Autocorrelation and Cross- Correlation Methods

Any physical quantity that varies with time is a signal. Examples from physiology are: an electrocardiogram (ECG), an electroencephalogram (EEG), an arterial pressure waveform, and a variation of someone’s blood glucose concentration along time. In Fig. 1, one can see examples of two signals: an electromyogram (EMG) in (a) and the corresponding force in (b). When a muscle contracts, it generates a force or torque while it generates an electrical signal that is the EMG (1). The experiment to obtain these two signals is simple. The subject is seated with one foot strapped to a pedal coupled to a force or torque meter. Electrodes are attached to a calf muscle of the strapped foot (m. soleus), and the signal is amplified. The subject is instructed to produce an alternating pressure on the pedal, starting after an initial rest period of about 2.5 s. During this initial period, the foot stays relaxed on the pedal, which corresponds to practically no EMG signal and a small force because of the foot resting on the pedal. When the subject controls voluntarily the alternating contractions, the random-looking EMG has waxing and waning modulations in its amplitude while the force also exhibits an oscillating pattern (Fig. 1). Biological signals vary as time goes on, but when they are measured, for example, by a computerized system, the measures are usually only taken at pre-specified times, usually at equal time intervals. In a more formal jargon, it is said that although the original biological signal is defined in continuous time, the measured biological signal is defined in discrete time. For continuous-time signals, the time variable t takes values either from 1 to þ1 (in theory) or in an interval between t1 and t2 (a subset of the real numbers, t1 indicating the time when the signal started being observed in the experiment and t2 the final time of observation). Such signals are indicated as y(t), x(t), w(t), and so on. On the other hand, a discrete-time signal is a set of measurements taken sequentially in time (e.g., at every millisecond). Each measurement point is usually called a sample, and a discrete-time signal is indicated by y(n), x(n), or w(n), where the index n is an integer that points to the order of the measurements in the sequence. Note that the time interval T between two adjacent samples is not shown explicitly in the y(n) representation, but this information is used whenever an interpretation is required based on continuous-time units (e.g., seconds). As a result of the low price of computers and microprocessors, almost any equipment used today in medicine or biomedical research uses digital signal processing, which means that the signals are functions of discrete time. From basic probability and statistics theory, it is known that in the analysis of a random variable (e.g., the height of a population of human subjects), the mean and the variance are very useful quantifiers (2). When studying the linear relationship between two random variables (e.g., the height and the weight of individuals in a population), the correlation coefficient is an extremely useful quantifier (2). The correlation coefficient between N measurements of pairs of random variables, such as the weight w and height h of human subjects, may be estimated by