Dielectric strength

2013-12-20T21:24:58Z

138.226.68.17: /* Breakdown field strength */

[[File:FIR Filter General.svg|thumb|250px|A general [[finite impulse response]] filter with ''n'' stages, each with an independent delay, ''d''<sub>i</sub>, and amplification gain, ''a''<sub>i</sub>.]]

In [[electronics]], [[computer science]] and [[mathematics]], a '''digital filter''' is a system that performs mathematical operations on a [[Sampling (signal processing)|sampled]], [[discrete-time]] [[Signal (electrical engineering)|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 signal]]s. 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 (see [[digital signal processing]]). In an analog filter, the input signal is "directly" manipulated by the circuit.

A digital filter system usually consists of an [[analog-to-digital converter]] to sample the input signal, followed by a microprocessor and some peripheral components such as memory to store data and filter coefficients etc. Finally a [[digital-to-analog converter]] to complete the output stage. Program Instructions (software) running on the microprocessor implement the digital filter by performing the necessary mathematical operations on the numbers received from the ADC. In some high performance applications, an [[Field-programmable gate array|FPGA]] or [[Application-specific integrated circuit|ASIC]] is used instead of a general purpose microprocessor, or a specialized DSP with specific paralleled architecture for expediting operations such as filtering.

Digital filters may be more expensive than an equivalent analog filter due to their increased complexity, but they make practical many designs that are impractical or impossible as analog filters. When used in the context of real-time analog systems, digital filters sometimes have problematic latency (the difference in time between the input and the response) due to the associated [[analog-to-digital]] and [[digital-to-analog]] conversions and [[anti-aliasing filter]]s, or due to other delays in their implementation.

Digital filters are commonplace and an essential element of everyday electronics such as [[radio]]s, [[cellphone]]s, and [[stereophonic sound|stereo receivers]].

==Characterization of linear digital filters==

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. As such, designing a filter consists of developing specifications appropriate to the problem (for example, a second-order low pass filter with a specific cut-off frequency), and then producing a transfer function which meets the specifications.

The [[transfer function]] for a linear, time-invariant, digital filter can be expressed as a transfer function in the [[Z-transform|''Z''-domain]]; if it is causal, then it has the form:

:<math>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}}}</math>

where the order of the filter is the greater of ''N'' or ''M''.
See [[Z-transform#Linear constant-coefficient difference equation|''Z''-transform's LCCD equation]] for further discussion of this [[transfer function]].

This is the form for a [[recursive filter]] with both the inputs (Numerator) and outputs (Denominator), which typically leads to an IIR [[infinite impulse response]] behaviour, but if the [[denominator]] is made equal to [[1 (number)|unity]] i.e. no feedback, then this becomes an FIR or [[finite impulse response]] filter.

===Analysis techniques===

A variety of mathematical techniques may be employed to analyze the behaviour of a given digital filter. Many of these analysis techniques may also be employed in designs, and often form the basis of a filter specification.

Typically, one characterizes filters by calculating how they will respond to a simple input such as an impulse. One can then extend this information to compute the filter's response to more complex signals.

====Impulse response====

The [[impulse response]], often denoted <math>h[k]</math> or <math>h_k</math>, is a measurement of how a filter will respond to the [[Kronecker delta]] function. For example, given a difference equation, one would set <math>x_0 = 1</math> and <math>x_k = 0</math> for <math>k \ne 0</math> and evaluate. The impulse response is a characterization of the filter's behaviour. Digital filters are typically considered in two categories: [[infinite impulse response]] (IIR) and [[finite impulse response]] (FIR).
In the case of linear time-invariant FIR filters, the impulse response is exactly equal to the sequence of filter coefficients:

:<math>\ y_n= \sum_{k=0}^{n-1} h_{k} x_{n-k}</math>

IIR filters on the other hand are recursive, with the output depending on both current and previous inputs as well as previous outputs. The general form of an IIR filter is thus:
:<math>\ \sum_{m=0}^{M-1} a_{m}y_{n-m} = \sum_{k=0}^{n-1} b_{k} x_{n-k}</math>

Plotting the impulse response will reveal how a filter will respond to a sudden, momentary disturbance.

====Difference equation ====

In [[discrete-time]] systems, the digital filter is often implemented by converting the [[transfer function]] to a [[Z-transform#Linear_constant-coefficient_difference_equation|linear constant-coefficient difference equation]] (LCCD) via the [[Z-transform]]. The discrete [[Frequency domain|frequency-domain]] transfer function is written as the ratio of two polynomials. For example:

:<math>H(z) = \frac{(z+1)^2} {(z-\frac{1}{2}) (z+\frac{3}{4})}</math>

This is expanded:
:<math>H(z) = \frac{z^2+ 2z +1} {z^2 +\frac{1}{4} z - \frac{3}{8}}</math>

and divided by the highest order of <math>z</math>:
:<math>
H(z) = \frac{1 + 2z^{-1} +z^{-2}} {1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}} = \frac{Y(z)}{X(z)}
</math>

The coefficients of the denominator, <math>a_{k}</math>, are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients, <math>b_{k}</math>. The resultant [[difference equation|linear difference equation]] is:
:<math>
y[n] = -\sum_{k=1}^{N} a_{k} y[n-k] + \sum_{k=0}^{M} b_{k} x[n-k]
</math>

or, for the example above:

:<math>
\frac{Y(z)}{X(z)} = \frac{1 + 2z^{-1} +z^{-2}} {1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}}
</math>

rearranging terms:

:<math>
\Rightarrow (1 +\frac{1}{4} z^{-1} - \frac{3}{8} z^{-2}) Y(z) = (1 + 2z^{-1} +z^{-2}) X(z)
</math>

then by taking the inverse ''z''-transform:

:<math>
\Rightarrow y[n] + \frac{1}{4} y[n-1] - \frac{3}{8} y[n-2] = x[n] + 2x[n-1] + x[n-2]
</math>

and finally, by solving for <math>y[n]</math>:

:<math>
y[n] = - \frac{1}{4} y[n-1] + \frac{3}{8} y[n-2] + x[n] + 2x[n-1] + x[n-2]
</math>

This equation shows how to compute the next output sample, <math>y[n]</math>, in terms of the past outputs, <math>y[n-p]</math>, the present input, <math>x[n]</math>, and the past inputs, <math>x[n-p]</math>. Applying the filter to an input in this form is equivalent to a Direct Form I or II realization, depending on the exact order of evaluation.

==Filter design==
{{main|Filter design}}
The design of digital filters is a deceptively complex topic.<ref>M. E. Valdez, [http://home.mchsi.com/~mikevald/Digfilt.html Digital Filters], 2001.</ref> Although filters are easily understood and calculated, the practical challenges of their design and implementation are significant and are the subject of much advanced research.

There are two categories of digital filter: the [[recursive filter]] and the [[nonrecursive filter]]. These are often referred to as [[infinite impulse response]] (IIR) filters and [[finite impulse response]] (FIR) filters, respectively.<ref>A. Antoniou, chapter 1</ref>

==Filter realization==

After a filter is designed, it must be ''realized'' by developing a signal flow diagram that describes the filter in terms of operations on sample sequences.

A given transfer function may be realized in many ways. Consider how a simple expression such as <math>ax + bx + c</math> could be evaluated – one could also compute the equivalent <math>x(a + b) + c</math>. In the same way, all realizations may be seen as "factorizations" of the same transfer function, but different realizations will have different numerical properties. Specifically, some realizations are more efficient in terms of the number of operations or storage elements required for their implementation, and others provide advantages such as improved numerical stability and reduced round-off error. Some structures are better for [[fixed-point arithmetic]] and others may be better for [[floating-point arithmetic]].

===Direct Form I===

A straightforward approach for IIR filter realization is [[Digital_biquad_filter#Direct_Form_1|Direct Form I]], where the difference equation is evaluated directly. This form is practical for small filters, but may be inefficient and impractical (numerically unstable) for complex designs.<ref>J. O. Smith III, [http://ccrma-www.stanford.edu/~jos/filters/Direct_Form_I.html Direct Form I]
</ref> In general, this form requires 2N delay elements (for both input and output signals) for a filter of order N.

[[File:Biquad filter DF-I.svg|400px]]

===Direct Form II===

The alternate [[Digital_biquad_filter#Direct_Form_2|Direct Form II]] only needs ''N'' delay units, where ''N'' is the order of the filter – potentially half as much as Direct Form I. This structure is obtained by reversing the order of the numerator and denominator sections of Direct Form I, since they are in fact two linear systems, and the commutativity property applies. Then, one will notice that there are two columns of delays (<math>z^{-1}</math>) that tap off the center net, and these can be combined since they are redundant, yielding the implementation as shown below.

The disadvantage is that Direct Form II increases the possibility of arithmetic overflow for filters of high Q or resonance.<ref>J. O. Smith III, [http://ccrma-www.stanford.edu/~jos/filters/Direct_Form_II.html Direct Form II]</ref> It has been shown that as Q increases, the round-off noise of both direct form topologies increases without bounds.<ref>L. B. Jackson, "On the Interaction of Roundoff Noise and Dynamic Range in Digital Filters," ''Bell Sys. Tech. J.'', vol. 49 (1970 Feb.), reprinted in ''Digital Signal Process'', L. R. Rabiner and C. M. Rader, Eds. (IEEE Press, New York, 1972).</ref> This is because, conceptually, the signal is first passed through an all-pole filter (which normally boosts gain at the resonant frequencies) before the result of that is saturated, then passed through an all-zero filter (which often attenuates much of what the all-pole half amplifies).

[[File:Biquad filter DF-II.svg|400px]]

===Cascaded second-order sections===

A common strategy is to realize a higher-order (greater than 2) digital filter as a cascaded series of second-order "biquadratric" (or "biquad") sections<ref>J. O. Smith III, [http://ccrma-www.stanford.edu/~jos/filters/Series_Second_Order_Sections.html Series Second Order Sections]</ref> (see [[digital biquad filter]]). Advantages of this strategy is that the coefficient range is limited. Cascading direct form II sections result in N delay elements for filter order of N. Cascading direct form I sections result in N+2 delay elements since the delay elements of the input of any section (except the first section) are a redundant with the delay elements of the output of the preceding section.

===Other forms===

{{Expand section|date=July 2012}}

Other forms include:
* Direct Form I and II transpose
* Series/cascade lower (typical second) order subsections
* Parallel lower (typical second) order subsections
** Continued fraction expansion
* Lattice and ladder
** One, two and three-multiply lattice forms
** Three and four-multiply normalized ladder forms
** ARMA structures
* State-space structures:
** optimal (in the minimum noise sense): <math>(N+1)^2</math> parameters
** block-optimal and section-optimal: <math>4N-1</math> parameters
** input balanced with Givens rotation: <math>4N-1</math> parameters<ref name=LMXH10>{{cite journal|last=Li|first=Gang|coauthors=Limin Meng, Zhijiang Xu, Jingyu Hua|title=A novel digital filter structure with minimum roundoff noise|journal=Digital Signal Processing|date=July 2010|volume=20|issue=4|pages=1000–1009|url=http://www.sciencedirect.com/science/article/pii/S1051200409001961}}</ref>
* Coupled forms: Gold Rader (normal), State Variable (Chamberlin), Kingsbury, Modified State Variable, Zölzer, Modified Zölzer
* Wave Digital Filters (WDF)<ref name=Fet86>{{cite journal|last=Fettweis|first=Alfred|title=Wave digital filters: Theory and practice|journal=Proceedings of the IEEE|date=Feb 1986|volume=74|issue=2|pages=270–327|url=http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1457726}}</ref>
* Agarwal–Burrus (1AB and 2AB)
* Harris–Brooking
* ND-TDL
* Multifeedback
* Analog-inspired forms such as Sallen-key and state variable filters
* Systolic arrays

==Comparison of analog and digital filters==
Digital filters are not subject to the component non-linearities that greatly complicate the design of analog filters. Analog filters consist of imperfect electronic components, whose values are specified to a limit tolerance (e.g. resistor values often have a tolerance of ±5%) and which may also change with temperature and drift with time. As the order of an analog filter increases, and thus its component count, the effect of variable component errors is greatly magnified. In digital filters, the coefficient values are stored in computer memory, making them far more stable and predictable.<ref name="dspguide">http://www.dspguide.com/ch21/1.htm</ref>

Because the coefficients of digital filters are definite, they can be used to achieve much more complex and selective designs – specifically with digital filters, one can achieve a lower passband ripple, faster transition, and higher stopband attenuation than is practical with analog filters. Even if the design could be achieved using analog filters, the engineering cost of designing an equivalent digital filter would likely be much lower. Furthermore, one can readily modify the coefficients of a digital filter to make an [[adaptive filter]] or a user-controllable parametric filter. While these techniques are possible in an analog filter, they are again considerably more difficult.

Digital filters can be used in the design of finite impulse response filters. Analog filters do not have the same capability, because finite impulse response filters require delay elements.

Digital filters rely less on analog circuitry, potentially allowing for a better [[signal-to-noise ratio]]. A digital filter will introduce noise to a signal during analog low pass filtering, analog to digital conversion, digital to analog conversion and may introduce digital noise due to quantization. With analog filters, every component is a source of thermal noise (such as [[Johnson noise]]), so as the filter complexity grows, so does the noise.

However, digital filters do introduce a higher fundamental latency to the system. In an analog filter, latency is often negligible; strictly speaking it is the time for an electrical signal to propagate through the filter circuit. In digital systems, latency is introduced by delay elements in the digital signal path, and by [[Analog-to-digital converter|analog-to-digital]] and [[digital-to-analog converter]]s that enable the system to process analog signals.

In very simple cases, it is more cost effective to use an analog filter. Introducing a digital filter requires considerable overhead circuitry, as previously discussed, including two low pass analog filters.

==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 band-pass filters) before converting the modified spectrum back into a time-series signal.

Another form of a digital filter is that of a [[state space (controls)|state-space]] model.
A well used state-space filter is the [[Kalman filter]] published by [[Rudolf Kalman]] in 1960.

Traditional linear filters are usually based on attenuation. Alternatively nonlinear filters can be designed, including energy transfer filters <ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> which allow the user to move energy in a designed way. So that unwanted noise or effects can be moved to new frequency bands either lower or higher in frequency, spread over a range of frequencies, split, or focused. Energy transfer filters complement traditional filter designs and introduce many more degrees of freedom in filter design. Digital energy transfer filters are relatively easy to design and to implement and exploit nonlinear dynamics.

== See also ==
*[[Bessel filter]]
*[[Butterworth filter]]
*[[Elliptical filter|Elliptical filter (Cauer filter)]]
*[[Linkwitz–Riley filter]]
*[[Chebyshev filter]]
*[[Sample (signal)]]
*[[Electronic filter]]
*[[Filter design]]
*[[Digital biquad filter|Biquad filter]]
*[[High-pass filter]], [[Low-pass filter]]
*[[Infinite impulse response]], [[Finite impulse response]]
*[[Bilinear transform]]

==References==

===General===
*A. Antoniou, ''Digital Filters: Analysis, Design, and Applications'', New York, NY: McGraw-Hill, 1993.
*J. O. Smith III, [http://ccrma-www.stanford.edu/~jos/filters/filters.html Introduction to Digital Filters with Audio Applications], Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, September 2007 Edition.
*S.K. Mitra, ''Digital Signal Processing: A Computer-Based Approach'', New York, NY: McGraw-Hill, 1998.
*A.V. Oppenheim and R.W. Schafer, ''Discrete-Time Signal Processing'', Upper Saddle River, NJ: Prentice-Hall, 1999.
*J.F. Kaiser, ''Nonrecursive Digital Filter Design Using the Io-sinh Window Function,'' Proc. 1974 IEEE Int. Symp. Circuit Theory, pp. 20–23, 1974.
*S.W.A. Bergen and A. Antoniou, ''Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function,'' EURASIP Journal on Applied Signal Processing, vol. 2005, no. 12, pp. 1910–1922, 2005.
*[[Thomas W. Parks|T.W. Parks]] and [[James H. McClellan|J.H. McClellan]], [http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1083419 Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase], IEEE Trans. Circuit Theory, vol. CT-19, pp. 189–194, Mar. 1972.
*[[L. R. Rabiner]], J.H. McClellan, and T.W. Parks, [http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1451724 FIR Digital Filter Design Techniques Using Weighted Chebyshev Approximation], Proc. IEEE, vol. 63, pp. 595–610, Apr. 1975.
*A.G. Deczky, [http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1162392 Synthesis of Recursive Digital Filters Using the Minimum p-Error Criterion], IEEE Trans. Audio Electroacoust., vol. AU-20, pp. 257–263, Oct. 1972.

===Cited===
<references/>

== External links ==
*[http://www.winfilter.20m.com/ WinFilter] – Free filter design software
*[http://www.pyspuc.com/index.cgi/demo Filtplot] – Free customizable digital filter design software built with python and boost (WinXP/Ubuntu 6.10/RHEL-4). Also with interactive web interface.
*[http://www.digitalfilterdesign.com/ DISPRO] – Free filter design software
*[http://www.falstad.com/dfilter/ Java demonstration of digital filters]
*[http://www.terdina.net/iir/iir_explorer.html IIR Explorer educational software]
*[http://math.fullerton.edu/mathews/c2003/ZTransformFilterMod.html Introduction to Filtering]
*[http://ccrma.stanford.edu/~jos/filters/filters.html Introduction to Digital Filters]
*[http://www.cs.tut.fi/~ts/ Publicly available, very comprehensive lecture notes on Digital Linear Filtering (see bottom of the page)]

{{DEFAULTSORT:Digital Filter}}
[[Category:Digital signal processing]]
[[Category:Synthesiser modules]]
[[Category:Signal processing filter|*]]

formulasearchengine - User contributions [en]

Dielectric strength