IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO 2, FEBRUARY zyx 1996 B. F. La Scala, R. R. Bitmead, and M. R. James, “Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem,” zyxwvutsrqpon Math. Control, Signals zyxwvutsrqp Syst., vol. 8, no. 1, pp. 1-27, 1995. B. G. Quinn, “Estimating frequency by interpolating using Fourier coef- ficients,” IEEE Trans. Signal Processing, vol. 42, no. 5, pp. 1264-1268, 1994. D. R. A. McMahon and R. F. Barrett, “An efficient method for the estimation of the frequency of a single tone in noise from the phases of discrete Fourier transforms,” Signal Processing, vol. 11, pp. 169-177, 1986. zyxwvutsrqpon S. M. Kay, Modern Spectral Estimation: Theory and Application. En- glewood Cliffs, NJ: Prentice Hall, 1988. E. J. Hannan, “The central limit theorem for time series regression,” Stochastic Processes Applications, vol. 9, pp. 28 1-289, 1979. A. H. Jazwinski, Stochastic Processes Filtering Theory. London, U.K.: Academic, 1970. B. G. Quinn, R. F. Barrett, and S. J. Searle, “The estimation and HMM tracking of weak narrowband signals,” in Proc. Int. Con$ zyxwvuts Acoust., Speech, Signal Processing, Adelaide, Australia, 1994. On the Uniform ADC Bit Precision and Clip Level Computation for a Gaussian Signal Naofal Al-Dhahir and John M. Cioffi zyxwvutsrq Abstruct- The problem of computing the required bit precision of analog-to-digital converters is revisited with emphasis on Gaussian sig- nals. We present two methods of analysis. The first method fixes the probability of overload and sets the dynamic range of the quantizer to accommodate the worst-case signal-to-quantization noise ratio (SQNR). The second method sets the clipping level of the quantizer to meet a desired overload distortion level, using knowledge of the input probability density function. New closed-form expressions relating the distorlion- minimizing clip level of the uniform quantizer and the input bit rate are derived and shown to give remarkably close results to the optimum ones obtained using numerical iterative procedures devised elsewhere. I. INTRODUCTION Amplitude quantization refers to the transformation of an analog sample that can take a continuum of values into one that c c only assume a finite set of levels. Inherent in the quantization process is the introduction of quantization noise that has two components: granular noise and overload distortion due to quantization errors that arise when the input signal amplitude lies within or outside the maximum input range of the quantizer, respectively. In this correspondence, the focus on Gaussian input signals is motivated by our interest in multicarrier modulation, which has emerged over the past few years as an excellent modulation scheme that achieves high performance at a practical implementation com- plexity [I]. Briefly, the multicarrier signal is the sum of N (a large Manuscript received November 8, 1994; revised August 17, 1995. This work was supported by NSF under Contract NCR-9203131, JSEP under Contract DAAL 03-91-C-0010 and by NASA under Contract NAG2-842. The associate editor coordinating the review of this paper and approving it for publication was Dr. Akihiko Sugiyama. zyxwvutsr N. Al-Dhahir was with Stanford University, Stanford CA 94305 USA. He is now with General Electric Research and Development Center, Schenectady, NY 12301 USA (e-mail: zyxwvutsrq aldhahir@birch.crd.ge.com). J. Cioffi is with Stanford University, Stanford CA 94305 USA. Publisher Item Identifier S 1053-587X(96)01659-3. number typically 256 or 512) independent low-rate quadrature ampli- tude modulated (QAM) subsignals (sometimes called subchannels). Although the number of bits and the input power allocated to each subchannel can be different in order to maximize the overall data rate, we shall assume identical QAM-constellation sizes and a flat power distribution across all subchannels.’ Direct application of the central limit theorem implies that the probability density function (pdf) of a multicarrier signal can be considered for all practical purposes to be approximately Gaussian. In other words where, without loss of generality, we have assumed that the input signal has unity power. In Section 11, we present two methods for computing the bit precision of the uniform quantizer for a multicmier input signal. The problem of optimizing the clip level (to minimize the total quantization distortion) of the uniform quantizer was first solved in [6] by resorting to iterative numerical procedures. In Section 111, under the common assumption of uniformly distributed granular noise, we present new closed-form analytical expressions that demonstrate very accurately the complicated functional interdependence between the optimum clip level and the input bit rate (in bits per symbol). Details of the derivation are given in Appendixes A and B. II. BIT PRECISION ANALYSIS Three sources contribute to the required number of the analog-to- digital converter (ADC) bits for a multicarrier signal. The first source is information bits that achieve a desired probability of error level in the presence of additive white Gaussian noise (AWGN). Second, extra quantization bits are needed to keep the quantization noise at a factor of a(a > 1) times smaller than the AWGN level. Finally, additional bits are added to accommodate the high peak-to-average ratio (PAR) of the multicarrier input signal. A. First Method For an M-point QAM signal constellation, from which a multi- carrier subsignal is derived, the signal-to-noise ratio (SNR) required to support b,,fo information bits per dimension (M = 2’*%.”fo) at a probability of error P, is given by [5] zyxw 2 where Q(x) dGf sr Ae-ydv. For the midrise uniform quantizer shown in Fig. 1, if we assume b output bits per dimension, then it is a simple matter to show that the SNR at the output of the uniform quantizer is given by [4] - S w = 6$ + 4.77 - 20 log,, (Ico~) : in decibels. (3) Using this method, the PAR factor Kor. is set large enough such that the probability that the magnitude of the input sample exceeds IiOL, It was shown in [7] that this assumption simplifies the analysis significantly and serves as a good approximation to the general case of nonuniform bit and power distnbuuons across the subchannels. 1053-587W96$05.00 0 1996 IEEE