Digital Filtering

Some particular types of filters that we will look at include low pass, high pass, and band pass filters which cut off portions of the frequency spectrum while allowing other frequencies to pass through the filter. In implementing these filters we will want to achieve the desired frequency response while otherwise distorting the signal as little as possible. We will concentrate our analysis on filters which are themselves linear time invariant systems. This will enable us to apply all of the techniques that we have previously developed for LTI systems to analyze the performance of the filter. However, it is also possible to construct more complicated nonlinear filters which are not LTI systems. In analyzing filters we will again encounter the concept of stability. A linear filter is stable if the corresponding LTI is stable. If we try to use an unstable filter, we’re likely to find that small amounts of noise in the input build up in the output to intolerable levels. Unstable filters are practically impossible to use. In some cases a filter is designed to process the signal in real time with little or no delay, while in other cases we must receive the entire signal before we can begin processing it. There is generally a preference for linear filters which are causal, since these filters can be implemented in real time. If an acausal filter