# digital filter

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## digital filter

[′dij·əd·əl ′fil·tər]
(electronics)
An electrical filter that responds to an input which has been quantified, usually as pulses.

## Digital filter

Any digital computing means that accepts as its input a set of one or more digital signals from which it generates as its output a second set of digital signals. While being strictly correct, this definition is too broad to be of any practical use, but it does demonstrate the possible extent of application of digital-filter concepts and terminology.

#### Capabilities

Digital filters can be used in any signal-manipulating application where analog or continuous filters can be used. Because of their utterly predictable performance, they can be used in exacting applications where analog filters fail because of time- or other parameter-dependent coefficient drift in continuous systems. Because of the ease and precision of setting the filter coefficients, adaptive and learning digital filters are comparatively simple and particularly effective to implement. As digital technology becomes more ubiquitous, digital filters are increasingly acknowledged as the most versatile and cost-effective solutions to filtering problems.

The number of functions that can be performed by a digital filter far exceeds that which can be performed by an analog, or continuous, filter. By controlling the accuracy of the calculations within the filter (that is, the arithmetic word length), it is possible to produce filters whose performance comes arbitrarily close to the performance expected of the perfect models. For example, theoretical designs that require perfect cancellation can be implemented with great fidelity by digital filters.

#### Linear difference equation

The digital filter accepts as its input signals numerical values called input samples and produces as its output signal numerical values called output samples. Each output sample at any particular sampling instant is a weighted sum of present and past input samples, and past output samples. If the sequence of input samples is xn, xn+1, xn+2, …, then the corresponding sequence of output samples would be yn, yn+1, yn+2,….

From this simple time-domain expression, a considerable number of definitions can be constructed. If the filter coefficients (the a's and the b's) are independent of the x's and y's, this digital filter is a linear filter. If the a's and b's are fixed, this is a linear time-invariant (LTI) filter. The order of the filter is given by the largest of the subscripts among the a's and b's, that is, the larger of M and N. If the b's are all zero (that is, if the output is the weighted sum of present and past input samples only), the digital filter is referred to as a nonrecursive (having no feedback) or finite impulse response (FIR) filter because the response of the filter to an impulse (actually a unit pulse) input is simply the sequence of the “a” coefficients. If any value of b is nonzero, the filter is recursive (having feedback) and is generally an infinite impulse response (IIR) filter.

If the digital filter under consideration is not a linear, time-invariant filter, the transfer function cannot be used.

#### Transfer functions

Although the time-domain difference equation is a useful description of a filter, as in the continuous-domain filter case, a powerful alternative form is the transfer function. The information content of the transfer function is the same as that of the difference equation as long as a linear, time-invariant system is under consideration. A difference equation is converted to transfer-function form by use of the z transform. The z transform is simply the Laplace transform adapted for sampled systems with some shorthand notation introduced.

#### Adaptive filters

So far only LTI filters have been discussed. An important class of variable-parameter filter change their coefficients to minimize an error criterion. These filters are called adaptive because they adapt their parameters in response to changes in the operating environment. An example is an FIR digital filter whose coefficients are continually adjusted so that the output will track a reference signal with minimum error. The performance criterion will be the minimization of some function of the error.

References in periodicals archive ?
The integrated configurable digital filter offers seven filter types and 13 different downsampling factors, providing users with a high level of flexibility to trade off bandwidth, filter response and noise performance for each application.
The LTC2380-24 no-latency 24-bit, 2-MS/s successive approximation register (SAR) analog-to-digital converter (ADC) features an integrated digital filter that averages 1 to 65,536 conversion results in real time, offering dynamic range of 101 dB at 1.
of Florida) introduces finite impulse response (FIR) and infinite impulse response (IIR) filters for altering a digital signal's attributes and applies MATLAB software to the design and analysis of digital filter solutions.
The efficiency of digital filter implemented on the programmable hardware, such as field programmable gate array (FPGA) chips, can be very low, because in the most of the available operating time the implemented digital filter is not used.
With a digital filter, the signal is digitized with an ADC, and the filtering is done in the digital domain.
The actual low-pass filtering is done by a digital filter with the ratio between the cut-off frequency and the Nyquist frequency of 1 :N.
First, the most common analog and digital filter design and implementation methods are covered.
Similarly, a 100-tap FIR digital filter (say, a 10MHz channel with over sampling) might completely fill a normal DSP or require an FPGA, but uses less than 5% of one picoArray.
The internal digital filter operates at 8x oversampling (96kHz sampling rate) or 4x oversampling (192kHz sampling rate), with two selectable roll-off performances: sharp roll-off or slow roll-off.
Despite these and other factors, you must still consider whether a digital filter is a viable alternative to an analog equivalent.
Deep within the controller, the digital filter, which consists of various mathematical operations, transforms incoming digital signals representing the state of the process.

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