0.6 Digital filtering and the dft  (Page 8/13)

Practical filtering

Filtering can be viewed as the process of emphasizing or attenuating certain frequencies within a signal. Linear time-invariant filtersare common because they are easy to understand and straightforward to implement. Whether in discreteor continuous time, a LTI filter is characterized by its impulse response (i.e., its output whenthe input is an impulse). The process of convolution aggregates the impulse responses from all theinput instants into a formula for the output. It is hard to visualize the action of convolution directlyin the time domain, making analysis in the frequency domain an important conceptual tool.The Fourier transform (or the DFT in discrete time) of the impulse response gives the frequency response,which is easily interpreted as a plot that shows how much gain or attenuation (or phase shift) each frequency undergoesby the filtering operation. Thus, while implementing the filter in the time domainas a convolution, it is normal to specify, design, and understand it in the frequency domain as a point-by-pointmultiplication of the spectrum of the input and the frequency response of the filter.

In principle, this provides a method not only of understanding the action of a filter, but also of designinga filter. Suppose that a particular frequency response is desired, say one that removes certain frequencies, while leaving othersunchanged. For example, if the noise is known to lie in one frequencyband while the important signal lies in another frequency band, then it is natural to design a filter that removes thenoisy frequencies and passes the signal frequencies. This intuitive notion translates directly into amathematical specification for the frequency response. The impulse response can then be calculated directlyby taking the inverse transform, and this impulse response defines the desired filter.While this is the basic principle of filter design, there are a number of subtleties that can arise, and sophisticated routines areavailable in M atlab that make the filter design process flexible, even if they are not foolproof.

Filters are classified in several ways:

• Lowpass filters (LPF) try to pass all frequencies below some cutoff frequency and remove all frequencies above.
• Highpass filters try to pass all frequencies above some specified value and remove all frequencies below.
• Notch (or bandstop) filters try to remove particular frequencies (usually in a narrow band) and to pass all others.
• Bandpass filters try to pass all frequencies in a particular range and to reject all others.

The region of frequencies allowed to pass through a filter is called the passband , while the region of frequencies removed is called the stopband . Sometimes there is a region between where it is relativelyless important what happens, and this is called the transition band .

By linearity, more complex filter specifications can be implemented as sums and concatenations of the above basic filter types.For instance, if $h_1\left[k\right]$ is the impulse response of a bandpass filter that passes only frequencies between100 and 200 Hz, and $h_2\left[k\right]$ is the impulse response of a bandpass filter that passes only frequencies between500 and 600 Hz, then $h\left[k\right]=h_1\left[k\right]+h_2\left[k\right]$ passes only frequencies between 100 and 200 Hz or between 500 and 600 Hz.Similarly, if $h_l\left[k\right]$ is the impulse response of a lowpass filter that passes all frequencies below 600 Hz, and $h_h\left[k\right]$ is the impulse response of a highpass filter that passes all frequencies above 500 Hz, then $h\left[k\right]=h_l\left[k\right]*h_h\left[k\right]$ is a bandpass filter that passes only frequencies between 500 and 600 Hz, where $*$ represents convolution.