Switched Capacitor Interfacing - CML Microcircuits

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Switched Capacitor Interfacing

Anti-Aliasing and Smoothing Filters

1. Introduction

Switched Capacitor Networks (SCNs) are sampled data systems and therefore are governed by the principles

of discrete time signal processing. In this context, “discrete time” means the signal is sampled prior to

processing. However, the amplitude is continuous – unlike Digital Signal Processing (DSP) systems where

both amplitude and time are discrete.

VLSI technology (generally CMOS) has facilitated complete integration of SCNs on a single monolithic chip.

Filters utilizing Switched Capacitor techniques (SCFs) are composed of switches, capacitors and opamps.

Notice resistors are not mentioned. The resistors normally present in active filters are approximated by

switched capacitors. By taking advantage of matching properties of CMOS capacitors, SCFs can be

fabricated with low tolerance levels (less than 0.1%) and virtually any order.

The wireless and wireline telecom industries have made extensive use of SCFs. Even with the move towards

digital technology and the ensuing popularity of DSP in the communications industry, SCNs are frequently

employed where cost and power consumption are critical. SCNs generally have relatively high sampling rates

(typically greater than 20 times the bandwidth of the interest). Due to their discrete time nature, aliasing and

output smoothing must be addressed. Often a simple single pole filter (R-C network) suffices, but the location

of these poles must be carefully thought out during the design process. The world is still analog.

The purpose of this paper is to address the need for anti-aliasing and smoothing filters for SCFs. First, the

basics of sampling will be reviewed. The fundamentals of SCNs will follow this, and finally the design of

anti-aliasing and smoothing filters for SCFs is addressed.

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2. Sampling [3]

Before a continuous signal is processed by a discrete time filter it must first be sampled. Referring to Figure

1, sampling is the process of instantaneously capturing the level of a continuous signal at some

predetermined rate. This predetermined rate is the sampling frequency. As the sampling rate increases

(Figure 1) the sampled signal begins to approximate the original continuous signal.

f(t)

tTs=1/fst

02T

s4Ts6Ts8Ts

(a)

fs(t)

04T

s8Ts12Ts16Ts

RECONSTRUCTION

(b)

fs(t) fr(t)

(c)

s2Ts3Ts4Ts

RECONSTRUCTION

fr(t)

fs(t)

Figure 1: Waveform Sampling:

(a) sampling process, (b) reconstruction with acceptable sampling rate,

On the other hand, as the sampling frequency decreases (Figure 1) the samples move further apart in time,

yielding a reconstructed signal that does not resemble the original. The limit on how far apart these samples

can become without losing information is the basis for Shannon’s sampling theorem.

Shannon’s original work on sampling and information theory [3] stated the sampling theorem as:

“If a function f(t) contains no frequencies higher than

cycles per second it is completely determined

by giving its ordinates at a series of points spaced (1/2

) seconds apart.”

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A formal derivation is obtained by convolving the Fourier transform of the continuous signal with the Fourier

transform of an infinite sequence of impulse functions.



)S(*)F(

)(FS















 dx)x(S)x(F

)(FS

Where: Fs()= sampled signal spectrum

F()= original signal spectrum

S()= spectrum for a sequence of impulses

=radian frequency (2

)

The result of the convolution is the original signal spectrum repeated at multiples of the sampling frequency as

shown in Figure 2. Notice that if the sampling frequency is less than twice the bandwidth of the original signal

then the replica centered at the sampling frequency will overlap and distort the original. This undesirable

phenomenon is aliasing. Aliasing will be avoided if, B2

Sf

Where:

f= sampling frequency , Hz

B = bandwidth of original signal, Hz

|M|

(a)

|M|

(b)

-fs

-2fsfs2fs

Figure 2: Frequency Spectrum of (a) continuous time signal, (b) sampled signal

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(a)

(b)

(c)

SAMPLING

RECONSTRUCTION

Figure 3: Time domain representation of sampling at fS = 2B,

(a) continuous signal, (b) sampled signal, (c) reconstructed signal

Shannon’s sampling theorem is mathematically sound although, in most cases, it is not practical to sample a

signal at exactly twice its highest frequency component. Figure 3 shows a sinewave sampled at 2B samples

per second, where B = 1/T and T is the period of the sinewave. Notice the difference between the

reconstructed wave shapes and the original unsampled signal. This type of error is not the result of aliasing,

yet it could cause deceptive results. A mathematician might claim that there is no information in a continuous

sinewave that never changes amplitude or frequency. This is a fact. However, if a signal is sampled at

exactly 2B, some of the original signal may be lost or distorted.

If the system designer is not concerned with the possibility of losing some of the original signal, and wanted to

sample at 2B, he or she would still be facing the problem of bandlimiting. Bandlimiting is necessary to avoid

aliasing. Figure 4 shows the spectrum of a sampled signal. Notice that for fs = 2B the bandlimiting filter must

have infinite attenuation at the frequency B. A filter that matches this requirement is physically unrealizable.

|M|

2fs

fs=2B

ANTIALIASING FILTER RESPONSE

DESIRED SIGNAL UNDESIRED SIGNAL

DUE TO ALIASING

Figure 4: Anti-aliasing filter required for sampling at exactly 2B

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3. SCN Concepts [1]

A single pole active filter (inverting integrator) is shown in Figure 5. This circuit is the primary building block

for most active filters.

vi(t) vo(t)

Figure 5: Active R-C Integrator

The output is defined by



0IO d)(v

)t(v

Hence, the output is an integrated version of the input, scaled by -1/RC. The pole frequency is determined by

the value of R and C.

RC21

p

f

For a pole frequency of 3000Hz, (5) implies RC=53.05s. A possible implementation, selecting C2 = 10pF

(relatively large capacitor for CMOS technology), requires R=5.305M. A monolithic resistor of this value

would consume excessive amounts of silicon and is subject to significant variation with process and

temperature. Integrated capacitors are also subject to variation with process and temperature. In addition,

capacitor variations are independent of resistor variations.

A more practical realization is shown in Figure 6.

vi(t)

vo(t)

φ1φ2

i(t)

Figure 6: SC Integrator

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Notice R is replaced by a capacitor and two switches. The function of the resistor is now approximated by

transferring charge from vi(t) to the inverting input of the opamp at intervals determined by clocks 1 and 2.

To ensure complete charge transfer 1 and 2 must not overlap as shown in Figure 7.

An approximate analysis of Figure 6 reveals the average current through the switches is

)v-(v







Where:

R

Looking back at the previous example for a pole frequency of 3000Hz, using T=10s (fclk=100kHz), yields

C1=1.885 pF. Hence, the integrator can be realized with two capacitors that track with process and

temperature. No longer is the absolute value of the capacitors important, only their ratio.

φ1

off

φ2

off

Figure 7: Non-overlapping clocks for SC Integrator

4. Anti-aliasing and Smoothing Filters for SCNs [2]

SCFs approximate active-RC filters by replacing resistors with switched capacitors. As the clock frequency

(sampling rate) tends towards infinity the SCF becomes equivalent to the continuous time filter. In other

words, higher clock frequency yields a better approximation of the desired response. Also, higher clock

frequency lessens the requirements of the anti-aliasing filter (AAF) and smoothing filter (SMF). However,

there are upper limits on clock frequency. Although SCFs have been implemented with clock frequencies

greater than 1MHz, typical telecom filters (BW less than 10kHz) are clocked at or below 250kHz.

One factor which opposes higher clock frequencies is large capacitor ratios. Referring to the example in

section 3.0,

using 1MHz in place of 100kHz, C1 becomes 0.1885 pf. The ratio between C1 and C2 is now 53.05 where it

was 5.305. Large capacitor ratios consume more silicon area and are more sensitive to process and

temperature variations. Hence, the final sampling rate is a compromise between SCF implementation

complexity and SMF/AAF requirements.

SMFs and AAFs can be integrated, but they are subject to the same problems associated with the continuous

time RC integrator discussed in section 3.0 and are generally implemented externally. In some cases, where

input and output signals are band limited externally, additional circuitry for AAFs and SMFs is not required. A

block diagram of a typical SCF system, including AAF and SMF is shown in Figure 8.

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AAF SMFSCF

fclk

IN OUT

Figure 8: SCF system with AFF and SMF

To determine AAF and SMF requirements the system designer must know the sampling frequency (fclk) and

the bandwidth (BW) or cutoff frequency (fco) of the SCF. For discussion purposes, assume fclk =100kHz and

fco =5kHz. Recall from section 2.0, if any signal energy is present at the input, with frequency

s greater than

1/2 fclk, it will be aliased to fclk - fs. Hence, an input with fs =99kHz will be aliased to 1kHz. This aliased

signal will appear as noise in the passband of the SCF. To eliminate or reduce the effect of this noise the

AAF must band limit the input signal to fclk - fco (see Figure 9).

|M|

fclk

2fclk

ANTIALIASING FILTER RESPONSE

fco f

5kHz 100kHz

fclk-fco

95kHz

fclk-fco

105kHz

Figure 9: AAF requirements for SCF system

To obtain 20 dB suppression of aliased noise the minimum requirement for the AAF is a first order section

(single pole) with it’s cutoff frequency set to 9.5kHz (20 dB/decade/pole). Placing the pole at 9.5kHz

minimizes the effect on the SCF passband response. To obtain further suppression more poles are

necessary.

The single pole AAF can be implemented with a simple RC at the input to the SCF. The values for can be

arrived at by using

p21

RC f



Where: fp = pole frequency

For the AAF response depicted in Figure 9, RC = 16.75ms. Setting C = 0.1f yields R = 168. The simple

RC network should be used with caution–capacitive loading can cause stability problems. Also, the value of

the resistor should be insignificant compared to the SCF input impedance.

A superior solution makes use of an additional opamp forming a damped integrator (see Figure 10). Often

this opamp will be available on the SCF chip.

The complete network, including R1, C1, R2 and C2, forms a bandpass filter with a frequency response

characteristic as shown in Figure 11.

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vi(t)

SCF

SCF CHIP

Figure 10: Active RC AAF

flp

fhp

20 dB/decade -20 dB/decade

SCF

RESPONSE

Gpb

Figure 11: AAF frequency response

The highpass cutoff (fhp) and lowpass cutoff (flp) are determined by

CR2 1

and CR2 1









The passband gain is set by

2C2R

Gpb 

C1 is not required if the input signal is biased properly (SCF internal bias = external bias).

To meet the AAF requirements shown in Figure 9 (assuming unity passband gain, flp = 9.5kHz and

fhp = 100 Hz) R1 = R2 = 100K, C2 = 168pf and C1 = 0.0159f.

SMF requirements are similar to those of the AAF. The SCF output is a sampled signal and therefore

contains replicas of the SCF response at multiples of the sampling frequency. If left unfiltered these replicas

appear as high frequency (clock) noise. In some cases this is tolerable. However, in systems where

minimum noise is critical the SMF is required.

5. Conclusion

Switched capacitor filters are sampled data systems and hence require anti-aliasing filters at their inputs and

smoothing filters at their outputs. The simplest form (and often sufficient) of these filters is the single pole RC

network. Other forms such as active RC networks and higher order filters can yield distinct benefits. In any

case, application of switched capacitor filters must be viewed at the system level in order to realize optimum

performance.

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6. REFERENCES

[1] R. Gregorian, K. W. Martin, and G. C. Temes, “Switched-Capacitor Circuit Design,” Proceedings of

the IEEE, pp. 941-966, August 1983.

[2] L. P. Huelsman and P. E. Allen, Introduction to the Theory and Design of Active Filters, McGraw-Hill,

New York, 1980.

[3] A. B. Jerri, “The Shannon Sampling Theorem - Its Various Extensions and Applications: A Tutorial

Review,” Proceedings of the IEEE, pp. 1565-1596, November 1977.