The central limit theorem and low-pass filters

The result of connecting many low-pass filters in series is considered. It is shown that if the impulse response of the filters is non-negative, then a version of the central limit theorem applies. Making use of the central limit theorem, we show that, as the number of filter sections gets large, the total delay of the cascaded filter system tends to the sum of the (approximate) time delays of each of the filters. Letting B_i denote the (approximate) bandwidth of the i^th filter, it is found that the bandwidth of the cascaded filter system tends to 1/√(1/B₁² + ··· + 1/B_N²). Using examples, it is shown that the rate at which the impulse response converges to a Gaussian is dependent on the nature of the impulse responses of the constituent low-pass filters.