Optimal FIR Filter Design based on Curve Fitting Approximation for Uncertain 2–1 Sigma–Delta...

Circuits Syst Signal ProcessDOI 10.1007/s00034-013-9671-7

Optimal FIR Filter Design based on Curve FittingApproximation for Uncertain 2–1 Sigma–DeltaModulator

Jun Peng · Jianhui Wang · Shuai Tan

Received: 11 April 2012 / Revised: 10 September 2013© Springer Science+Business Media New York 2013

Abstract Uncertain components in the integrators of 2–1 Sigma–Delta modulatorscannot eliminate the first stage quantization noise completely, and the signal-to-noiseratio in analogue-to-digital converters is not reduced perfectly either. In order to solvethe matching problem, older filter designs based on convex optimization are mathe-matically complicated, computationally intensive and not so efficient in application.In this paper, we propose a design method based on curve fitting approximation foruncertain linearized model of the modulator which is simple in principle and practicalin application. Simulation results show that the optimal filter has better performanceon multiple validations when compared to other modulator filters.

Keywords Sigma–Delta modulator · Curve fitting · Approximation · Optimal filter

1 Introduction

Recently, the requirement of the signal bandwidth reached far beyond the 10 MHz forthe analog-to-digital converter due to the emerging telecom standards. The Sigma–Delta Modulator (SDM) has been widely used in wireless and telecom systems owingto its linearity, robustness and power efficiency. However, in some applications, such

J. Peng · J. Wang · S. Tan (B)School of Information Science and Engineering, Northeastern University, Shenyang, 110004, Chinae-mail: [email protected]

J. Penge-mail: [email protected]

J. Wange-mail: [email protected]

J. PengElectrical and Computer Engineering, Purdue University, West Lafayetee, IN 47906, US

mailto:[email protected]



Circuits Syst Signal Process

as the software radio, it is difficult to achieve a wider signal bandwidth with a singleloop, and the accuracy of SDM will be dramatically reduced even though high ordertechniques are used [3]. An efficient scheme is to implement a cascaded Sigma–Delta(CSD) architecture [15] which is designed with a digital correction filter at the finalstage which is used to eliminate the effects of quantization noise resulting from theearly stage [6].

Nevertheless, such architecture also requires accurate matching of the analog loopfilters with a correction filter of the final stage [4]. A typical mismatch of 5 % betweenthem will results in leakage of −26 dB of the quantization noise of the first stage intothe output [17]. Design methods which guarantee performance over the uncertaintiesin the analogue components provide an alternative to the matching problem [1]. Therehave been several methods to minimize the mismatch of uncertain SDM. The statespace LMI design method [6] and the Central Polynomial LMI Method [14] havebeen applied for uncertain modulators. These two methods are complex in mathe-matics and intensive in computation. The method proposed by McKernan [14] hasgood performance for low-frequencies but the approximation is poor for θ > 0.1.

In this paper, we propose a method based on curve fitting which is simple, intuitiveand practical. An optimal FIR filter is derived by this method which minimizes thenoise transfer function (NTF) at all frequencies.

The remainder of this paper is organized as follows. In Sect. 2, the 2–1 Sigma–Delta modulator is presented, and the problem to be solved is introduced. In Sect. 3,a low order approximation based on the curve fitting method is derived, and optimalparameters are obtained by an analytical algorithm in Sect. 4. In Sect. 5, a simulationwith this method is presented, and Sect. 6 concludes with a related discussion.

2 Problem Description

As we discussed before, in order to get higher bandwidth and accuracy, the key pointof the cascaded SDM problem has turned to be the design of a matching filter whichcan minimize the effect of the first stage quantization noise [12]. As shown in ideal-ized linear forms in Fig. 1, the first and second stages are of second and first order,and the outputs are Y1(z) and Y2(z), respectively. The architecture is also known asthe 2–1 Sigma–Delta Modulator [16], and the stages are cascaded such that just thefirst stage quantization noise E1(z) is processed by the simpler second stage. Thematching filter F(z) is placed in the output of the second stage, to cancel the firststage quantization noise at the final subtraction for output Y3(z).

From Fig. 1, we obtain the outputs of the first and second stage as

Y1(z) = 1

1 + H2(z)z−1 + H1(z)H2(z)z−1E1(z)

+ H1(z)H2(z)

1 + H2(z)z−1 + H1(z)H2(z)z−1U(z) (1)

and

Y2(z) = H3(z)

1 + H3(z)z−1E1(z) + 1

1 + H3(z)z−1E2(z). (2)


Fig. 1 2–1 Sigma–deltamodulator

Thus, the output Y3(z) is

Y3(z) = (T11(z) − T12(z)F (z)

)E1(z) − T12(z)F (z)

H3(z)E2(z) + T11(z)H1(z)H2(z)U(z),

(3)where

T11(z) = 1

1 + H2(z)z−1 + H1(z)H2(z)z−1, (4)

T12(z) = H3(z)

1 + H3(z)z−1. (5)

And the noise transfer function (NTF) [5] of the first stage quantization is

NTF1(z)�= T11(z) − T12(z)F (z). (6)

In theory, the noise coming from the first stage can be completely eliminated bychoosing the nominal filter

F0(z) = T11(z)

T12(z). (7)

But in practice, the integrator parameters have some variations, leading to uncertaintyof the integrator. We suppose that the transfer functions of the integrators Hi(z) (i =1,2) are given by

Hi(z) = 1 − ai

1 − (1 − bi)z−1i = 1,2, (8)

where the uncertainties satisfy:

0 ≤ ai ≤ a+i , 0 ≤ bi ≤ b+

i i = 1,2. (9)

Here the noise from E2(z) to output Y3(z) is not as large as E1(z), and the variationsin H3(z) are assumed to be insignificant, so we can henceforth take T12(z) = 1 in thesecond stage. So

NTF1(z) = T11(z) − F(z) (10)


and the nominal theoretical matching filter is

Fnom(z) = T11(z) = z2 − 2z + 1

z2. (11)

The goal is to find an alternative such that the effect of the quantization noise E1(z)

to the output Y3(z) is minimized [8].

3 Curve Fitting and Approximation

As mentioned above, the optimal filter we design will minimize the maximum of thefirst stage quantization noise transfer function at all frequencies and small parameteruncertainties. The model matching problem is turned into reducing NTF1(z).

In terms of the uncertainties, Eq. (8) can be substituted into Eq. (4), leading to

T11(z) = n0 + n1z + z2

d0 + d1z + z2. (12)

Here

n0 = (1 − b1)(1 − b2), n1 = b1 + b2 − 2,

d0 = (1 − b1)(a2 − b2), d1 = b1 + b2 + a2(a1 − 2) − a1.

The form of T11(z) must be suitable for the derivation of the optimal filter, so letz = eiθ = cos(θ) − i sin(θ). In the interval [0,π ], we adapt the polynomial curvefitting method to best fit to a series of data points of sine and cosine functions, inthe least squares sense. In order to calculate easily, the polynomial is a row vector oflength 3 containing the coefficients in descending powers [11]:

{sin(θ) = A1θ

2 + B1θ + C1

cos(θ) = A2θ2 + B2θ + C2.

(13)

Keeping in mind the initial values and for easy calculation, the C1 and C2 values areset to 0 and 1, respectively. By the curve fitting method [7], we get the parametersAi , Bi (i = 1,2) of the quadratic polynomial approximation of sin(θ) = A1θ

2 +B1θ and cos(θ) = A2θ

2 + B2θ + 1. The standard errors (Root-mean-square-error)of these two polynomial approximations are 0.02417 and 0.112 at all frequencies,respectively. Even at a low frequency, the polynomial curve fitting method shows itssuperiority when compared to a second order approximation which was applied in alow frequency filter [14] design, as shown in Table 1.

Then,

z = cos(θ) + i sin(θ) = 1 + B2θ + iB1θ + A2θ2 + iA1θ

2,

z2 = cos(2θ) + i sin(2θ) = 1 + 2B2θ + 2iB1θ + 4A2θ2 + 4iA1θ

2.(14)


Table 1 Validation and comparison results

Curve fitting 2nd order

Sine Cosine Sine Cosine

Root-mean-square-error 0.01166 0.01058 0.08502 0.02173

Coefficient of determination 0.9986 0.9988 0.9238 0.995

Substituting into Eq. (12) and keeping the second order of T11(θ) in ai , bi and θ , theapproximation of T11(θ) is

T ′11(θ) = p + qB2θ + 2A2θ

2 + i(qB1θ + 2A1θ

2). (15)

Here b1b2 = p,b1 + b2 = q .Let the optimal FIR filter be

Fopt(z) = m0 + m1z + m2z2

z2. (16)

Similarly, for T ′11(θ) we get

Fopt(θ) = aiθ2 + bθ2 + ciθ + dθ + e, (17)

where

a = 4m0(2B1B2 − A1) + m1(4B1B2 − 3A1),

b = 4m0(B2

2 − A2 − B21

) + m1(2B2

2 − 3A2 − 2B21

),

c = (−2m0 − m1)B1,

d = (−2m0 − m1)B2,

e = m0 + m1 + m2.

(18)

4 Optimization

We wish to minimize the maximum of NTF1(θ) in the range of uncertainties of themodulator parameter, which is a typical optimization problem [13]:

mina,b,c,d,e

∥∥NTF1(θ)∥∥∞

b1∈[0,b+1 ],b2∈[0,b+

2 ]∀θ ∈ [0,π]. (19)

From Eqs. (15) and (17), we get

NTF1(θ) = p + qB2θ + 2A2θ2 + i

(qB1θ + 2A1θ

2) − (aiθ2 + bθ2 + ciθ + dθ + e

)

= (p − e) + (qB2 − d)θ + (2A2 − b)θ2 + (qB1 − c)iθ + (2A1 − a)iθ2.

(20)


Thus, problem (19) is equivalent to

mina,b,c,d,e

(max

b1∈[0,b+1 ],b2∈[0,b+

2 ]∣∣NTF1(θ)

∣∣2)

∀θ ∈ [0,π]. (21)

And the corresponding Hessian matrix is

Hp,q

(∣∣NTF(θ)∣∣2) =

(2 2B2θ

2B2θ 2B21θ2 + 2B2

2θ2

). (22)

For θ > 0, the Hessian matrix is positive definite and |NTF1(θ)|2 is convex withrespect to p and q , and due to the theory of convex optimization, its maxima lie atthe end of the domain of p and q [9].

Thus, the problem turns into

mina,b,c,d,e

f (a · · · e)

subject to f (a · · · e) = g(a · · · e)(23)

where f (a · · · e) = |NTF1(θ, b1 = b2 = 0)|2 and g(a · · · e) = |NTF1(θ, b1 = b+1 , b2 =

b+2 )|2.

We use the method of Lagrange multipliers to get the minimum:

F = f (a · · · e) + λ(f (a · · · e) − g(a · · · e)) (24)

where λ is the Lagrange multiplier.We set

h = (2A2 − b)θ2 − dθ − e

k = (2A1 − a)θ2 − cθ.(25)

Equation (24) can be simplified into

F = (h2 + k2) − λ

[(β2 + 2h)β2 + (β1 + 2k)β1

](26)

where β2 = p+ + q+B2θ,β1 = q+B1θ .We get a solution when the following set of equations is satisfied:

⎧⎪⎪⎨

⎪⎪⎩

∂F∂h

= 2h − 2λβ2 = 0,

∂F∂k

= 2k − 2λβ1 = 0,

∂F∂λ

= (β2 + 2h)β2 + (β1 + 2k)β1 = 0.

(27)

The solution set is⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

a = 2A1,

b = 2A2,

c = 0.5q+B1,

d = 0.5q+B2,

e = 0.5p+.

(28)


5 Simulation

For e = 0.01, we get the parameters of the optimal filter by solving⎡

⎢⎢⎢⎣

8B1B2 − 4A1 4B1B2 − 3A1 0

4B22 − 4A2 − 4B2

1 2B22 − 3A2 − 2B2

1 0

−2 −1 01 1 1

⎤

⎥⎥⎥⎦

⎡

⎣m0m1m2

⎤

⎦ =

⎡

⎢⎢⎣

2A12A2

0.5q+0.5p+

⎤

⎥⎥⎦ . (29)

The least squares method is applied to solve the overdetermined equations [2]

AT Ax = AT b (30)

where

A =

⎡

⎢⎢⎢⎣

8B1B2 − 4A1 4B1B2 − 3A1 0

4B22 − 4A2 − 4B2

1 2B22 − 3A2 − 2B2

1 0

−2 −1 01 1 1

⎤

⎥⎥⎥⎦

,

x =⎡

⎣m0m1m2

⎤

⎦ , b =

⎡

⎢⎢⎣

2A12A2

0.5q+0.5p+

⎤

⎥⎥⎦ .

For θ ∈ (0,π), a set of parameters of approximation is got by the curve fitting method⎧⎪⎪⎨

⎪⎪⎩

A1 = −0.04446,

B1 = 1.003,

A2 = −0.4975,

B2 = −1.916 × 10−4.

(31)

Substituting (31) into (30), the corresponding filter is

Fopt(z) = 1.0132z2 − 2.01755z + 1.0044. (32)

For comparison, numerical simulations have been performed using MATLAB forthe nominal filter, low frequency filter, and the optimal filter we proposed [10]. TheOSR is set at 64 and the frequency of a test input signal is 0.3 · bw, where bw is thebandwidth. It is set at 1.8 × 104. The sampling frequency is fs = 2 · bw · OSR.

For e = 0.01, the maximum noise transfer functions versus frequency for the mod-ulator filtered by the nominal, low frequency, and optimal filters are shown in Fig. 2.A maximum low-frequency noise of −90.5 dB is given by the optimal filter as com-pared to −86 dB and −79.9 dB for the low frequency filter and the nominal filter,respectively. Obviously, the optimal filter achieves better performance than the othertwo, due to its superior goodness-of-fit obtained with the curve fitting method. Thecomparisons of the average maximum noise transfer functions of the three filters atlow and high frequencies are given in Table 2.


Fig. 2 Maximum NTF1 of anuncertain modulator usingnominal, low frequency, andoptimal filters

Table 2 Average NTF of thethree filters at low and highfrequencies

Optimalfilter

Low frequencyfilter

Nominalfilter

Low frequency −80.1 dB −75.2 dB −69.0 dB

High frequency −29.6 dB −24.2 dB −25.9 dB

Fig. 3 Minimum SNRperformance comparisons of anuncertain modulator usingnominal, low frequency, andoptimal filters

At a high frequency which is above 0.932, the nominal filter achieves lower max-imum NTF compared to the low frequency filter due to the inaccurate second orderapproximation method applied in the low frequency filter design.

Figure 3 shows the minimum SNR performance of these three filters for e = 0.01.The proposed optimal filter has the largest SNR, for input between −60 and −10 dB.For a high input level which above −10 dB, the SNR of the three filters are as givenin Table 3.

The advantage of the optimal filter is not so obvious for input levels above −8 dB,and the SNR of the optimal filter is at least 5 dB lower than that of the nominal filterfor input levels around −2 dB due to lack of explicit treatment of nonlinearities in thefilter design. Another reason is that the advantage of the curve fitting is not so greatwhen input levels are small. Figure 4 validates our conclusion.

It can be seen that the nominal and low frequency filters give better calibration thanthe optimal filter when small uncertainties are present, especially for uncertaintiesless than 0.003. For uncertainties between 0.003 and 0.005, the SNR of the optimal


Table 3 SNR of the three filtersfor high input levels High input levels (dBFS)

−8 −6 −4 −2

Optimal filter 81.5 83.5 81.5 69.7

Low frequency filter 79.5 82.1 82.2 74.1

Nominal filter 74.5 77.2 77.2 74.7

Fig. 4 SNR performancecomparisons for differentuncertainties using nominal, lowfrequency, and optimal filters

filter is larger than that of the nominal filter and smaller than for the low frequencyfilter. However, when there are large uncertainties, the optimal filter outperforms thelow frequency and nominal filters.

In other words, the optimal filter provides a cheap SD modulator to be chosen com-pared to more a precise and expensive SD modulator for nominal and low frequencyfilter.

6 Conclusion

In this paper, we have developed a novel optimal FIR filter for uncertain 2–1 Sigma–Delta modulator via the curve fitting method which minimizes the first stage quantiza-tion noise over all frequencies. Simulations show that our optimal filter outperformsthe low frequency and nominal filters in terms of a lower noise transfer function andhigher SNR for a range of larger uncertainties. These features make the method weproposed suitable for making cheap SD modulators to satisfy high performance con-verters without high complexity and sensitivity.

Acknowledgement This work is supported by the Fundamental Research Funds for the Central Uni-versities of China (Project No. N120304004) and China Postdoctoral Science Foundation (Project No.2013M530937).

References

1. S. Bhattacharyya, H. Chapellat, L. Keel, Robust Control: the Parametric Approach (Prentice Hall,Upper Saddle River, 1995)


2. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)3. D.P. Brown, On limit cycles of second-order sigma-delta modulators with constant input. Circuits

Syst. Signal Process. 14(5), 675–688 (1995)4. G. Cauwenberghs, G.C. Temes, Adaptive digital correction of analog errors in MASH ADCs. I. Off-

line and blind on-line calibration. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 47(7),621–628 (2000)

5. J. Chung, D. Blaser, Transfer function method of measuring in-duct acoustic properties. I. Theory. J.Acoust. Soc. Am. 68, 907 (1980)

6. Y. Fuwen, M. Gani, An H∞ approach for robust calibration of cascaded sigma delta modulators, in45th IEEE Conference on Decision and Control 2006 (2006)

7. P.G. Guest, Numerical Methods of Curve Fitting (Cambridge University Press)8. U. Gustavsson, T. Eriksson, C. Fager, Quantization noise minimization in sigma–delta modulation

based RF transmitter architectures. IEEE Trans. Circuits Syst. I, Regul. Pap. 57(12), 3082–3091(2010)

9. A. Hjrungnes, Complex-Valued Matrix Derivatives: with Applications in Signal Processing and Com-munications (Cambridge University Press)

10. V.K. Ingle, J.G. Proakis, Digital Signal Processing Using MATLAB, 3rd edn. (Cengage Learning,Stamford, 2010)

11. H. Kellerer, V.A. Strusevich, Fully polynomial approximation schemes for a symmetric quadraticknapsack problem and its scheduling applications. Algorithmica 57(4), 769–795 (2010)

12. J. Marttila, M. Allen, M. Valkama, Quadrature sigma-delta modulators for cognitive radio-I/Q imbal-ance analysis and complex multiband principle. Circuits Syst. Signal Process. 30(4), 775–797 (2011)

13. D. Mayne, Y.-C. Ho, Q.-C. Zhao, Q.-S. Jia, Ordinal optimization: soft optimization for hard problems.J. Optim. Theory Appl. 145(3), 613–615

14. J. McKernan et al., Robust filter design for uncertain 2–1 sigma–delta modulators via the centralpolynomial method. IEEE Signal Process. Lett. 15, 737–740 (2008)

15. A. Rusu, M. Ismail, Sigma-delta solutions for future wireless handields, in 13th IEEE InternationalConference on Electronics, Circuits and Systems, 2006. ICECS ’06 (2006)

16. R. Schreier, G.C. Temes, J. Wiley, Understanding Delta-Sigma Data Converters, vol. 74 (IEEE Press,Piscataway, 2005)

17. C. Zhiheng, Y. Shouli, A digital background calibration method for mash sigma–delta; modulators byusing coefficient estimation, in IEEE International Symposium on Circuits and Systems, 2005. ISCAS2005 (2005)

Optimal FIR Filter Design based on Curve Fitting Approximation for Uncertain 2–1 Sigma–Delta...

Documents

Transcript of Optimal FIR Filter Design based on Curve Fitting Approximation for Uncertain 2–1 Sigma–Delta...