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Analysis, Design, and Order Estimation of Least-Squares FIR Equalizers for Bandwidth Extension of ADCs

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Abstract

In modern mixed-signal systems, it is important to build the conversion components with a flat frequency response over their full Nyquist frequency band. However, with increasing circuit speed, it is becoming more difficult to achieve this, due to limitations of the analog front-end circuits. This paper considers finite-length impulse-response (FIR) filters, designed in the least-squares sense, for the bandwidth extension of analog-to-digital converters, which is one of the most important applications in frequency response equalization. The main contributions of this paper are as follows: Firstly, based on extensive simulations, filter order-estimation expressions of the least-squares designed equalizers are derived. It appears to be the first time that order-estimation expressions are presented for any least-squares designed FIR filter. These expressions accurately estimate the order required for given specifications on the targeted extended bandwidth systems. Secondly, based on the derived order-estimation expressions, systematic design procedures are presented, which contribute to reducing the design time. Finally, a relation between the dynamic-range degradation and the system parameters is also derived and verified in the paper.

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Notes

  1. Noncausal filters are considered in the design for convenience. The corresponding causal filter implementation is obtained by introducing an appropriate amount of delay elements. Further, even-order filters are assumed throughout the paper, which matches a desired noncausal-filter frequency response that equals unity when the impulse response is nonzero for \(k\in [-N/2,\, N/2]\). Naturally, one may alternatively design odd-order filters, after appropriate minor modifications in the equations, but such fine details have no essential effect on the results to be derived, and they are therefore left out.

  2. Alternatively, one can design \(H\left( e^{j\omega T_{s}}\right) \) to approximate \(1/Q_{c}(j\omega )\), but this gives an additional frequency-dependent weight of \(1/Q_{c}(j\omega )\) to the error \(Q_{c}(j\omega )H\left( e^{j\omega T_{s}}\right) -1\).

  3. Since it is more convenient and direct to measure the dynamic degradation in terms of number of bits, D, \(D_p\), and \(D_s\) are introduced and derived as (21)–(23). In case \(D_s\) is given, \(L_s\) can be directly obtained from (23)

  4. It is natural to use \(W_{r}=L_{r}\) as the first attempt which stems from the minimax design, where \(W_{r}\) is set to the ripple ratio [22].

  5. The values \(10^{-7}\) and \(10^{-3}\) of \(L_{p}\) correspond to, approximately, \(-50\) to \(-15\) dBc of the passband ripple, respectively. If greater values than \(10^{-3}\) are utilized, the filter order required goes to zero (as illustrated in Figs. 4 and 6), and they are therefore excluded. Smaller values than \(10^{-7}\) for the passband error power are also of less interest in practical applications, as it is rare to have requirements on extremely small frequency response errors in the passband.

  6. It is firstly found that the order estimates have different function forms for \(W_{r}>1\) and \(W_{r}<1\), which are given in (29) and (30), respectively. Thus, it is straightforward to divide \(W_{r}\) into three regions as \(W_{r}>1\), \(W_{r}=1\), and \(W_{r}<1\). Furthermore, we have not found a reasonably simple order-estimation expression that covers the whole parameter range of \(W_{r}\). Therefore, we divide the whole range into five regions.

  7. Due to the discontinuity, we exclude the regions of (1, 1.01) and (0.99, 1). In practical applications, values of \(W_{r}\) close to unity are not of interest anyhow, as one would then use \(W_{r}=1\).

  8. In this example, the single-parameter algorithm needs only a few designs, typically less than 10, to obtain the filter with order of \(N_{sp}\), whereas the dual-parameter algorithm typically takes hundreds of designs to achieve the minimal order \(N_{opt}\) and \(W_{r,opt}\). Further, the number of designs for the dual-parameter algorithm is dependent on the regulating accuracy of \(W_r\).

  9. It is obvious that the dual-parameter search algorithm requires more time to find the optimal filter order and the weighting ratio, whereas the single-parameter search algorithm only needs to regulate and determine the filter order. Therefore, the number of search iterations is obviously reduced, and the design time is dependent on the number of search iterations.

References

  1. J. Hancock, Advantages and disadvantages of using DSP filtering on oscilloscope waveforms. Application Note 1494, Agilent Technologies (2004), https://www.eetimes.com/document.asp?doc_id=1275751

  2. S.C. Chapra, R.P. Canale, Numerical Methods for Engineers (McGraw-Hill, Maidenherd, 2012)

    Google Scholar 

  3. S. Dhabu, A.P. Vinod, A new time-domain approach for the design of variable FIR filters using the spectral parameter approximation technique. Circuits Syst. Signal Process. 36(5), 2154–2165 (2017)

    Article  Google Scholar 

  4. Z. Hao, Y. Peng, Research on analog bandwidth enhancement technology for digital storage oscilloscope. in Proceedings of IEEE International Conference on Electronic Measurement and Instruments (2011), pp. 217–220

  5. L. Hars, Frequency response compensation with DSP. IEEE Signal Process. Mag. 20(4), 91–95 (2003)

    Article  Google Scholar 

  6. X. Huang, Y. Wang, Z. Yan, H. Xian, M. Liu, Closed-form FIR filter design with accurately controllable cut-off frequency. Circuits Syst. Signal Process. 36(2), 721–741 (2017)

    Article  MATH  Google Scholar 

  7. K. Ichige, M. Iwaki, R. Ishii, Accurate estimation of minimum filter length for optimum FIR digital filters. IEEE Trans. Circuits Syst. II 47(10), 1008–1016 (2000)

    Article  Google Scholar 

  8. H. Johansson, P. Löwenborg, A least-squares filter design technique for the compensation of frequency-response mismatch errors in time-interleaved A/D converters. IEEE Trans. Circuits Syst. II: Express Briefs 55(11), 1154–1158 (2008)

    Article  Google Scholar 

  9. W.J. Kenneth, Digital Signal Processing (DSP) in Oscilloscopes. Application Note (LeCroy Corporation, New York, 2010)

    Google Scholar 

  10. N. Kurosawa, H. Kobayashi, K. Maruyama, H. Sugawara, K. Kobayashi, Explicit analysis of channel mismatch effects in time-interleaved ADC systems. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 48, 261–271 (2001)

    Article  Google Scholar 

  11. W.R. Lee, L. Caccetta, K.L. Teo, V. Rehbock, A unified approach to multistage frequency-response masking filter design using the WLS technique. IEEE Trans. Signal process. 54(9), 3459–3467 (2006)

    Article  MATH  Google Scholar 

  12. S.M. Louwsma, A.J.M. Van Tuijl, M. Vertregt, B. Nauta, A 1.35 GS/s, 10 b, 175 mw time-interleaved AD converter in 0.13 \(\mu \)m CMOS. IEEE J. Solid-State Circuits 43(4), 778–786 (2008)

    Article  Google Scholar 

  13. B. Murmann, C. Vogel, H. Koeppl, Digitally enhanced analog circuits: system aspects. in Proceedings of IEEE International Symposium on Circuits and Systems (2008), pp. 560–563

  14. A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing (Prentice Hall, Englewood Cliffs, 1989)

    MATH  Google Scholar 

  15. J.J. Pickerd, DSP in high performance oscilloscopes. TektronixTM White Paper (2005)

  16. L. Rabiner, J. McClellan, T. Parks, FIR digital filter design technique using weighted-Chebyshev approximation. IEEE Proc. 63(4), 595–610 (1975)

    Article  Google Scholar 

  17. Z.U. Sheikh, H. Johansson, A class of wide-band linear-phase FIR differentiators using a two-rate approach and the frequency-response masking technique. IEEE Trans. Circuits Syst. I: Reg. Pap. 58(8), 1827–1839 (2011)

    Article  MathSciNet  Google Scholar 

  18. Z.U. Sheikh, A. Eghbali, H. Johansson, Linear-phase FIR digital differentiator order estimation. in Proceedings of European Conference on Circuit Theory and Design (Linköping, Sweden, 2011), pp. 29–31

  19. J.-J. Shyu, S.-C. Pei, C.-H. Chan, Y.-D. Huang, Minimax design of variable fractional-delay FIR digital filters by iterative weighted least-squares approach. IEEE Signal Process. Lett. 15, 693–696 (2008)

    Article  Google Scholar 

  20. P.P. Vaidyanathan, S.-M. Phoong, Y.-P. Lin, Signal Processing and Optimization for Transceiver Systems (Cambridge University Press, Cambridge, 2010)

    Book  MATH  Google Scholar 

  21. R.J. Van de Plassche, CMOS Integrated Analog-to-Digital and Digital-to-Analog Converters (Kluwer Academic Publishers, Dordrecht, 2003)

    Book  MATH  Google Scholar 

  22. Y. Wang, H. Johansson, H. Xu, J. Diao, Minimax design and order estimation of FIR filters for bandwidth extension of ADCs. in Proceedings of IEEE International Symposium on Circuits and Systems (2016), pp. 2186–2189

  23. L. Wanhammar, H. Johansson, Digital Filters Using Matlab (Linköping University, Linköping, 2011)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61701509 and 61704191).

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Authors and Affiliations

  1. College of Electronic Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Yinan Wang, Nan Li & Qingjiang Li

  2. Division of Communication Systems, Department of Electrical Engineering, Linköping University, 581 83, Linköping, Sweden

    Håkan Johansson

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  1. Yinan Wang
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Correspondence to Yinan Wang.

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Wang, Y., Johansson, H., Li, N. et al. Analysis, Design, and Order Estimation of Least-Squares FIR Equalizers for Bandwidth Extension of ADCs. Circuits Syst Signal Process 38, 2165–2186 (2019). https://doi.org/10.1007/s00034-018-0958-6

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