![]() If the resulting filter is not causal, it can be made causal by introducing an appropriate time-shift (or delay). A standard approach is to leave this requirement until the final step. In order to be implementable, any time-dependent filter (operating in real time) must be causal: the filter response only depends on the current and past inputs. The goal of the design process is then to realize a filter which tries to meet both these contradicting design goals as much as possible. The latter condition can be realized by considering a very narrow function as the wanted impulse response of the filter even though this function has no relation to the desired frequency function. For example, we may want both a specific frequency function of the filter and that the resulting filter have a small effective width in the signal domain as possible. In some cases it may even be relevant to consider a frequency function and impulse response of the filter which are chosen independently from each other. However, in certain applications it may be the filter's impulse response that is explicit and the design process then aims at producing as close an approximation as possible to the requested impulse response given all other requirements. That means that any requirement on the frequency function is a requirement on the impulse response, and vice versa. There is a direct correspondence between the filter's frequency function and its impulse response: the former is the Fourier transform of the latter.
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