Digital Calibration of a Continuous-Time Cascaded Σ∆ Modulator based on Variance Derivative Estimation

Robert Rutten, Lucien J. Breems Philips Research

Eindhoven, the Netherlands

Gunnar Wetzker Philips Semiconductors

Eindhoven, the Netherlands

Abstract— This paper presents a digital calibration system for a continuous-time cascaded Σ∆ modulator. The calibration algorithm matches the digital noise cancellation filter coefficients with the integrator time constants of the analog loopfilters, resulting in an optimal signal-to-quantization noise ratio. The calibration method that is proposed is based on a two-points variance estimation of the idling output datastream when zero input signal is applied. This calibration is highly insensitive to the large spread of the variance, due to the pseudo random nature of a Σ∆ modulator. The main advantages of this calibration method are the very high accuracy and very fast calibration speed, while the additional hardware costs are negligible. A 0.18µm CMOS prototype test chip has been fabricated and the measured calibration time is 0.6ms for a 10MHz, 67dB dynamic range continuous-time cascaded Σ∆ modulator.

I. INTRODUCTION Modern wireless communication systems require A/D

converters with increasing bandwidth and resolution to support new standards with higher data rates. A Σ∆ modulator satisfies this high-resolution requirement, while the effective bandwidth is often restricted to a few megahertz because of the required oversampling ratio. A Σ∆ modulator with an additional cascaded stage offers better dynamic range at a relative low oversampling ratio. This paper presents a cascaded Σ∆ modulator consisting of two 4-bit Σ∆ stages with 2nd order continuous-time loopfilters with automatic calibration (Fig. 1). The continuous-time loopfilters shape the quantization noise outside the band of interest and perform anti-aliasing filtering of the input signal X. This input signal X is fed to the first stage, while the second stage encodes the quantization error signal Q1 of the first stage. Both output signals Y1 and Y2 contain the quantization error signal Q1, however with different transfer function characteristics. An additional Noise Cancellation Filter (NCF) equalizes both characteristics and after combining streams Y1 and Y2’, the quantization error Q1 is cancelled [1]. This cancellation of Q1 is only effective when the digital NCF transfer function accurately matches with the analog transfer function. However, both stages of the

cascaded modulator have continuous-time loopfilters with coefficients based on analog component values (resistors and capacitors), which absolute values suffer from chip to chip variations up to ±20%. To compensate for this mismatch between the digital NCF and the analog loopfilter, calibration is needed.

fs

+

fs

fs

+ -+

+ -+

++ - ++Y12X

Y2’

Y1

decimation filter

Y

fs

fcal

calibration

DAC1

DAC2

DAC12Q1

Quant.

Quant.

( )sH1

( )sH2NCF

w(s) w(z)

Y2

fs

+

fs

fs

+ -+

+ -+

++ - ++Y12X

Y2’

Y1

decimation filter

Y

fs

fcal

calibration

DAC1

DAC2

DAC12Q1

Quant.

Quant.

( )sH1

( )sH2NCF

w(s) w(z)

Y2

Figure 1: Cascaded continuous-time Σ∆ modulator.

II. Σ∆ MODULATOR The 2nd order loopfilters of the cascaded Σ∆ modulator consist of two integrators with unity gain frequencies ω1 and ω2, and one feedback coefficient γ (Fig. 2). The output w1(s) is obtained by feedforward summation of the integrator output signals with weight factors c1 and c2.

+s1ω

s2ω c2

c1

+

γ

w1(s)+s

1ωs

2ω c2

c1

+

γ

w1(s)

Figure 2: Second-order integrator filter with feedforward summation and

local resonator feedback.

1-4244-0303-4/06/$20.00 ©2006 IEEE. 199

The transfer function H1(s) of the loopfilter equals:

2112

11212

1

11 )(

)()(ωωγωωω

++

==s

sccsDsNsH (1)

Likewise, the transfer function H2(s) of loopfilter H2 equals:

4322

33434

2

22 )(

)()(ωωγωωω

++

==s

sccsDsNsH (2)

where Nx and Dx represent the numerator and denominator of the transfer function Hx, c3, c4 are the feedforward scaling factors, w3 and w4 are the unity gain frequencies and y2 equals the feedback coefficient. The quantizer output signals Y1 and Y2 are discrete-time signals and can be written as a function of the signals X, Q1 and Q2 in the discrete-time domain:

)()()(

)()()()(

)()(11

11

11

11 zX

zNzDzNzQ

zNzDzDzY

++

+=

)()()(

)()()()(

)()( 122

22

22

22 zQ

zNzDzNzQ

zNzDzDzY

++

+= (3)

where Dx(z) and Nx(z) are discrete-time transfer functions, representing Dx(s) and Nx(s) respectively in the digital domain. In this paper, this time-discrete representation is calculated by the bilinear transformation. This transformation is accurate for relatively low frequencies, which is therefore appropriate for a Σ∆ modulator, since the frequency band of interest is also low compared to the sample frequency.

III. NOISE CANCELATION FILTER According to (3), both output signals Y1 and Y2 contain quantization error Q1, but with different transfers. The transfer functions of Q1 can be made equal if Y2 is filtered with the quotient of both transfer functions according to:

)()()()(

)()()(

11

22

2

1

zNzDzNzD

zNzDzHNCF +

+⋅−= (4)

where HNCF(z) equals the transfer function of the NCF filter. To simplify this equation, the loopfilter functions are chosen such that D2(z) + N2(z) = D1(z) + N1(z), which leads to:

)()()(

2

1

zNzDzH NCF −= (5)

Note that the derivation (4)-(5) in this paper corrects the slightly different derivation of HNCF in [1].

When the output of the NCF Y2’ is added to Y1, the quantization error Q1 is cancelled from the output signal Y12. Finally, Y12 is fed to the decimation filter that filters of the quantization noise outside the signal band. After simplification of (5), HNCF equals:

( ) ( )( ) ( )1

33434

21211

21

121114)( 1 −

−−

−++

++−−= −

zczczzzH NCF

ωωωωωγ (6)

The unity gain frequencies ωx values are dependent on

RC-values in the loopfilter design. The matching between those RC values is approximately 0.1-1%, while the absolute inaccuracy can be up to 20%. According to [1], the unity gain variation effect on the pole positions in (6) is negligible for frequencies in the band of interest. Therefore, the calibration algorithm does not compensate for the pole position variations. The zero location variations however, have a large effect on the filter transfer function for the frequencies in the band of interest. The zero locations are given by the numerator of (6) and can be rewritten as:

( )12484)( 2211

2 ++++−= zzzzzN NCF ωωγ (7)

The inaccuracy of the RC times is denoted with the parameter δ:

( )δωω +⋅= 1xdx (8)

where ωxd is the RC value which is the desired value and δ is the deviation on the RC value. Note that δ differs for each ωxd in one Σ∆ modulator, but this on-chip deviation is much smaller compared to batch to batch deviations. The NCF compensates for those batch to batch deviation, not for on-chip deviations. Substituting (8) into (7) yields:

( )( )22211

2 112484)( δωωγ +++++−+= zzzzzN ddNCF (9)

The NCF filter is implemented as an adaptive FIR filter, followed by an IIR filter as shown in Fig. 3.

++

4

Z-1

+++

1

+g1

Y2

IIR

2 1

Z-1Z-1

Z-1

-8 4

γ1 ωd1 ωd2

Y2’-1

Figure 3. Block diagram of NCF filter

200

The upper part is a fixed FIR filter, while the lower FIR filter is the adaptive part with one variable parameter setting g1. The IIR filter adds the two poles at the nominal locations, given by the denominator of (6). The parameter g1 is an adaptive gain and equals:

21 )1( δ+=g (10)

in the optimal case. For simplicity, g1 is implemented as a linear variable gain factor. So, if a maximum spread of 20% is assumed, δ varies between -0.2 and 0.2, and g1 ranges from 0.64 to 1.44.

IV. CALIBRATION

In this section the calibration method is described that determines the optimal NCF parameter g1 setting. With the optimal setting of g1, the quantization noise is lowest and the noise in the signal band is dominated by the thermal noise. The calibration algorithm minimizes the noise power in the signal band of interest. The amount of noise can be expressed by the variance var of the decimated output samples Y when the Σ∆ modulator is idling (zero input signal) [2]. By minimizing this variance, the optimal NCF parameter can be determined.

The number of samples of Y that is used to determine the variance should be as small as possible for fast calibration. However, the variance value is very inaccurate, when a small number of samples is used. This is illustrated by the following example. The modulator of Fig 1. has been simulated for 768.000 output samples Y, while the NCF gain g1 is set at 0.7. The output data stream Y is then divided into 1500 different subsets of 512 samples. For each subset the variance is calculated. The distribution of the calculated variance values is shown by the triangles in Fig. 4. The simulation is repeated for a g1 setting of 0.74 and the distribution is shown by the squares in Fig. 4. From this figure it can be observed that both distribution curves overlap. This means that two simulation results at different gain settings, do not always give a consistent prediction towards the optimal setting. This is due to the pseudo random nature of the quantization noise of a Σ∆ modulator. Therefore, straightforward calibration based on calculating the absolute variance is not reliable, and could converge to a wrong gain setting g1.

Fig. 5 shows calculated variance values as a function of the filter setting g1 (g1 varies between 0.41 and 1.54 with 256 linear steps). Each variance is represented by a dot and again based on a unique set of 512 output samples Y1,Y2. As was expected from the distribution plot of Fig. 4, the dots are not on a monotonic line, especially when g1 is far from the optimum (1.23 in this example). Thus, a linear control algorithm cannot be applied. Increasing the number of data samples per variance calculation improves the smoothness of the curve, but also increases calibration time while monotonicity still cannot be guaranteed.

The solid line in Fig. 5 shows the variance as function of the filter setting g1 when the same sets of 512 data samples Y1, Y2

are used for each variance calculation. The dashed line shows another variance calculation for a different fixed set of data samples. Clearly, when a fixed set of data samples is used, the variance is a monotonic function of g1.

This monotonic curve can be used in a linear control algorithm to converge to the optimal g1 setting gopt. The drawback of this calibration is that it requires a memory to store data streams Y1 and Y2. This is due to the fact that for each variance calculation the same dataset must be used to guarantee monotonicity.

Figure 4: Variance histogram for two NCF parameter settings

Figure 5: Signal variance calculation per 512 decimated output samples vs filter setting g1.

To eliminate the need for the memory, a different calibration method is proposed. With this method the derivative of the variance is estimated as a function of parameter setting g1. The derivative estimator is based on a two-points variance calculation on the same datasets Y1, Y2 with different NCF filter setting g1. The difference between those two variance calculations gives a very accurate and consistent prediction for the direction towards the optimal filter setting gopt. This direction is used as an input for the calibration algorithm.

201

Fig. 6 shows the block diagram of the calibration algorithm. In calibration mode, datastream Y2 is passed through two NCF filters with different filter setting g+∆ and g-∆. The results are both added to Y1. After adding, both signals are decimated by two decimation filters. With the subsequent blocks the variances of the decimated output signals Yg+∆ and Yg-∆ are determined. The constants g2, g3 and g4 determine the speed and accuracy of the variance calculation. The resulting variances (for example var1m and var2m on the solid line in Fig. 5) are substracted from each other, which results in dvarm. After p number of decimated output samples, dvarm is fed to an integrator. The filter setting g1 is updated and a new calibration cycle starts. Two new variances var1m+1 and var2m+1 are calculated, using the new filter settings. The calibration can be stopped when the gain setting g1 is toggling around its optimal value.

++z-1

NCF + +

X

+ +z-1 z-1- -

+ ++z-1 z-1-

+

+

Y2

Y1

+∆

NCF

-

X

+-

decimation filter

fs/8 fs/8fs

fs/8 fs/8

fs

fs/(8*p)

fs /2 fs /4 fs /8

decimation filter

fs /2 fs /4 fs /8

+-

∆+g

∆−g

var1

var2

dvar

Yg+∆

Yg-∆

Variance estimator 1

Variance estimator 2

g2 g3

g2 g3

g4g1 HIntegrator

++++z-1z-1

NCFNCF ++ ++

X

++ ++z-1z-1 z-1z-1- -

++ ++++z-1z-1 z-1z-1-

++

++

Y2

Y1

++∆

NCFNCF

-

X

++-

decimation filter

fs/8 fs/8fs

fs/8 fs/8

fs

fs/(8*p)

fs /2 fs /4 fs /8

decimation filter

fs /2 fs /4 fs /8

+-

∆+g

∆−g

var1

var2

dvar

Yg+∆

Yg-∆

Variance estimator 1

Variance estimator 2

g2 g3

g2 g3

g4g1 HIntegrator

Figure 6: Derivative variance estimation and calibration system.

V. MEASUREMENTS The proposed calibration system of Fig. 6 has been

implemented in a 0.18µm CMOS prototype chip, as shown in Fig. 8. The A/D converter is used in a quadrature configuration, therefore two NCF’s and two decimation filters are implemented for the I-path and the Q-path. The area of these filters equals 0.55 mm2. During calibration, both NCF’s and decimation filters are used for calibrating one path, so the additional hardware for the calibration system only consists of two variance calculators and an integrator. The additional area for this calibration system equals 0.085 mm2. Fig. 7 shows the NCF setting during calibration. First, the I-path is calibrated. The settling time is 0.6ms. Next, the Q-path is calibrated, which also takes 0.6ms. The Q-path Σ∆ modulator is located close to the digital NCF and decimation filters and suffers from some substrate noise, therefore, the gain setting of this path is slightly varying.

Figure 7: Calibration measurement of I & Q path, using datasets of 512 decimated output samples.

Figure 8: Chip microphotograph of Σ∆ modulator including digital calibration and post-processing.

VI. CONCLUSIONS This paper has presented a calibration algorithm that

realizes an accurate matching of the digital Noise Cancellation Filter coefficients and the integrator time constants of the analog loopfilters of a continuous-time cascaded Σ∆ modulator to achieve optimal dynamic range. The calibration method uses a two-points variance estimation of the in-band noise to minimize the in-band quantization noise. The derivative variance estimation makes the calibration very robust and fast. The additional hardware cost in a quadrature system is limited to only two variance estimators and an integrator.

REFERENCES [1] L. Breems, R. Rutten, G. Wetzker, “Cascaded Continuous- Time Σ∆

Modulator With 67-dB Dynamic Range in 10-MHz Bandwidth,’” Solid-State Circuits, IEEE Journal of, Volume: 39, pp. 2152- 2160, 12 December 2004.

[2] Jae Hoon Shim, In-Cheol Park, and Beomsup Kim, “A Hybrid Delta-Sigma Modulator with Adaptive Calibration”, Proc. of ISCAS 2003, pp. 1025-1028, May 2003.

Σ∆ Modulator I & Q

NCF’s & DecimatorsCalibration

202