Complex filters

A filter operates on a signal to change the amplitude of specific frequency (or period) components above or below a certain specified frequency, the cut-off frequency. A low-pass filter cuts off or attenuates higher frequencies than the cut-off frequency, whereas a high-pass filter attenuates lower frequencies than the cut-off frequency.

In a band-pass filter, a high-pass and a low-pass filter are combined, with the cut-off frequency of the high-pass filter lower than the cut-off frequency of the low-pass filter, so that frequencies between these cut-off frequencies are not attenuated. The spectrum of such filters is attenuated at the low and high ends, but not in the middle.

In a notch filter, a high-pass and a low-pass filter are also combined, but the cut-off frequency of the high-pass filter is higher than the cut-off frequency of the low-pass filter, so that the frequencies between the two cut-off frequencies are attenuated. The spectrum of a notch filter is attenuated in the centre and not at the low and high ends.

Filters are modelled in one of two ways:

1. In the time domain, as the convolution operation. The signal which results from passing a unit pulse through the filter represents the impulse response of the filter; the length of the resulting signal is the length of the input signal (zero in the case of the unit pulse) added to the length of the impulse response; the amplitude of the impulse response decays in time more or less rapidly. The length of the impulse response may be thought of as the `echo' or delay effect of the filter. For each point in time, the operation of convolution involves the reflection of the signal (the `echo') in the filter window, multiplication of the reflected signal with the impulse response, and obtaining the sum of the product sequence for this window.
2. In the frequency domain, as the multiplication operation: the spectrum of the impulse response of the filter is multiplied with the spectrum of the signal which falls into the filter window, and transformed back into the frequency domain. In some implementation contexts, it may well be more efficient to perform the Fourier transformation, multiplication and inverse Fourier transformation than the convolution operation.

The problem of efficient implementation of the convolution operation is a central point in DSP design.

Second order and higher order filters can be constructed by cascading simple filters, or they may be defined directly.