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An FIR filter
An FIR filter

In electronics, a digital filter is any electronic filter that works by performing digital mathematical operations on an intermediate form of a signal. A finite impulse response (FIR filter is a type of a Digital filter. Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted This is in contrast to older analog filters which work entirely in the analog realm and must rely on physical networks of electronic components (such as resistors, capacitors, transistors, etc. An analogue filter handles analogue signals or continuous-time signals whether Electric potential, sound waves, or mechanical |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals ) to achieve the desired filtering effect.

Digital filters can achieve virtually any filtering effect that can be expressed as a mathematical function or algorithm. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation The two primary limitations of digital filters are their speed (the filter can't operate any faster than the computer at the heart of the filter), and their cost. However as the cost of integrated circuits has continued to drop over time, digital filters have become increasingly commonplace and are now an essential element of many everyday objects such as radios, cellphones, and stereo receivers. Microchipsjpg|right|thumb|200px|Microchips ( EPROM memory with a transparent window showing the integrated circuit inside Radio is the transmission of signals by Modulation of electromagnetic waves with frequencies below those of visible Light. Stereophonic sound, commonly called stereo, is the reproduction of Sound, using two or more independent audio channels through a Symmetrical

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Digital vs. analog filters

Advantages

Digital filters can realize characteristics not practically achievable with analog filters; for example, creating a 1000 Hz low-pass filter which can achieve near-perfect transmission of a 999 Hz input while entirely blocking a 1001 Hz signal. A low-pass filter is a filter that passes low- Frequency signals but Attenuates (reduces the Amplitude of signals with frequencies Practical analog filters cannot discriminate between such closely spaced signals.

Digital filters have the potential to attain much better signal-to-noise ratios than analog filters because random analog noise occurs only during the analog-to-digital conversion(ADC). Signal-to-noise ratio (often abbreviated SNR or S/N) is an Electrical engineering concept also used in other fields (such as scientific Measurements An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit which converts continuous signals to Multi-stages analog filters have random noise injected at each stage--causing errors to propagate down the system.

Disadvantages

The ADC stage creates deterministic quantization error which is due to digital storage and computation limitations. The difference between the actual analog value and quantized digital value due is called quantization error. This also creates sample rate limitation which is generally lower than (cf. In Signal processing, sampling is the reduction of a Continuous signal to a Discrete signal. Nyquist sampling theorem) analog filters. The Nyquist–Shannon sampling theorem is a fundamental result in the field of Information theory, in particular Telecommunications and Signal processing

Types of digital filters

Many digital filters are based on the Fast Fourier transform, a mathematical algorithm that quickly extracts the frequency spectrum of a signal, allowing the spectrum to be manipulated (such as to create pass-band filters) before converting the modified spectrum back into a time-series signal. Familiar concepts associated with a Frequency are colors musical notes radio/TV channels and even the regular rotation of the earth

The transfer function for a typical linear digital filter can be expressed as a transform in the Z-domain, as:

H(z) = \frac{B(z)}{A(z)} = \frac{{b_{0}+b_{1}z^{-1}+b_{2}z^{-2} + \cdots + b_{N}z^{-N}}}{{1+a_{1}z^{-1}+a_{2}z^{-2} + \cdots +a_{M}z^{-M}}}

where M is the order of the filter. A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant) In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real See Z-transform's LCCD equation for further discussion of this transfer function. In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant)

This form is for a recursive filter, which typically leads to infinite impulse response behaviour, but if the denominator is unity, then this is the form for a finite impulse response filter. In Signal processing, a recursive filter is a type of filter which re-uses one or more of its outputs as an input IIR redirects here For the conference company IIR see Informa. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity A finite impulse response (FIR filter is a type of a Digital filter.

Another form of a digital filter is that of a state-space model. In Control engineering, a state space representation is a mathematical model of a physical system as a set of input output and state variables related by first-order Differential A well used state-space filter is the Kalman filter published by Rudolf Kalman in 1960. The Kalman filter is an efficient Recursive filter that estimates the state of a Dynamic system from a series of noisy measurements Rudolf Emil Kálmán (born May 19 1930) is a Hungarian - American mathematical system theorist and a Professor Emeritus at the Year 1960 ( MCMLX) was a Leap year starting on Friday (link will display full calendar of the Gregorian calendar.

Converting the transfer function to a difference equation

In discrete-time systems, the digital filter is often implemented by converting the transfer function to a linear constant-coefficient difference equation (LCCD) via the Z-transform. A discrete signal or discrete-time signal is a Time series, perhaps a signal that has been sampled from a continuous-time signal. A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant) In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real The discrete frequency-domain transfer function is written as the ratio of two polynomials. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency For example:

H(z) = \frac{(z+1)^2} {(z-\frac{1}{2}) (z+\frac{3}{4})}

This is expanded:

H(z) = \frac{z^2+ 2z +1} {z^2 +\frac{1}{4} z - \frac{3}{8}}

and divided by the highest order of z:

H(z) = \frac{1 + 2z^{-1} +z^{-2}} {1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}} = \frac{Y(z)}{X(z)}

The coefficients of the denominator, ak, are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients, bk. The resultant linear difference equation is:

y[n] = -\sum_{k=1}^{N} a_{k} y[n-k] + \sum_{k=0}^{M} b_{k} x[n-k]

or, for the example:

y[n] = x[n] + 2x[n-1] + x[n-2] - \frac{1}{4} y[n-1] + \frac{3}{8} y[n-2]

This equation shows how to compute the next output sample, y[n], in terms of the past outputs, y[n − p], the present input, x[n], and the past inputs, x[n − p]. "Difference equation" redirects here It should not be confused with a Differential equation. In this form, the filter is amenable to numerical simulation via straightforward iteration. Iteration means the act of repeating Mathematics Iteration in mathematics may refer to the process of iterating a function, or to the techniques used

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See also

External links

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