Analog and Digital Technologies
Filters

In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature.

Most often, this means removing some frequencies and not others in order to suppress interfering signals and reduce background noise. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist.

The drawback of filtering is the loss of information associated with it. Signal combination in Fourier space is an alternative approach for removal of certain frequencies from the recorded signal.

There are many different types of classifying filters and these overlap in many different ways; there is no simple hierarchical classification. Filters may be:

- analog or digital
- discrete-time (sampled) or continuous-time
- linear or non-linear
- infinite impulse response (IIR) or finite impulse response (FIR) type of discrete-time or digital filter.

Some terms used to describe and classify linear filters:

- Low-pass filter – low frequencies are passed, high frequencies are attenuated.
- High-pass filter – high frequencies are passed, low frequencies are attenuated.
- Band-pass filter – only frequencies in a frequency band are passed.
- Band-stop filter or band-reject filter – only frequencies in a frequency band are attenuated.
- Notch filter – rejects just one specific frequency – an extreme band-stop filter.

Filters can be built in a number of different technologies. The same transfer function can be realized in several different ways, that is the mathematical properties of the filter are the same but the physical properties are quite different. Often the components in different technologies are directly analogous to each other and fulfill the same role in their respective filters.

- Electronic filters were originally entirely passive consisting of resistance, inductance and capacitance. Active technology makes design easier and opens up new possibilities in filter specifications.
- Digital filters operate on signals represented in digital form. The essence of a digital filter is that it directly implements a mathematical algorithm, corresponding to the desired filter transfer function, in its programming or microcode.

## Linear analogue filters

Linear continuous-time circuit is perhaps the most common meaning for filter in the signal processing world, and simply "filter" is often taken to be synonymous. These are filters that are designed to remove certain frequencies and allow others to pass.

Such a filter is, of necessity, a linear filter. Any non-linearity will result in the output signal containing components of frequency which were not present in the input signal.

The modern design methodology for linear continuous-time filters is called network synthesis. Some important filter families designed in this way are:

- Chebyshev filter, has the best approximation to the ideal response of any filter for a specified order and ripple.
- Butterworth filter, has a maximally flat frequency response.
- Bessel filter, has a maximally flat phase delay.

The difference between these filter families is that they all use a different polynomial function to approximate to the ideal filter response. This results in each having a different transfer function.

Especially in the field of telecommunications, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with transmission lines.

## Digital filters

In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal.

This is in contrast to the other major type of electronic filter, the analog filter, which is an electronic circuit operating on continuous-time analog signals. An analog signal may be processed by a digital filter by first being digitized and represented as a sequence of numbers, then manipulated mathematically, and then reconstructed as a new analog signal. In an analog filter, the input signal is directly manipulated by the circuit.

A digital filter is characterized by its transfer function, or equivalently, its difference equation. Mathematical analysis of the transfer function can describe how it will respond to any input.

### Finite impulse response filter

Finite impulse response filter, called FIR filter – is a filter whose impulse response is of finite duration, because it settles to zero in finite time.

FIR filters have no feedback so output signal depends only on samples of input signal. In case we have N samples, N –1 is filter level.

Difference equation is defined as:

Transfer function is defined as:

The main advantages of FIR filters are:

- Because there is no feedback, errors are no compounded by summed iterations.
- Simpler implementation.
- Stability. This is because there is no feedback so all poles* are located at the origin. The system is stable if the absolute value of each pole (poles are roots of denominators of filter transfer function) is less than one.
- It’s easily designed to have linear phase. This is guaranteed by making impulse response symmetric or anti symmetric.

*Note: poles are roots of denominators of filter transfer function.

Impulse response (linear phase frequency characteristic)

To design a filter means to select the coefficients such that the system has specific characteristics. The required characteristics are stated in filter specifications. Most of the time filter specifications refer to the frequency response of the filter. There are different methods to find the coefficients from frequency specifications:

- Intuitive method
- Frequency sampling method
- Window design method
- Weighted least square design

Impulse response, magnitude and phase frequency characteristic of FIR filter

### Infinite impulse response filter

Infinite impulse response filter, called IIR filter – is a filter whose impulse response is function that is non-zero over an infinite length of time.

Filter output sample is given as a sum of N samples of the input signal weighted with ak coefficients, and samples of output signal weighted with bk coefficients. This is obvious also from difference equation defining IIR filter:

Difference equation defining IIR filter is:

and transfer function is defined as:

Transfer function is fraction of two polynomials and so stability is not always guaranteed. As was mentioned before, the system is stable if the absolute value of each pole is less than one. There are several methods how to stabilize non stabile filter, as PLSI algorithm or using all-pass filter.

To design a filter means to select the coefficients such that the system has specific frequency characteristic. It means for IIR filters to determine level of numerator and denominators and coefficients ak and bk. The design methods of IIR filters can be divided into two groups.

Into first group belong straightforward methods:

- Intuitive method
- Frequency sampling method
- Prony method

To the other group is based on analog filters design method. When a digital IIR filter is going to be implemented, an analog filter (e.g. Chebyshev filter, Butterworth filter, Elliptic filter) is first designed and then is converted to a digital filter by applying discretization techniques. Here belong methods as:

- Bilinear transform
- Impulse invariance.

Impulse response, magnitude and phase frequency characteristic of IIR filter (exponential function)
Impulse response, magnitude and phase frequency characteristic of IIR filter (sinc function)

The main advantage IIR filters have over FIR filters is that through recursion they use fewer taps. Therefore in digital signal processing applications IIR filters use fewer computing resources than an equivalent FIR filter.

A disadvantage of IIR filters is they can be unstable. The implementation of IIR filters is more complicated than the implementation of FIR filters.