Approximate LMS tuning of continuous time filters: convergence and sensitivity analysis D.T. Westwick, B. Maundy and R. Salmeh Abstract: The quality factor of continuous time filters may be tuned using a master/slave arrangement in which an analogue implementation of the LMS algorithm is used to tune the master filter. Two such schemes are analysed, and their sensitivities to the effects of parasitic capacitances are determined theoretically and then compared. The two schemes are shown to have similar sensitivities to parasitically induced errors in the magnitude frequency response of the master filter, whereas only one of them is shown to be immune to phase errors in the master filter. The theoretical results are verified using a behavioural simulation. Approximate hardware costs of the two tuning schemes are also compared. 1 Introduction The subject of filter tuning has received a lot of attention in the literature lately, mainly because as filters for high frequencies are developed, the roles of parasitic capacitances become increasingly significant in the design of accurate filters. It is well known that in the case of high frequency OTA-C filters, depending on the topology, parasitics can affect filter passband regions, quality factor Q, and poles. Methods such as the introduction of controlled phase shifts can reduce losses, but never eliminate them completely [1] . In light of the increasing role of parasitics and drift among components, automatic tuning must be employed to obtain a desired quality factor, Q ¼ Q d . Several approaches have been used in the past, including phase locked loop (PLL) methods [2] , vector loop tuning [3], adaptive tuning, [4, 5] , correlation tuning [6], and orthogonal tuning [7]. For low Q (Qo50), the master/slave approach has been the dominant design choice, with different tuning methods being used to compensate for excess phase shifts. In [1] , filters based on the master/slave technique have been successfully demonstrated with Q values up to 50 at RF and IF frequencies. In [8, 9] the problem of tuning a high-Q bandpass filter was addressed by using the magnitude locked loop method in conjunction with a modified least mean squares (LMS) algorithm. In [9] , the authors added an extra phase locked loop around the modified LMS Q- tuning loop to provide greater accuracy in the centre frequency tuning. However, for even higher Q values (Q450), other techniques are required [6, 7]. In [10], the question of whether an approach similar to [8] could be taken to tune high-frequency low-pass or high-pass filters was addressed. While, generally speaking, the Q’s required for low-pass filters will be much lower than those required for bandpass filters, the questions naturally arise as to whether or not it is possible, and if so, of what accuracy can be achieved. The added difficulty with these filters is in the 901 phase shift that they have at resonance (unlike bandpass filters). In [10], Salmeh and Maundy used an all- pass filter to compensate for this added phase shift in the master filter, and tuned both the Q of the master filter and the time constant of the all-pass filter with an analogue LMS algorithm. A block diagram of this system is shown in Fig. 1. In this paper, the LMS scheme proposed by Salmeh and Maundy [10] , which is suitable for low-Q high-frequency filters is analysed in detail, and compared with the original approach of Stevenson and Sanchez-Sinencio [8]. Parasitic capacitances in the master/slave filters introduce zeros into their transfer functions, and hence modify the gain and phase at resonance. In addition, at high frequencies the problem is exacerbated and gets progressively worse as more Q is demanded from the filter. The effects of these non-idealities on the performance of the adaptive filter is determined. It is shown that the scheme proposed in [10] is insensitive to phase errors due to parasitics, while that Fig. 1 Block diagram representation of adaptive tuning scheme for analogue low-pass filters, proposed in [10] The authors are with the Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 E-mail: westwick@enel.ucalgary.ca r IEE, 2005 IEE Proceedings online no. 20045003 doi:10.1049/ip-cds:20045003 Paper first received 20th April and in revised form 6th July 2004 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 1