Question

The squaring operator defined in Eq.(1) is quite sensitive to large amplitudes of the signal. To get around this problem, an average magnitude function is sometimes used to provide a measure of the amplitude variations. The computation of the average magnitude function involves the computation of the absolute value of the signal sample: y[n]=∣x[n]∣. where x[n] and y[n] are the input and output signals. Is the absolute operator linear? Is it timeinvariant? Is it causal?

Answer 1

The absolute operator, represented by the function ∣x[n]∣, is not linear. For a function to be linear, it must satisfy two properties: additivity and homogeneity. Additivity means that if we have two input signals x1[n] and x2[n], the output of the function applied to the sum of these signals should be equal to the sum of the outputs of the function applied to each individual signal. Homogeneity means that if we scale the input signal by a constant factor, the output of the function should also be scaled by the same factor.

However, the absolute operator does not satisfy these properties. Taking the absolute value of the sum of two signals is not equivalent to the sum of the absolute values of the individual signals. In mathematical terms, ∣x1[n] + x2[n]∣ ≠ ∣x1[n]∣ + ∣x2[n]∣. Similarly, scaling the input signal by a constant factor does not scale the output by the same factor.