ADAPTIVE TECHNIQUES Spectral estimation using an adaptive oscillator Prof. Lloyd J. Griffiths and Kevin B. McGregor Indexing term: Signal processing Abstract: The paper presents experimental results and comments relating to an adaptive oscillator structure proposed recently by Griffiths. A possible lattice structure for the new adaptive filter is also given. 1 Background The adaptive filtering structure of interest to this paper is shown in Fig. 1. With the switch in the up position the struc- ture operates as an adaptive linear prediction filter. An itera- tive algorithm is used to update the g t coefficients such that (e 2 (n)) is minimised. The algorithm may be either the LMS method due to Widrow et al. [1] or a weighted least-squares method such as those studied by Medaugh [2]. e(n) adaptive algorithm Fig. 1 Filter structure used in the adaptive frequency tracking oscillator The properties of this structure have been well studied and have been reported previously [1, 3]. Sinusoidal components which present a high signal/noise ratio (SNR) in x(n) are eliminated owing to the fact that the zeros of h(z) lie near the unit circle at angles corresponding to the frequencies of the sinusoids. Frequency estimates derived from the zero locations are then very accurate. At lower SNRs, however, the zeros move towards z = 0 on nonradial arcs, and frequency estimates based on the angle of the zeros become biased. To deal with this problem, a modification of the linear predictor which involves placing the switch S in the lower position was proposed. In this case the structure may be viewed as an oscillator defined by the following equations: gl (n)x(n-l) = x(n) gi(n+ 1) = g t (n) + ne(n)x(n-l) (1) (2) The first is a time-varying Z,fh-order difference equation in the estimate x(n) and the second is the LMS algorithm which is minimising the output power (e 2 (n)). A quasi-steady-state analysis is made in Reference 4. The filter coefficients are assumed to be close to their steady-state values, such that ifjOO —£i- Thus, for x(n) to be a stationary sequence, the roots of the characteristic polynomial of the filter7j must satisfy ?, I = 0 or 1 for all / (3) Further, the analysis showed that any root that has a magni- tude of unity must lie at a frequency corresponding to the Paper 2424F, first received 1st April and in revised form 15th October 1982 The authors are with the Department of Electrical Engineering, Univer- sity of Colorado, Boulder, CO 80309, USA 246 input sinusoids. Minimising the squared error ensures that the oscillator amplitude and phase at this frequency match x(n). Fig. 2 illustrates the root behaviour for a single-sinuoid example. The short broken lines represent the locus of roots for the switch in the upward position as the SNR is varied. switch in upward position switch in downward position unit circle Fig. 2 Root locations for the adaptive oscillator for single-input- sinusoid case 2 Experimental results Several experiments were performed implementing the filter as given in Fig. 1. All experiments were performed on the CDC Cyber 172 at the University of Colorado. The switch was kept in the upper position until approximate steady state was reached. At that point, the switch was changed to the downward position. The first example employed a second-order filter. Fig. 3 shows the output of this filter for an input signal given by x(ri) = A sin (2irf c n) + w(n) (4) 40 80 120 160 200 240 280 320 360 400 iteration n Fig. 3 Error output of the second-order oscillator IEEPROC, Vol. 130, Pt. F, No. 3, APRIL 1983