THE EFFECT OF SAMPLING AND QUANTIZATION ON FREQUENCY ESTIMATION Anders Høst-Madsen Dept of Information and Communications Kwangju Institute of Science and Technology 572 Sangam-dong, Kwangsan-ku, Kwangju 506-712 Korea Peter H¨ andel Signal Processing Royal Institute of Technology S-100 44 Stockholm Sweden ABSTRACT The effect of sampling and quantization on frequency estimation for a single sinusoid is investigated. Cram´ er-Rao bound for 1 bit quantization is derived, and compared with the limit of infinite quantization. It is found that 1 bit quantization gives a slightly worse performance, however, with a dramatic increase of vari- ance at certain frequencies. This can be avoided by using 4 times oversampling. The effect of sampling when using non-ideal anti- aliasing lowpass filters is therefore investigated. Cram´ er-Rao lower bounds are derived, and the optimal filters and sampling frequen- cies are found. Finally, fast estimators for 1 bit sampling, in par- ticular correlation based estimators, are derived. The paper is con- cluded with simulation results for 4 times oversampled 1 bit quan- tization. 1. INTRODUCTION We consider the classical problem of estimating the frequency, phase and amplitude of a single complex sinusoid in additive, white Gaussian noise. Thus, the continuous time observed signal is (1) where is continuous time white Gaussian noise (WGN) with power , , are the unknown parameters, and . Usually, the signal is processed digitally. In order to do the digital processing, the signal must be sampled and quantized. In most cases, analyses of accuracy do not consider this process, al- though it can considerably influence the accuracy. In this paper we will consider the influence of sampling and quantization, and in particular optimization of sampling and quantization with respect to accuracy and in a trade off with complexity. Prior to sampling, the signal is transmitted through an ana- log anti-aliasing filter. We here assume that the signal has been stationary for so long time prior to the start of the sampling pro- cess (or that it has a smooth envelope), that we can disregard the transient response. Thus, if the antialising filter has frequency re- sponse and the sampling time is the sampled signal is 1 (2) The authors’ email: madsen@dic.kjist.ac.kr, ph@s3.kth.se, respec- tively. 1 We denote all quantities associated with the continuous time signal by and all quantities associated with the discrete time signal by . Here is additive Gaussian noise, which is not necessarily white. We will return to the specific characteristics of the discrete time noise below. After sampling, the signal is quantized, i.e., rounded to one of a finite number of levels. If the quantization is very fine, e.g., 12 bits precision, the quantization can be disregarded or treated as another source of additive noise. However, some applications deal with very high frequency signals (Giga Hertz range) and fine quan- tization is impossibly or economically infeasible. We therefore consider coarse quantization, in particular single-bit quantization, with a signal given by, (3) Apart from being simple to implement, 1 bit quantization also has the advantage that no gain control is needed and, as we will see below, that very efficient algorithms for processing of one bit sam- ples can be made. The classical case is the case with infinitely fine quantization, ideal low-pass antialising filters, and . In this case, the Fisher information matrix for the unknown parameters is given by [4] (4) which can be explicitly summed and inverted to give the Cram´ er- Rao bound (CRB) (5) Furthermore, the maximum likelihood estimator (MLE) can be im- plemented by finding the maximum peak of the DFT of [4]. 2. CRAM ´ ER-RAO BOUND FOR ONE-BIT QUANTIZATION At first we will consider the case of ideal low-pass anti-aliasing filters and one bit quantization. Let and . The pdf of is then given by (6) where . We get a similar expression for the pdf of , just with instead of . Notice that this pdf is (continuously)