A linear$-$phase $256-$tap FIR digital filter is to be implemented as part of an audio

processing system operating at a sampling rate of $48$ kHz$.$ Calculate the number of

multiplies and additions required to process a $10-$second duration of a $($real$)$ audio

signal, if the filter is implemented using a standard transversal filter structure.

Calculate the possible saving in the number of multiplies required to process the same

duration of input signal, if the filter is implemented using fast convolution in the

frequency domain, taking other possible computational savings into account. Each

frame consists of $10$ milliseconds of signal, zero$-$padded with a further $32$ zero$-$valued

samples to permit the use of the FFT$.$ Hamming windowing is also applied. You may

assume that the coefficients of the Hamming window are pre$-$computed, and that there

is overlap of $50\%$ between successive frames. You may ignore any additional overhead

associated with overlap$-$add processing of the FFT output.