Uncertainty Propagation in Finite Impulse Response Filters: Evaluating the Gaussian Assumption

A common assumption in signal processing is that underlying data numerically conforms to a Gaussian distribution. It is commonly utilized in signal processing to describe unknown additive noise in a system and is often justified by citing the central limit theorem for sums of random variables, although the central limit theorem applies only to sums of independent identically distributed random variables. However, many linear operations in signal processing take the form of weighted sums, which transforms the random variables such that their distributions are no longer identical. One such operation is a finite impulse response (FIR) filter. FIR filters are commonly used in signal processing applications as a pre-processing step. FIR output noise is generally assumed to be Gaussian. This article examines the FIR output response in the presence of uniformly distributed quantization noise. We express the FIR output uncertainty in terms of the input quantization uncertainty and filter coefficients. We show that the output uncertainty cannot be assumed to be Gaussian, but depending on the application a Gaussian estimation may still be useful. Then, we show through detailed numerical simulations that the output uncertainty distribution of the filter can be estimated through its most dominant coefficients.