By superposition, we may readily generalize complex sinewave analysis to the case in which is an arbitrary superposition of input sinusoids:
where is the amplitude response, and the phase response of the LTI filter. We have thus shown by superposition that, given any input signal , the output spectrum is equal to the input spectrum multiplied by the frequency response , where the frequency response can be measured one frequency at a time using a sinusoidal input signal.
In contrast to the polar representation of frequency response , the real and imaginary parts do not have such intuitively appealing individual interpretations. Consequently, the polar form is usually preferred for expressing filter responses as a function of frequency.