gain for all of them results to be the same. This concept results in the fact that the transfer function of a sampled-data system is periodical with period equal to Fs, and it is symmetrical in its period.

Switched-Capacitor Filters-0167

Switched-Capacitor Filters-0168

For instance, in Figure 62.3 the frequency response amplitude for a low-pass filter is shown for frequencies higher than Fs.

As stated above, to avoid the aliasing effect to corrupt the signal, it is necessary to limit the input signal bandwidth. This function is performed by the AA filter, which is placed in front of the SC filter and operates in the continuous-time domain (Figure 62.4). From a practical point of view, the poles of the SC filter are typically much smaller than Fs/2 and the frequency response is required to be accurate only in the passband. In contrast, the AA filter transfer function is not required to be accurate. Thus the AA filter is usually implemented with active-RC filter.

A staircase signal is produced at the output of the SC network. If a continuous-time output waveform is needed, a continuous-time smoothing filter must be added. The overall SC filter-processing chain results are given in Figure 62.5. Of course, in some cases the input signal spectrum is already limited to Fs/2 and then the AAF is not necessary, while in other cases the final smoothing filter is no more necessary, as in the case when the SC filter is used in front of a sampled-data system (for instance an ADC).

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