# Digital filter

A digital filter is a mathematical filter to the manipulation of a signal such as to block or pass a predetermined frequency range. The difference from the analog filter lies in the realization: Analog filters are constructed with passive electronic components such as capacitors, inductors, resistors or active with operational amplifiers. Digital filter can be realized with logic devices such as ASICs, FPGAs, or in the form of a sequential program of a signal processor.

- 4.1 advantages
- 4.2 Disadvantages of digital filters

- 5.1 Linear frequency filter
- 5.2 Distorted frequency filter
- 5.3 multirate filter

## Properties

Another key feature of process digital filter no continuous signals, but only time and value discrete signals. A discrete-time signal is periodic in the time- sequence of only individual pulses which represent the signal waveform over time, to the respective samples. The sample is a discrete value, since the digital number representation has only a finite resolution.

The filter behavior of digital filters is easier to reproduce. Also can be certain types of filters, such as to realize the so-called FIR - filter as a digital filter and not only as an analog filter circuit. Digital filters in combination with analog - to-digital converters and digital - to-analog converters also replace increasingly been realized purely analog filter structures. Digital filters are the basis of digital signal processing and are used for example in communications technology.

Continuous filter transfer functions and formed therefrom analog filters as Butterworth filters, Bessel filters, Chebyshev or elliptic filter can be prepared by adjusting the filter transfer function to the finite, discrete spectrum in the form of digital IIR filters replicate with suitably chosen filter coefficients.

## Mathematical definition

An abstract digital filter is an operator, the time-discrete digital signals just such assigns again. Often, the description will be assumed for simplicity that the signal has real numbers as values , i.e., the quantization of the sample values (i.e., the rounding on one of a finite number of values of the bit map ) of the digital signal is disregarded. A discrete-time signal x is a figure that each point of said discrete, equidistant amount

Assigns a number. It may also be the consequence of its function values

Be specified. The bracket notation is the preferred in computer science with index in mathematics.

The basic operation of a ( finite, non-recursive ) filtering operation is as follows: At each time point, or point of the grid, an environment is fixed for obvious points in time, eg every two points before and after. The form of this environment is constant over time. If the ambient only previous points in time, the filter is called causal.

Now is the time to each tuple of values in its environment before. In this tuple one same function is always applied, for example, maximum generation, averaging, weighted averages, ... If this function is linear, so the filter is called linear, or nonlinear.

Consider a family of signals resulting from time shift apart, and generates the family of the transformed signals by the filter, then the filtered signals differ by exactly the same time offset each other. The filter is time-invariant. Signal transformations with these properties are referred to as LTI systems, English for Linear Time Invariant. Looking at the discrete signal a sequence of coefficients of a Fourier series expansion, i.e., the signal values as the Fourier integrals, so can an LTI system, the amplitudes | change and relative to the input signal in the phase arg at each frequency | f (s) ( f (s) ) to turn.

### Convolution operators as LTI systems

A convolution operator is given a sequence f of coefficients in the discrete signal x acting by convolution:

This sum is well defined in the following cases:

This is called

- X bounded if K < xn < K for a K and all n ∈ ℤ,
- X " quadratsummierbar " when the number of the absolute squares converged,
- Fn f finally, when there is a finite subset of I ℤ so ≠ 0 for n ∈ I,
- F absolutely summable if the series converges to the amounts,
- F of a limited frequency response, when the Fourier series to f converges almost everywhere and ( essential ) is limited.

As you think about it, the impulse response of the convolution operator is the result of f in all these cases

For a finite filter is called the amount of I are also support the difference between starting and end points of the support is called the length of the filter. The elements of the carrier are often referred to as taps, their number being one greater than the length of the signal. Only this first case corresponds to the finite described in the introduction. The set I defines the environment, which is used to determine of the filtered values , the elements of F to define a linear function of the values of that environment.

The absolutely summable filter sequences f of the second case are not only limited, but also a steady frequency response. This gives the amplitude change for the elementary vibrations with. These are limited, so defined and

Ideal frequency selective filters take in its frequency response at only the values 0 and 1. The jumps occur can be difficult with the constant frequency responses absolutely summable and even worse with the approximate polynomial frequency responses of finite filter.

For the Fourier series, which only exist in the third case, all (as L ² functions), the following relationship applies:

The square sum of e (x) is referred to as the " energy " of the signal. Due to the Parseval identity

Can be achieved by means of a frequency-selective filter is an orthogonal decomposition of the signal.

### Finite special cases

, The support of the filter F of finite length, the filter is referred to as an FIR system for finite impulse response FIR ( Finite Impulse Response Data Sheet ). These filters are also referred to as non-recursive or without feedback implementable

, The support of the filter F no finite length, the filter is referred to as an IIR system for infinite impulse response IIR ( Infinite Impulse Response Data Sheet ). Among these are a class of filters F, which are referred to as recursive or implemented with feedback, which can be represented as the ratio of finite filter, i.e., there are two finite sequences a and b, so that the convolution product of a * f = b. Only those endless filter can be implemented at all.

## Implementations

## Advantages and disadvantages of digital filters

Digital filters play an important role in communication technology. Do you have over analog filters the important advantage to abide by their specifications exactly.

### Benefits

- No fluctuations due to tolerance of the components
- No aging of the components
- No manual adjustment in production necessary therefore quicker final testing of equipment
- Possible filter functions that are difficult or impossible to implement with analog filters, such as filters with linear phase.

### Disadvantages of digital filters

- Limited frequency range ( by periodic continuation of the spectrum)
- Limited range of values ( by Wertequantisierung )
- By internal rounding, truncation and limiting operations to word length limit point digital filters in practice on quantization noise and other non-linear effects, which manifest themselves primarily in recursive filters of higher order and a finer quantization, use of floating point numbers, matched filter structures such as the use of may require wave digital filters.
- For non- electrical input and output variables of additional effort for the conversion.

## Classification of digital filters

### Linear frequency filter

Based on the structure of two classes of digital filters can be distinguished:

A second distinction can be made on the basis of the impulse response:

FIR filters are always stable, even those with recursive elements. The reason is that the non-recursive forms have only zeros and poles in the trivial origin in the transfer function and the non-trivial poles lie in recursive forms of the FIR filter is always the unit circle. Zeros are subject regarding the stability criterion is not restricted in its location in the pole-zero diagram. If they are all inside the unit circle, then one speaks of a minimum-phase system, there is at least one outside, so it is a nichtminimalphasiges system. The design of an FIR filter, a windowing is performed to reduce the leakage effect in most cases.

IIR filters are only stable if all poles are within the unit circle. Are simple poles on the unit circle, then the system is conditionally stable, i.e., in response to the input signal. Once two or more poles is at the same point of the unit circle, or even a pole outside the unit circle, there is an unstable filter.

The advantage of IIR filters is that they have in the transfer function in addition to the zeros and poles and thus allow higher filter grades. The calculation of the IIR filter is compared with that of a complex FIR filter, and should also include a stability study of the quantized coefficients. A reliable method for determining coefficients of an IIR filter provides the method of Prony.

Is performed virtually the coefficient determination with programs such as MATLAB.

### Distorted frequency filter

( based on the low-pass - low-pass transform)

A distinction of this filter is no longer possible by means of the impulse response.

- WFIR warped FIR filter - are stable. These filter based on an FIR filter, but which is frequency- distorted. You always have an infinite impulse response.
- Wiir warped IIR filters - are also only stable if all poles are within the unit circle. They also belong to the frequency- warped filters. You can not be implemented directly because a coefficient mapping is required to remove delay-free loops.

### Multirate filter

They are used to convert between different sampling rates and avoid the occurrence of mirror spectra aliasing. Examples of multi-rate filter are CIC filter.