An adaptive filter is a digital filter that can adjust its coefficients to give the best match t An
adaptive filter is a digital filter that can adjust its coefficients to give the best match to a given
desired signal. When an adaptive filter operates in a changeable environment the filter
coefficients can adapt in response to changes in the applied input signals. Adaptive filters
depend on recursive algorithms to update their coefficients and train them to near the optimum
solution. An everyday example of adaptive filters is in the telephone system where, impedance
mismatches causing echoes of a signal are a significant source of annoyance to the users of the
system. The adaptive signal process is here to estimate and generate the echo path and
compensate for it. To do this the echo path is viewed as an unknown system with some impulse
response and the adaptive filter must mimic this response.
Adaptive Filters are generally implemented in the time domain which works well in most
scenarios however in many applications the impulse response become long, and increasing the
complexity of the filter beyond a level where it can no longer be implemented efficiently in the
time domain. An example of acoustic echo cancellation applications is in hands free telephony
system. However there exists an alternative solution and that is to implement the filters in the
frequency domain. The Discrete Fourier Transform or Fast Fourier Transform (FFT) allows the
conversion of signals from the time domain to the frequency domain in an efficient manner.
Despite the efficiency of the FFT the overhead involved in converting the signals to the
frequency domain does place a restriction on the use of the algorithm. When the impulse
response of the unknown system and hence the impulse response of the filter is long enough
however this is not an issue since the computational cost of the conversion is much less than that
of the time domain algorithm. The actual filtering of the signals requires little computational
cost in the frequency domain. Investigation of the so-called crossover point, the point where the
frequency domain implementation becomes more efficient than the time domain implementation
is important to establish the point where frequency domain implementation becomes practica
