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I/O Controller Hub

2009-11-02T17:15:02Z

200.55.139.25: /* ICH */

'''I/O Controller Hub''' ('''ICH'''), also known as '''Intel 82801''', is an Intel [[southbridge (computing)|southbridge]] on [[motherboard]]s with Intel [[chipset]]s ([[Intel Hub Architecture]]). As with any other southbridge, the ICH is used to connect and control peripheral devices.

{| class="wikitable" style="text-align:center; vetical-align:center;"
|-
! Southbridge
! USB 2.0 ports
|-
| ICH5
| 8
|-
| ICH8
| 10
|-
| ICH9
| 12
|-
| ICH10
| 12
|-
| PCH
| 14
|}

== ICH ==

The first version of the ICH was released in June 1999, day of birthday of Osmel along with the [[Intel 810]] northbridge. While its predecessor, the [[Intel 82371|82371]], was connected to the northbridge through an internal [[Peripheral Component Interconnect|PCI]] bus with 133 [[megabyte|MB]]/s, the ICH used a proprietary interface (called by Intel ''Hub Interface'') that linked it to the northbridge through an 8-[[bit]] wide, 266 MB/s bus.

The Hub Interface was a point-to-point connection between different components on the motherboard. Another design decision was to substitute the rigid North-South axis on the motherboard with a star structure.

Note that, along with the ICH, Intel evolved other uses of the "Hub" terminology. Thus, the northbridge became the [[Memory Controller Hub]] (MCH) or if it had integrated graphics (like the Intel 810), the Graphics and Memory Controller Hub (GMCH).

Other ICH features include:
* PCI Rev 2.2 compliant with support for 33 MHz PCI operations.
* [[Advanced Configuration and Power Interface|ACPI]] Support
* Integrated [[IDE]] controller for [[Ultra ATA]] support
* System Management Bus ([[SMBus]]) with support for [[I2C]] devices
* [[AC'97]] 2.1 Compliant Link
* [[Low Pin Count]] interface

The ICH came in two flavors: <ref>http://developer.intel.com/design/chipsets/datashts/29065503.pdf</ref>
*82801AA (ICH) - Ultra ATA/66 support, 6 PCI slots, [[Alert on LAN]] support
*82801AB (ICH0) - Ultra ATA/33 support, 4 PCI slots, no Alert on LAN

== ICH2 ==

In early 2000 Intel had suffered a significant setback with the i820 northbridge. Customers were not willing to pay the high prices for RDRAM and either bought i810 or i440BX motherboards or changed to the competition. The hastily developed 82815 northbridge for PC-133 SDRAM became Intel's method to recover in the middle class segment.

The ICH1 or the new ICH2 (360 pins) could be placed to the side of the 82815. For the first time a [[Fast Ethernet]] chip (82559) was integrated into the southbridge, depending upon an external PHY chip.

The [[AT Attachment|PATA]] interface was accelerated to ATA/100 and the number of [[Universal Serial Bus|USB]] connections was doubled to four. The sound chip now controlled six channels.

There was also a mobile variant called the ICH2-M.

The following variations existed:
*82801BA (ICH2)
*82801BAM (ICH2-M) Mobile

== ICH3 ==

In 2001, Intel delivered ICH3, which was available in two versions. The server version, ICH3-S, running with the E7501 Northbridge. And the mobile version, ICH3-M, which worked with the i830 and i845 northbridges. There is no version for desktop motherboards.

In comparison with the ICH2, the changes were limited. The PATA Controller now supported "Native Mode", up to six USB-1.1 devices, [[System Management Bus|SMBus]] 2.0, and the newest SpeedStep version, which allowed power-saving devices to be switched off during operation. The chip had 421 pins.

This has the following variations:
*82801CA (ICH3-S) Server
*82801CAM (ICH3-M) Mobile

== ICH4 ==

The ICH4 was Intel's southbridge for the year 2002. The most important innovation was the support of USB 2.0 on all six ports. Sound support was improved and corresponded the newest AC97 specification, version 2.3. Like the preceding generation, the ICH4 had 421 pins.

This has the following variations:
*82801DB (ICH4) Base
*82801DBM (ICH4-M) Base Mobile

== ICH5 ==

In 2003, and in conjunction with the i865 and i875 northbridges, the ICH5 was created. A [[Serial ATA|SATA]] host controller was integrated. The ICH5R variant additionally supported [[RAID]] 0 on SATA ports. Eight USB-2.0 ports were available. The chip had full support for [[Advanced Configuration and Power Interface|ACPI]] 2.0. It had 460 pins.

Since 1999 the 266 MB/s hub interface was assumed to be a bottleneck. In the new chip generation, Intel therefore offered an optional port for a Gigabit Ethernet Controller directly attached to the MCH.

The goal of this [[Intel Communication Streaming Architecture|CSA]] technology was to reduce the latencies for Gigabit LAN by direct memory access and to free up bandwidth on the Hub interface between ICH and MCH for non removable disk and PCI data traffic.

Since mid-2004, the large motherboard manufacturers noticed an increased complaint ratio with motherboards equipped with ICH5. A cause was the insufficient [[Electrostatic discharge|ESD]] tolerance of certain ICH5 steppings.

In particular, when connecting USB devices via front panels, the chips died by discharges of static electricity. Intel reacted to the problem by shipping ICH5 with increased ESD tolerance. Effective ESD preventive measures on USB ports are difficult and costly, since they can impair signal quality of the USB-2.0 high-speed signals. Many motherboard manufacturers had omitted the necessary high-quality safety devices for front panel connectors for cost reasons.

This has the following variations:
*82801E (C-ICH) Communications
*82801EB (ICH5) Base
*82801ER (ICH5R) RAID
*82801EBM (ICH5-M) Base Mobile
*6300ESB (ESB) Enterprise Southbridge

== ICH6 ==

ICH6 was Intel's first [[PCI Express]] southbridge. It made four PCI Express x1 ports available. Faster x16-Ports were accommodated in the MCH. The bottleneck Hub interface was replaced by a new "Direct Media Interface" (in reality a PCI Express x4 link) with 1 [[gigabyte|GB]]/s per direction. Support for [[Intel High Definition Audio]] was included. In addition, [[AC97]] and the classical PCI 2.3 were still supported.

Two additional SATA ports were added, and one PATA channel was removed. The ICH6R variant supported RAID modes 0, 1, 0+1 and the Intel specific "Matrix RAID".

ICH6R and ICH6-M had [[Advanced Host Controller Interface|AHCI]] SATA controllers implemented. The chips had 652 pins. Originally Intel had planned to bring two further variants under the names ICH6W and ICH6RW to the market, which should contain a software Access Point for a [[Wireless LAN]]. These chips are published.

This has the following variations:
*82801FB (ICH6) Base
*82801FR (ICH6R) RAID
*82801FBM (ICH6-M) Base Mobile
*631xESB/632xESB (ESB2) Enterprise Southbridge

== ICH7 ==

The ICH7 started to ship in mid-2005 together with Intel's newest high-end MCH, the i955X. Two additional PCI express x1-Ports, an accelerated SATA Controller for up to 300 MB/s data transmission rate, as well as support for Intel's "Active Management Technology" were added.

The ICH7R additionally supported RAID 5.

This has the following variations:
*82801GB (ICH7) Base
*82801GR (ICH7R) RAID
*82801GDH (ICH7DH) Digital Home
*82801GBM (ICH7-M) Base Mobile
*82801GHM (ICH7-M DH) Home Mobile
*82801GU (ICH7-U) Ultra-mobile

== ICH8 ==

ICH8 is offered in several different versions and is the complement to the 965 class MCH chips. The non-mobile ICH8 does not have a traditional PATA interface, and just one AC97. In practice, most baseboard manufacturers would like to still support and offer PATA appropriate connection types over additional chips of [[JMicron]] or [[Marvell Technology Group|Marvell]].

As the first, ICH controls the ICH8 [[Serial ATA#External SATA|eSATA]] and Gigabit Ethernet (before accommodated in the MCH). The base version possesses only four SATA 2.0 ports.

The ICH8R (RAID) as well as the remaining chips have the possibility of attaching six SATA devices. Additionally the ICH8DH (Digital Home) has Quick Resume and can be used together with the P965 and/or G965 in [[Intel Viiv]]-certified systems.

The counterpart to the ICH8DO (Digital Office) is the Q965 MCH, which together provide [[Intel vPro]] compatibility.

This has the following variations:
*82801HB (ICH8) Base
*82801HR (ICH8R) RAID
*82801HDH (ICH8DH) Digital Home
*82801HDO (ICH8DO) Digital Office
*82801HBM (ICH8M) Base Mobile
*82801HEM (ICH8M-E) Enhanced Mobile

== ICH9 ==

The ICH9 came out in May 2007 in the [[Intel P35|P35]] chipset. It removes all PATA support. In practice, many baseboard manufacturers continue providing PATA using an additional chip. Only the ICH9R, ICH9DH, ICH9DO chip have AHCI support<ref>[http://www.intel.com/support/chipsets/imst/sb/CS-012304.htm Intel ICHx chip that have AHCI support], Intel Website, accessed April 12, 2008.</ref>

This part has the following variations:
*82801IB ICH9 Base (ICH9) neither [[Advanced Host Controller Interface|AHCI]] or [[RAID]] support
*82801IR ICH9 RAID (ICH9R) with [[Advanced Host Controller Interface|AHCI]] and [[RAID]] Support
*82801IH ICH9 Digital Home (ICH9DH) with [[Advanced Host Controller Interface|AHCI]] and no [[RAID]] Support
*82801IO ICH9 Digital Office (ICH9DO) with [[Advanced Host Controller Interface|AHCI]] and [[RAID]] Support

== ICH10 ==
Intel launched the ICH10 southbridge in June 2008 with the [[Intel P45]] (Eaglelake) chipset. Datasheet <ref>[http://www.intel.com/assets/pdf/datasheet/319973.pdf Intel I/O Controller Hub 10 (ICH10) Family Datasheet]</ref> and errata <ref>[http://www.intel.com/assets/pdf/specupdate/319974.pdf Intel I/O Controller Hub 10 (ICH10) Family Specification Update]</ref> information is now available.

ICH10 implements the 10Gbit/s bidirectional [[Direct Media Interface|DMI]] interface to the "northbridge" device. It supports various interfaces to "low-speed" peripherals, and it supports a suite of housekeeping functions.
===Peripheral support===
*Six PCIe version 1.1 ports, four of which can be configured as either 4x1 or 1x4.
*PCI bus
*Six SATA 3 Gbit/s ports in either legacy IDE or AHCI mode. can support external eSATA
*[[Intel High Definition Audio]]
*Integrated gigabit LAN.
*Six USB 2.0 controllers
ICH10 does not offer direct PATA or LPT support. Notably there is support of 'hot-swap' functionality. ICH10 also offers reduced load on CPU and decreased power consumption.

ICH10R is a RAID variant which also supports a new technology called “Turbo Memory”. This allows the use of flash memory on a motherboard for fast caching.

== See also ==
*[[List of Intel chipsets]]
*[[Platform Controller Hub]] (PCH)
*[[Intel 82371]] (PIIX)
*[[Northbridge (computing)]]
*[[Southbridge (computing)]]

== References ==
{{Reflist}}

== External links ==
*[http://www.intel.com/design/chipsets/datashts/273598.htm Intel 82801E Communications I/O Controller Hub (C-ICH) Datasheet]
*[http://www.intel.com/design/intarch/datashts/290550.htm 82371FB (PIIX) and 82371SB (PIIX3) PCI ISA IDE Xcelerator]
*[http://www.intel.com/design/intarch/datashts/290562.htm Intel(R) 82371AB PCI-TO-ISA/IDE Xcelerator (PIIX4) Datasheet]
*[http://www.intel.com/design/chipsets/datashts/290655.htm Intel 82801AA (ICH) and Intel 82801AB (ICH0) I/O Controller Hub]
*[http://www.intel.com/design/chipsets/datashts/290687.htm Intel 82801BA I/O Controller Hub 2 (ICH2) and Intel 82801BAM I/O Controller Hub 2 Mobile (ICH2-M) Datasheet]
*[http://www.intel.com/design/chipsets/e7500/datashts/290733.htm Intel 82801CA I/O Controller Hub 3 (ICH3-S) Datasheet]
*[http://www.intel.com/design/chipsets/datashts/290716.htm Intel 82801CAM I/O Controller Hub 3 (ICH3-M) Datasheet]
*[http://www.intel.com/design/chipsets/datashts/290744.htm Intel 82801DB I/O Controller Hub 4 (ICH4) Datasheet]
*[http://www.intel.com/design/mobile/datashts/252337.htm Intel 82801DBM I/O Controller Hub 4 Mobile (ICH4-M) Datasheet]
*[http://www.intel.com/design/chipsets/datashts/252516.htm Intel 82801EB I/O Controller Hub 5 (ICH5) and Intel 82801ER I/O Controller Hub 5 R (ICH5R) Datasheet]
*[http://www.intel.com/design/intarch/datashts/300641.htm Intel 6300ESB I/O Controller Datasheet]
*[http://www.intel.com/design/chipsets/datashts/301473.htm Intel I/O Controller Hub 6 (ICH6) Family Datasheet] For Intel 82801FB ICH6, 82801FR ICH6R, 82801FBM ICH6-M I/O Controller Hubs
*[http://www.intel.com/design/chipsets/datashts/313082.htm Intel 631xESB/632xESB I/O Controller Hub Datasheet]
*[http://www.intel.com/design/chipsets/datashts/307013.htm Intel I/O Controller Hub 7 (ICH7) Family Datatsheet] For the Intel 82801GB ICH7, 82801GR ICH7R, 82801GDH ICH7DH, 82801GBM ICH7-M, 82801GHM ICH7-M DH, and 82801GU ICH7-U I/O Controller Hubs
*[http://www.intel.com/design/chipsets/datashts/313056.htm Intel I/O Controller Hub 8 (ICH8) Family Datasheet] For the Intel I82801HB ICH8, 82801HR ICH8R, 82801HH ICH8DH, 82801HO ICH8DO, 82801HBM ICH8M and 82801HEM ICH8M-E I/O Controller Hubs
*[http://www.intel.com/design/chipsets/datashts/316972.htm Intel I/O Controller Hub 9 (ICH9) Family Datasheet] For the Intel 82801IB ICH9, 82801IR ICH9R, 82801IH ICH9DH, 82801IO ICH9DO I/O Controller Hubs

[[Category:Intel products]]

[[de:I/O Controller Hub]]
[[fr:Intel I/O Controller Hub]]
[[ja:I/O コントローラー・ハブ]]

Z-transform

2007-05-23T14:19:20Z

200.55.139.25: /* External links */

In [[mathematics]] and [[signal processing]], the '''Z-transform''' converts a discrete [[time-domain]] signal, which is a [[sequence]] of [[real number]]s, into a [[complex number|complex]] [[frequency-domain]] representation.

The Z-transform and [[advanced Z-transform]] were introduced (under the Z-transform name) by [[E. I. Jury]] in [[1958]] in ''Sampled-Data Control Systems'' (John Wiley & Sons). The idea contained within the Z-transform was previously known as the "generating function method".

The (unilateral) Z-transform is to discrete-time signals what the one-sided [[Laplace transform]] is to continuous-time signals.

==Definition==

The Z-transform, like many other integral transforms, can be defined as either a ''one-sided'' or ''two-sided'' transform.

===Bilateral Z-transform===

The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal ''x[n]'' is the function ''X(z)'' defined as

:<math>X(z) = Z\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \ </math>

where ''n'' is an integer and ''z'' is, in general, a [[complex number]]:
:<math> z= A e^{j\phi} </math>
:where ''A'' is the magnitude of ''z'', and φ is the [[angular frequency]] (in [[radians]] per [[sample (signal)|sample]]).

===Unilateral Z-transform===

Alternatively, in cases where ''x''[''n''] is defined only for ''n'' ≥ 0, the ''single-sided'' or ''unilateral'' Z-transform is defined as

:<math>X(z) = Z\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n} \ </math>

In [[signal processing]], this definition is used when the signal is [[causal system|causal]].

An important example of the unilateral Z-transform is the [[probability-generating function]], where the component <math>x[n]</math> is the probability that a discrete random variable takes the value <math>n</math>, and the function <math>X(z)</math> is usually written as <math>X(s)</math>, in terms of <math>s = z^{-1}</math>. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.

==Inverse Z-transform==

The ''inverse'' Z-transform is

:<math> x[n] = Z^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz \ </math>

where <math> C \ </math> is a counterclockwise closed path encircling the origin and entirely in the [[Radius of convergence|region of convergence]] (ROC). The contour or path, <math> C \ </math>, must encircle all of the poles of <math> X(z) \ </math>.

A special case of this [[contour integral]] occurs when <math> C \ </math> is the unit circle (and can be used when the ROC includes the unit circle). The inverse Z-transform simplifies to the [[Discrete-time_Fourier_transform#Inverse_transform|inverse discrete-time Fourier transform]]:

:<math> x[n] = \frac{1}{2 \pi} \int_{-\pi}^{+\pi} X(e^{j \omega}) e^{j \omega n} d \omega \ </math> .

The Z-transform with a finite range of ''n'' and a finite number of uniformly-spaced ''z'' values can be computed efficiently via [[Bluestein's FFT algorithm]]. The [[Discrete-time Fourier transform|discrete-time Fourier transform]] (DTFT) (not to confuse with the [[discrete Fourier transform]] (DFT)) is a special case of such a Z-transform obtained by restricting ''z'' to lie on the unit circle.

==Region of convergence==
The [[Radius of convergence|region of convergence]] (ROC) is the set of points in the complex plane for which the Z-transform summation converges.

:<math>ROC = \left\{ z : \left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right| < \infty \right\} </math>

===Example 1 (No ROC)===
Let <math>x[n] = 0.5^n\ </math>. Expanding <math>x[n]\ </math> on the interval <math>(-\infty, \infty)\ </math> it becomes

:<math>x[n] = \{..., 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, ...\} = \{..., 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, ...\}\ </math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} < \infty\ </math>

There are no such values of <math>z\ </math> that satisfy this condition.

===Example 2 ([[causal system|causal]] ROC)===
[[Image:Region of convergence 0.5 causal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle and the circle <math>\left|z\right| = 0.5</math> is shown as a dashed black circle]]

Let <math>x[n] = 0.5^n u[n]\ </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]\ </math> on the interval <math>(-\infty, \infty)\ </math> it becomes

:<math>x[n] = \{..., 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, ...\}\ </math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}0.5^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{0.5}{z}\right)^n = \frac{1}{1 - 0.5z^{-1}}\ </math>

The last equality arises from the infinite [[geometric series]] and the equality only holds if <math>\left|0.5 z^{-1}\right| < 1\ </math> which can be rewritten in terms of <math>z\ </math> as <math>\left|z\right| > 0.5\ </math>. Thus, the ROC is <math>\left|z\right| > 0.5\ </math>. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
 

===Example 3 ([[Anticausal system|anticausal]] ROC)===
[[Image:Region of convergence 0.5 anticausal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle and the circle <math>\left|z\right| = 0.5</math> is shown as a dashed black circle]]

Let <math>x[n] = -(0.5)^n u[-n-1]\ </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]\ </math> on the interval <math>(-\infty, \infty)\ </math> it becomes

:<math>x[n] = \{..., -(0.5)^{-3}, -(0.5)^{-2}, -(0.5)^{-1}, 0, 0, 0, ...\}\ </math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = -\sum_{n=-\infty}^{-1}0.5^nz^{-n} = -\sum_{n=-\infty}^{-1}\left(\frac{z}{0.5}\right)^{-n}\ </math>
:<math>= -\sum_{m=1}^{\infty}\left(\frac{z}{0.5}\right)^{m} = -\frac{0.5^{-1}z}{1 - 0.5^{-1}z} = \frac{z}{z - 0.5} = \frac{1}{1 - 0.5z^{-1}}\ </math>

Using the infinite [[geometric series]], again, the equality only holds if <math>\left|0.5^{-1}z\right| < 1\ </math> which can be rewritten in terms of <math>z\ </math> as <math>\left|z\right| < 0.5\ </math>.
Thus, the ROC is <math>\left|z\right| < 0.5\ </math>. In this case the ROC is a disc centered at the origin and of radius 0.5.

What differentiates this example from the previous example is ''only'' the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
 

===Examples conclusion===
Examples 2 & 3 clearly show that the Z-transform <math>X(z)\ </math> of <math>x[n]\ </math> is unique when and only when specifying the ROC. Creating the [[pole-zero plot]] for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will ''never'' contain poles.

In example 2, the causal system yields an ROC that includes <math>\left| z \right| = \infty\ </math> while the anticausal system in example 3 yields an ROC that includes <math>\left| z \right| = 0\ </math>.

[[Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring <math>0.5 < \left| z \right| < 0.75\ </math>]]
In systems with multiple poles it is possible to have an ROC that includes neither <math>\left| z \right| = \infty\ </math> nor <math>\left| z \right| = 0\ </math>. The ROC creates a circular band. For example, <math>x[n] = 0.5^nu[n] - 0.75^nu[-n-1]\ </math> has poles at 0.5 and 0.75. The ROC will be <math>0.5 < \left| z \right| < 0.75\ </math>, which includes neither the origin nor infinity. Such a system is called a [[mixed-causality system|mixed-causality]] system as it contains a causal term <math>0.5^nu[n]\ </math> and an anticausal term <math>-(0.75)^nu[-n-1]\ </math>.

The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., <math>\left| z \right| = 1\ </math>) then the system is stable. In the above systems the causal system is stable because <math>\left| z \right| > 0.5\ </math> contains the unit circle.

If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous <math>x[n]\ </math>) you can determine a unique <math>x[n]\ </math> provided you desire the following:
* Stability
* Causality

If you need stability then the ROC must contain the unit circle.
If you need a causal system then the ROC must contain infinity.
If you need an anticausal system then the ROC must contain the origin.

The unique <math>x[n]\ </math> can then be found.

==Properties==

{| class="wikitable"
|+ '''Properties of the z-transform'''
!
! Time domain
! Z-domain
! ROC
|-
! Notation
| <math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math>
| <math>X(z)=\mathcal{Z}\{x[n]\}</math>
| ROC: <math>r_2<|z|<r_1 \ </math>
|-
! Linearity
| <math>a_1 x_1[n] + a_2 x_2[n]\ </math>
| <math>a_1 X_1(z) + a_2 X_2(z) \ </math>
| At least the intersection of ROC1 and ROC2
|-
! Time shifting
| <math>x[n-k]\ </math>
| <math>z^{-k}X(z) \ </math>
| ROC, except <math>z=0\ </math> if <math>k>0\,</math> and <math>z=\infty</math> if <math>k<0\ </math>
|-
! Scaling in the z-domain
| <math>a^n x[n]\ </math>
| <math>X(a^{-1}z) \ </math>
| <math>|a|r_2<|z|<|a|r_1 \ </math>
|-
! Time reversal
| <math>x[-n]\ </math>
| <math>X(z^{-1}) \ </math>
| <math>\frac{1}{r_1}<|z|<\frac{1}{r_2} \ </math>
|-
! Conjugation
| <math>x^*[n]\ </math>
| <math>X^*(z^*) \ </math>
| ROC
|-
! Real part
| <math>\operatorname{Re}\{x[n]\}\ </math>
| <math>\frac{1}{2}\left[X(z)+X^*(z^*) \right]</math>
| ROC
|-
! Imaginary part
| <math>\operatorname{Im}\{x[n]\}\ </math>
| <math>\frac{1}{2j}\left[X(z)-X^*(z^*) \right]</math>
| ROC
|-
! Differentiation
| <math>nx[n]\ </math>
| <math> -z \frac{dX(z)}{dz}</math>
| ROC
|-
! [[Convolution]]
| <math>x_1[n] * x_2[n]\ </math>
| <math>X_1(z)X_2(z) \ </math>
| At least the intersection of ROC1 and ROC2
|-
! [[Cross-correlation|Correlation]]
| <math>r_{x_1,x_2}(l)=x_1[l] * x_2[-l]\ </math>
| <math>R_{x_1,x_2}(z)=X_1(z)X_2(z^{-1})\ </math>
| At least the intersection of ROC of X1(z) and X2(<math>z^{-1}</math>)
|-
! Multiplication
| <math>x_1[n]x_2[n]\ </math>
| <math>\frac{1}{j2\pi}\oint_C X_1(v)X_2(\frac{z}{v})v^{-1}\mathrm{d}v \ </math>
| At least <math>r_{1l}r_{2l}<|z|<r_{1u}r_{2u} \ </math>
|-
! [[Parseval's theorem|Parseval's relation]]
| <math>\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n]\ </math>
| <math>\frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\frac{1}{v^*})v^{-1}\mathrm{d}v \ </math>
|
|}

* '''Initial value theorem'''
::<math>x[0]=\lim_{z\rightarrow \infty}X(z) \ </math>, If <math>x[n]\,</math> causal

* '''Final value theorem'''
:: <math>x[\infty]=\lim_{z\rightarrow 1}(z-1)X(z) \ </math>, Only if poles of <math>(z-1)X(z) \ </math> are inside the unit circle

==Table of common Z-transform pairs==

{| class="wikitable"
|-
! !! Signal, <math>x[n]</math> !! Z-transform, <math>X(z)</math> !! ROC
|-
| 1 || <math>\delta[n] \, </math> || <math>1\, </math> ||<math> \mbox{all }z\, </math>
|-
| 2 || <math>\delta[n-n_0] \,</math> || <math> z^{-n_0} \, </math> || <math> z \neq 0\,</math>
|-
| 3 || <math>u[n] \,</math> || <math> \frac{1}{1-z^{-1} }</math> || <math>|z| > 1\,</math>
|-
| 4 || <math>- u[-n-1] \,</math> || <math> \frac{1}{1 - z^{-1}}</math> ||<math>|z| < 1\,</math>
|-
| 5 || <math>n u[n] \,</math> || <math> \frac{z^{-1}}{( 1-z^{-1} )^2}</math> || <math>|z| > 1\,</math>
|-
| 6 || <math> - n u[-n-1] \,</math> || <math> \frac{z^{-1} }{ (1 - z^{-1})^2 }</math> ||<math> |z| < 1 \,</math>
|-
| 7 || <math>n^2 u[n] \,</math> || <math> \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| > 1\,</math>
|-
| 8 || <math> - n^2 u[-n - 1] \,</math> || <math> \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| < 1\,</math>
|-
| 9 || <math>n^3 u[n] \,</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| > 1\,</math>
|-
| 10 || <math>- n^3 u[-n -1] \,</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| < 1\,</math>
|-
| 11 || <math>a^n u[n] \,</math> || <math> \frac{1}{1-a z^{-1}}</math> ||<math> |z| > |a|\,</math>
|-
| 12 || <math>-a^n u[-n-1] \,</math> || <math> \frac{1}{1-a z^{-1}}</math> ||<math>|z| < |a|\,</math>
|-
| 13 || <math>n a^n u[n] \,</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> || <math>|z| > |a|\,</math>
|-
| 14 || <math>-n a^n u[-n-1] \,</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> ||<math> |z| < |a|\,</math>
|-
| 15 || <math>n^2 a^n u[n] \,</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| > |a|\,</math>
|-
| 16 || <math>- n^2 a^n u[-n -1] \,</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| < |a|\,</math>
|-
| 17 || <math>\cos(\omega_0 n) u[n] \,</math> || <math> \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1\,</math>
|-
| 18 || <math>\sin(\omega_0 n) u[n] \,</math> || <math> \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1\,</math>
|-
| 19 || <math>a^n \cos(\omega_0 n) u[n] \,</math> || <math> \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math> |z| > |a|\,</math>
|-
| 20 || <math>a^n \sin(\omega_0 n) u[n] \,</math> || <math> \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math> |z| > |a|\,</math>
|}

==Relationship to Laplace==
The bilateral Z-transform is simply the [[two-sided Laplace transform]] of the ideal sampled function

::<math> x(t) \sum_{n=-\infty}^{\infty} \delta(t-nT) = \sum_{n=-\infty}^{\infty} x[n] \delta(t-nT) \ </math>

where <math> x(t) \ </math> is the continuous-time function being sampled, <math> x[n]=x(nT) \ </math> the nth sample, <math> T \ </math> is the sampling period, and with the substitution: <math> z = e^{sT} \ </math>.

Likewise the unilateral Z-transform is simply the one-sided [[Laplace transform]] of the ideal sampled function. Both assume that the sampled function is zero for all negative time indices.

The [[Bilinear transform]] is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa. To do this, you can use the following substitutions in <math> H(s) </math> or <math> H(z) </math> :

<math> s =\frac{2}{T} \frac{z-1}{z+1} </math> from Laplace to z (Tustin transformation);

<math> z =\frac{2+sT}{2-sT} </math> from z to Laplace.

==Relationship to Fourier==
The Z-transform is a generalization of the [[discrete-time Fourier transform]] (DTFT). The DTFT can be found by evaluating the Z-transform <math>X(z)\ </math> at <math>z=e^{j\omega}\ </math> or, in other words, evaluated on the unit circle. In order to determine the [[frequency response]] of the system the Z-transform must be evaluated on the unit circle, meaning that the system's region of convergence must contain the unit circle. Otherwise, the DTFT of the system does not exist.

==Linear constant-coefficient difference equation==
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the [[autoregressive moving-average]] equation.

:<math>\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q}\ </math>

Both sides of the above equation can be divided by <math>\alpha_0 \ </math>, if it is not zero, normalizing <math>\alpha_0 = 1\ </math> and the LCCD equation can be written

:<math>y[n] = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}\ </math>

This form of the LCCD equation is favorable to make it more explicit that the "current" output <math>y[{n}]\ </math> is a function of past outputs <math>y[{n-p}]\ </math>, current input <math>x[{n}]\ </math>, and previous inputs <math>x[{n-q}]\ </math>.

===Transfer function===
Taking the Z-transform of the equation (using linearity and time-shifting laws) yields

:<math>Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q}\ </math>

and rearranging results in

:<math>H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-p}\alpha_{p}} = \frac{\beta_0 + z^{-1} \beta_1 + z^{-2} \beta_2 + ... + z^{-M} \beta_M}{\alpha_0 + z^{-1} \alpha_1 + z^{-2} \alpha_2 + ... + z^{-N} \alpha_N}\ </math>

===Zeros and poles===
From the [[fundamental theorem of algebra]] the [[numerator]] has M [[root (mathematics)|roots]] (corresponding to [[Zero (complex analysis)|zeros]] of H) and the [[denominator]] has N roots (corresponding to [[Pole (complex analysis)|poles]]). Rewriting the [[transfer function]] in terms of poles and zeros

:<math>H(z) = \frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})...(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})...(1 - p_N z^{-1})}\ </math>

Where <math>q_k\ </math> is the <math>k^{th}\ </math> zero and <math>p_k\ </math> is the <math>k^{th}\ </math> pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the [[pole-zero plot]].

In simple words, zeros are the solutions to the equation obtained by setting the numerator equal to zero, while poles are the solutions to the equation obtained by setting the denominator equal to zero.

In addition, there may also exist zeros and poles at <math>z=0</math> and <math>z=\infty</math>. If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.

By factoring the denominator, [[partial fraction]] decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the [[impulse response]] and the linear constant coefficient difference equation of the system.

===Output response===
If such a system <math>H(z)\ </math> is driven by a signal <math>X(z)\ </math> then the output is <math>Y(z) = H(z)X(z)\ </math>. By performing [[partial fraction]] decomposition on <math>Y(z)\ </math> and then taking the inverse Z-transform the output <math>y[n]\ </math> can be found. In practice, it is often useful to fractionally decompose <math>\frac{Y(z)}{z}\ </math> before multiplying that quantity by <math>z\ </math> to generate a form of <math>Y(z)\ </math> which has terms with easily computable inverse Z-transforms.

==See also==
* [[Zeta function regularization]]
* [[Advanced Z-transform]]
* [[Formal power series]]
* [[Laplace transform]]

==Bibliography==
* Eliahu Ibrahim Jury, ''Theory and Application of the Z-Transform Method'', Krieger Pub Co, 1973. ISBN 0-88275-122-0.

* Refaat El Attar, ''Lecture notes on Z-Transform'', Lulu Press, Morrisville NC, 2005. ISBN 1-4116-1979-X.

==External links==
* [http://mathworld.wolfram.com/Z-Transform.html Mathworld's entry on the Z-transform]
* [http://www.dsprelated.com/comp.dsp/keyword/Z_Transform.php Z-Transform threads in Comp.DSP]
* [http://math.fullerton.edu/mathews/c2003/ZTransformIntroMod.html Z-Transform Module by John H. Mathews]
* [http://www.ling.upenn.edu/courses/ling525/z.html University of Penn Z-Transform Intro]
* [http://franklin.vp.googlepages.com/ejercicios University Centre Jos'e Antonio Echeverria,Cuba Z-transform class notes]
{{DSP}}

[[Category:Transforms]]

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Z-transform

2007-05-23T14:16:56Z

200.55.139.25: /* External links */

In [[mathematics]] and [[signal processing]], the '''Z-transform''' converts a discrete [[time-domain]] signal, which is a [[sequence]] of [[real number]]s, into a [[complex number|complex]] [[frequency-domain]] representation.

The Z-transform and [[advanced Z-transform]] were introduced (under the Z-transform name) by [[E. I. Jury]] in [[1958]] in ''Sampled-Data Control Systems'' (John Wiley & Sons). The idea contained within the Z-transform was previously known as the "generating function method".

The (unilateral) Z-transform is to discrete-time signals what the one-sided [[Laplace transform]] is to continuous-time signals.

==Definition==

The Z-transform, like many other integral transforms, can be defined as either a ''one-sided'' or ''two-sided'' transform.

===Bilateral Z-transform===

The ''bilateral'' or ''two-sided'' Z-transform of a discrete-time signal ''x[n]'' is the function ''X(z)'' defined as

:<math>X(z) = Z\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \ </math>

where ''n'' is an integer and ''z'' is, in general, a [[complex number]]:
:<math> z= A e^{j\phi} </math>
:where ''A'' is the magnitude of ''z'', and φ is the [[angular frequency]] (in [[radians]] per [[sample (signal)|sample]]).

===Unilateral Z-transform===

Alternatively, in cases where ''x''[''n''] is defined only for ''n'' ≥ 0, the ''single-sided'' or ''unilateral'' Z-transform is defined as

:<math>X(z) = Z\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n} \ </math>

In [[signal processing]], this definition is used when the signal is [[causal system|causal]].

An important example of the unilateral Z-transform is the [[probability-generating function]], where the component <math>x[n]</math> is the probability that a discrete random variable takes the value <math>n</math>, and the function <math>X(z)</math> is usually written as <math>X(s)</math>, in terms of <math>s = z^{-1}</math>. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.

==Inverse Z-transform==

The ''inverse'' Z-transform is

:<math> x[n] = Z^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz \ </math>

where <math> C \ </math> is a counterclockwise closed path encircling the origin and entirely in the [[Radius of convergence|region of convergence]] (ROC). The contour or path, <math> C \ </math>, must encircle all of the poles of <math> X(z) \ </math>.

A special case of this [[contour integral]] occurs when <math> C \ </math> is the unit circle (and can be used when the ROC includes the unit circle). The inverse Z-transform simplifies to the [[Discrete-time_Fourier_transform#Inverse_transform|inverse discrete-time Fourier transform]]:

:<math> x[n] = \frac{1}{2 \pi} \int_{-\pi}^{+\pi} X(e^{j \omega}) e^{j \omega n} d \omega \ </math> .

The Z-transform with a finite range of ''n'' and a finite number of uniformly-spaced ''z'' values can be computed efficiently via [[Bluestein's FFT algorithm]]. The [[Discrete-time Fourier transform|discrete-time Fourier transform]] (DTFT) (not to confuse with the [[discrete Fourier transform]] (DFT)) is a special case of such a Z-transform obtained by restricting ''z'' to lie on the unit circle.

==Region of convergence==
The [[Radius of convergence|region of convergence]] (ROC) is the set of points in the complex plane for which the Z-transform summation converges.

:<math>ROC = \left\{ z : \left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right| < \infty \right\} </math>

===Example 1 (No ROC)===
Let <math>x[n] = 0.5^n\ </math>. Expanding <math>x[n]\ </math> on the interval <math>(-\infty, \infty)\ </math> it becomes

:<math>x[n] = \{..., 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, ...\} = \{..., 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, ...\}\ </math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} < \infty\ </math>

There are no such values of <math>z\ </math> that satisfy this condition.

===Example 2 ([[causal system|causal]] ROC)===
[[Image:Region of convergence 0.5 causal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle and the circle <math>\left|z\right| = 0.5</math> is shown as a dashed black circle]]

Let <math>x[n] = 0.5^n u[n]\ </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]\ </math> on the interval <math>(-\infty, \infty)\ </math> it becomes

:<math>x[n] = \{..., 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, ...\}\ </math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}0.5^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{0.5}{z}\right)^n = \frac{1}{1 - 0.5z^{-1}}\ </math>

The last equality arises from the infinite [[geometric series]] and the equality only holds if <math>\left|0.5 z^{-1}\right| < 1\ </math> which can be rewritten in terms of <math>z\ </math> as <math>\left|z\right| > 0.5\ </math>. Thus, the ROC is <math>\left|z\right| > 0.5\ </math>. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".
 

===Example 3 ([[Anticausal system|anticausal]] ROC)===
[[Image:Region of convergence 0.5 anticausal.svg|thumb|250px|ROC shown in blue, the unit circle as a dotted grey circle and the circle <math>\left|z\right| = 0.5</math> is shown as a dashed black circle]]

Let <math>x[n] = -(0.5)^n u[-n-1]\ </math> (where <math>u</math> is the [[Heaviside step function]]). Expanding <math>x[n]\ </math> on the interval <math>(-\infty, \infty)\ </math> it becomes

:<math>x[n] = \{..., -(0.5)^{-3}, -(0.5)^{-2}, -(0.5)^{-1}, 0, 0, 0, ...\}\ </math>

Looking at the sum

:<math>\sum_{n=-\infty}^{\infty}x[n]z^{-n} = -\sum_{n=-\infty}^{-1}0.5^nz^{-n} = -\sum_{n=-\infty}^{-1}\left(\frac{z}{0.5}\right)^{-n}\ </math>
:<math>= -\sum_{m=1}^{\infty}\left(\frac{z}{0.5}\right)^{m} = -\frac{0.5^{-1}z}{1 - 0.5^{-1}z} = \frac{z}{z - 0.5} = \frac{1}{1 - 0.5z^{-1}}\ </math>

Using the infinite [[geometric series]], again, the equality only holds if <math>\left|0.5^{-1}z\right| < 1\ </math> which can be rewritten in terms of <math>z\ </math> as <math>\left|z\right| < 0.5\ </math>.
Thus, the ROC is <math>\left|z\right| < 0.5\ </math>. In this case the ROC is a disc centered at the origin and of radius 0.5.

What differentiates this example from the previous example is ''only'' the ROC. This is intentional to demonstrate that the transform result alone is insufficient.
 

===Examples conclusion===
Examples 2 & 3 clearly show that the Z-transform <math>X(z)\ </math> of <math>x[n]\ </math> is unique when and only when specifying the ROC. Creating the [[pole-zero plot]] for the causal and anticausal case show that the ROC for either case does not include the pole that is at 0.5. This extends to cases with multiple poles: the ROC will ''never'' contain poles.

In example 2, the causal system yields an ROC that includes <math>\left| z \right| = \infty\ </math> while the anticausal system in example 3 yields an ROC that includes <math>\left| z \right| = 0\ </math>.

[[Image:Region of convergence 0.5 0.75 mixed-causal.svg|thumb|250px|ROC shown as a blue ring <math>0.5 < \left| z \right| < 0.75\ </math>]]
In systems with multiple poles it is possible to have an ROC that includes neither <math>\left| z \right| = \infty\ </math> nor <math>\left| z \right| = 0\ </math>. The ROC creates a circular band. For example, <math>x[n] = 0.5^nu[n] - 0.75^nu[-n-1]\ </math> has poles at 0.5 and 0.75. The ROC will be <math>0.5 < \left| z \right| < 0.75\ </math>, which includes neither the origin nor infinity. Such a system is called a [[mixed-causality system|mixed-causality]] system as it contains a causal term <math>0.5^nu[n]\ </math> and an anticausal term <math>-(0.75)^nu[-n-1]\ </math>.

The stability of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., <math>\left| z \right| = 1\ </math>) then the system is stable. In the above systems the causal system is stable because <math>\left| z \right| > 0.5\ </math> contains the unit circle.

If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous <math>x[n]\ </math>) you can determine a unique <math>x[n]\ </math> provided you desire the following:
* Stability
* Causality

If you need stability then the ROC must contain the unit circle.
If you need a causal system then the ROC must contain infinity.
If you need an anticausal system then the ROC must contain the origin.

The unique <math>x[n]\ </math> can then be found.

==Properties==

{| class="wikitable"
|+ '''Properties of the z-transform'''
!
! Time domain
! Z-domain
! ROC
|-
! Notation
| <math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math>
| <math>X(z)=\mathcal{Z}\{x[n]\}</math>
| ROC: <math>r_2<|z|<r_1 \ </math>
|-
! Linearity
| <math>a_1 x_1[n] + a_2 x_2[n]\ </math>
| <math>a_1 X_1(z) + a_2 X_2(z) \ </math>
| At least the intersection of ROC1 and ROC2
|-
! Time shifting
| <math>x[n-k]\ </math>
| <math>z^{-k}X(z) \ </math>
| ROC, except <math>z=0\ </math> if <math>k>0\,</math> and <math>z=\infty</math> if <math>k<0\ </math>
|-
! Scaling in the z-domain
| <math>a^n x[n]\ </math>
| <math>X(a^{-1}z) \ </math>
| <math>|a|r_2<|z|<|a|r_1 \ </math>
|-
! Time reversal
| <math>x[-n]\ </math>
| <math>X(z^{-1}) \ </math>
| <math>\frac{1}{r_1}<|z|<\frac{1}{r_2} \ </math>
|-
! Conjugation
| <math>x^*[n]\ </math>
| <math>X^*(z^*) \ </math>
| ROC
|-
! Real part
| <math>\operatorname{Re}\{x[n]\}\ </math>
| <math>\frac{1}{2}\left[X(z)+X^*(z^*) \right]</math>
| ROC
|-
! Imaginary part
| <math>\operatorname{Im}\{x[n]\}\ </math>
| <math>\frac{1}{2j}\left[X(z)-X^*(z^*) \right]</math>
| ROC
|-
! Differentiation
| <math>nx[n]\ </math>
| <math> -z \frac{dX(z)}{dz}</math>
| ROC
|-
! [[Convolution]]
| <math>x_1[n] * x_2[n]\ </math>
| <math>X_1(z)X_2(z) \ </math>
| At least the intersection of ROC1 and ROC2
|-
! [[Cross-correlation|Correlation]]
| <math>r_{x_1,x_2}(l)=x_1[l] * x_2[-l]\ </math>
| <math>R_{x_1,x_2}(z)=X_1(z)X_2(z^{-1})\ </math>
| At least the intersection of ROC of X1(z) and X2(<math>z^{-1}</math>)
|-
! Multiplication
| <math>x_1[n]x_2[n]\ </math>
| <math>\frac{1}{j2\pi}\oint_C X_1(v)X_2(\frac{z}{v})v^{-1}\mathrm{d}v \ </math>
| At least <math>r_{1l}r_{2l}<|z|<r_{1u}r_{2u} \ </math>
|-
! [[Parseval's theorem|Parseval's relation]]
| <math>\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n]\ </math>
| <math>\frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\frac{1}{v^*})v^{-1}\mathrm{d}v \ </math>
|
|}

* '''Initial value theorem'''
::<math>x[0]=\lim_{z\rightarrow \infty}X(z) \ </math>, If <math>x[n]\,</math> causal

* '''Final value theorem'''
:: <math>x[\infty]=\lim_{z\rightarrow 1}(z-1)X(z) \ </math>, Only if poles of <math>(z-1)X(z) \ </math> are inside the unit circle

==Table of common Z-transform pairs==

{| class="wikitable"
|-
! !! Signal, <math>x[n]</math> !! Z-transform, <math>X(z)</math> !! ROC
|-
| 1 || <math>\delta[n] \, </math> || <math>1\, </math> ||<math> \mbox{all }z\, </math>
|-
| 2 || <math>\delta[n-n_0] \,</math> || <math> z^{-n_0} \, </math> || <math> z \neq 0\,</math>
|-
| 3 || <math>u[n] \,</math> || <math> \frac{1}{1-z^{-1} }</math> || <math>|z| > 1\,</math>
|-
| 4 || <math>- u[-n-1] \,</math> || <math> \frac{1}{1 - z^{-1}}</math> ||<math>|z| < 1\,</math>
|-
| 5 || <math>n u[n] \,</math> || <math> \frac{z^{-1}}{( 1-z^{-1} )^2}</math> || <math>|z| > 1\,</math>
|-
| 6 || <math> - n u[-n-1] \,</math> || <math> \frac{z^{-1} }{ (1 - z^{-1})^2 }</math> ||<math> |z| < 1 \,</math>
|-
| 7 || <math>n^2 u[n] \,</math> || <math> \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| > 1\,</math>
|-
| 8 || <math> - n^2 u[-n - 1] \,</math> || <math> \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} </math> || <math>|z| < 1\,</math>
|-
| 9 || <math>n^3 u[n] \,</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| > 1\,</math>
|-
| 10 || <math>- n^3 u[-n -1] \,</math> || <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} </math> || <math>|z| < 1\,</math>
|-
| 11 || <math>a^n u[n] \,</math> || <math> \frac{1}{1-a z^{-1}}</math> ||<math> |z| > |a|\,</math>
|-
| 12 || <math>-a^n u[-n-1] \,</math> || <math> \frac{1}{1-a z^{-1}}</math> ||<math>|z| < |a|\,</math>
|-
| 13 || <math>n a^n u[n] \,</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> || <math>|z| > |a|\,</math>
|-
| 14 || <math>-n a^n u[-n-1] \,</math> || <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> ||<math> |z| < |a|\,</math>
|-
| 15 || <math>n^2 a^n u[n] \,</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| > |a|\,</math>
|-
| 16 || <math>- n^2 a^n u[-n -1] \,</math> || <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> || <math>|z| < |a|\,</math>
|-
| 17 || <math>\cos(\omega_0 n) u[n] \,</math> || <math> \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1\,</math>
|-
| 18 || <math>\sin(\omega_0 n) u[n] \,</math> || <math> \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> ||<math> |z| >1\,</math>
|-
| 19 || <math>a^n \cos(\omega_0 n) u[n] \,</math> || <math> \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math> |z| > |a|\,</math>
|-
| 20 || <math>a^n \sin(\omega_0 n) u[n] \,</math> || <math> \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> ||<math> |z| > |a|\,</math>
|}

==Relationship to Laplace==
The bilateral Z-transform is simply the [[two-sided Laplace transform]] of the ideal sampled function

::<math> x(t) \sum_{n=-\infty}^{\infty} \delta(t-nT) = \sum_{n=-\infty}^{\infty} x[n] \delta(t-nT) \ </math>

where <math> x(t) \ </math> is the continuous-time function being sampled, <math> x[n]=x(nT) \ </math> the nth sample, <math> T \ </math> is the sampling period, and with the substitution: <math> z = e^{sT} \ </math>.

Likewise the unilateral Z-transform is simply the one-sided [[Laplace transform]] of the ideal sampled function. Both assume that the sampled function is zero for all negative time indices.

The [[Bilinear transform]] is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa. To do this, you can use the following substitutions in <math> H(s) </math> or <math> H(z) </math> :

<math> s =\frac{2}{T} \frac{z-1}{z+1} </math> from Laplace to z (Tustin transformation);

<math> z =\frac{2+sT}{2-sT} </math> from z to Laplace.

==Relationship to Fourier==
The Z-transform is a generalization of the [[discrete-time Fourier transform]] (DTFT). The DTFT can be found by evaluating the Z-transform <math>X(z)\ </math> at <math>z=e^{j\omega}\ </math> or, in other words, evaluated on the unit circle. In order to determine the [[frequency response]] of the system the Z-transform must be evaluated on the unit circle, meaning that the system's region of convergence must contain the unit circle. Otherwise, the DTFT of the system does not exist.

==Linear constant-coefficient difference equation==
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the [[autoregressive moving-average]] equation.

:<math>\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q}\ </math>

Both sides of the above equation can be divided by <math>\alpha_0 \ </math>, if it is not zero, normalizing <math>\alpha_0 = 1\ </math> and the LCCD equation can be written

:<math>y[n] = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}\ </math>

This form of the LCCD equation is favorable to make it more explicit that the "current" output <math>y[{n}]\ </math> is a function of past outputs <math>y[{n-p}]\ </math>, current input <math>x[{n}]\ </math>, and previous inputs <math>x[{n-q}]\ </math>.

===Transfer function===
Taking the Z-transform of the equation (using linearity and time-shifting laws) yields

:<math>Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q}\ </math>

and rearranging results in

:<math>H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-p}\alpha_{p}} = \frac{\beta_0 + z^{-1} \beta_1 + z^{-2} \beta_2 + ... + z^{-M} \beta_M}{\alpha_0 + z^{-1} \alpha_1 + z^{-2} \alpha_2 + ... + z^{-N} \alpha_N}\ </math>

===Zeros and poles===
From the [[fundamental theorem of algebra]] the [[numerator]] has M [[root (mathematics)|roots]] (corresponding to [[Zero (complex analysis)|zeros]] of H) and the [[denominator]] has N roots (corresponding to [[Pole (complex analysis)|poles]]). Rewriting the [[transfer function]] in terms of poles and zeros

:<math>H(z) = \frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})...(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})...(1 - p_N z^{-1})}\ </math>

Where <math>q_k\ </math> is the <math>k^{th}\ </math> zero and <math>p_k\ </math> is the <math>k^{th}\ </math> pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the [[pole-zero plot]].

In simple words, zeros are the solutions to the equation obtained by setting the numerator equal to zero, while poles are the solutions to the equation obtained by setting the denominator equal to zero.

In addition, there may also exist zeros and poles at <math>z=0</math> and <math>z=\infty</math>. If we take these poles and zeros as well as multiple-order zeros and poles into consideration, the number of zeros and poles are always equal.

By factoring the denominator, [[partial fraction]] decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the [[impulse response]] and the linear constant coefficient difference equation of the system.

===Output response===
If such a system <math>H(z)\ </math> is driven by a signal <math>X(z)\ </math> then the output is <math>Y(z) = H(z)X(z)\ </math>. By performing [[partial fraction]] decomposition on <math>Y(z)\ </math> and then taking the inverse Z-transform the output <math>y[n]\ </math> can be found. In practice, it is often useful to fractionally decompose <math>\frac{Y(z)}{z}\ </math> before multiplying that quantity by <math>z\ </math> to generate a form of <math>Y(z)\ </math> which has terms with easily computable inverse Z-transforms.

==See also==
* [[Zeta function regularization]]
* [[Advanced Z-transform]]
* [[Formal power series]]
* [[Laplace transform]]

==Bibliography==
* Eliahu Ibrahim Jury, ''Theory and Application of the Z-Transform Method'', Krieger Pub Co, 1973. ISBN 0-88275-122-0.

* Refaat El Attar, ''Lecture notes on Z-Transform'', Lulu Press, Morrisville NC, 2005. ISBN 1-4116-1979-X.

==External links==
* [http://mathworld.wolfram.com/Z-Transform.html Mathworld's entry on the Z-transform]
* [http://www.dsprelated.com/comp.dsp/keyword/Z_Transform.php Z-Transform threads in Comp.DSP]
* [http://math.fullerton.edu/mathews/c2003/ZTransformIntroMod.html Z-Transform Module by John H. Mathews]
* [http://www.ling.upenn.edu/courses/ling525/z.html University of Penn Z-Transform Intro]
* [http://franklin.vp.googlepages.com/clases University Centre Jos'e Antonio Echeverria,Cuba Z-transform class notes]
{{DSP}}

[[Category:Transforms]]

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Relay

2007-05-22T06:23:24Z

200.55.139.25: /* Applications */

{{Otheruses1|the electronic component}}
[[Image:Relay.jpg|thumb|right|Automotive style miniature relay]]
A '''relay''' is an electrical [[switch]] that opens and closes under the control of another electrical circuit. In the original form, the switch is operated by an [[magnet|electromagnet]] to open or close one or many sets of contacts. It was invented by [[Joseph Henry]] in [[1835]]. Because a relay is able to control an output circuit
of higher power than the input circuit, it can be considered, in a broad sense, to be a form of an electrical [[amplifier]].

== Operation ==
When a [[Current (electricity)|current]] flows through the [[coil]], the resulting [[magnetic field]] attracts an armature that is mechanically linked to a moving contact. The movement either makes or breaks a connection with a fixed contact. When the current to the coil is switched off, the armature is returned by a force approximately half as strong as the magnetic force to its relaxed position. Usually this is a [[spring (device)|spring]], but gravity is also used commonly in industrial motor starters. Most relays are manufactured to operate quickly. In a low voltage application, this is to reduce noise. In a high voltage or high current application, this is to reduce [[arcing]].

If the coil is energized with DC, a diode is frequently installed across the coil, to dissipate the energy from the collapsing magnetic field at deactivation, which would otherwise generate a spike of voltage and might cause damage to circuit components. If the coil is designed to be energized with AC, a small copper ring can be crimped to the end of the solenoid. This "shading ring" creates a small out-of-phase current, which increases the minimum pull on the armature during the AC cycle.<ref>Mason, C. R., ''Art & Science of Protective Relaying'', Chapter 2, GE Consumer & Electrical [http://www.geindustrial.com/Multilin/notes/artsci/index.htm], </ref>

By analogy with the functions of the original electromagnetic device, a solid-state relay is made with a [[thyristor]] or other solid-state switching device. To achieve electrical isolation, a [[light-emitting diode]] (LED) is used with a photo transistor.

== Types of relay ==
[[Image:Relay2.jpg|thumb|right|Small relay as used in electronics]]

===Latching relay===
A '''latching relay''' has two relaxed states (bistable). These are also called 'keep' relays. When the current is switched off, the relay remains in its last state. This is achieved with a solenoid operating a ratchet and cam mechanism, or by having two opposing coils with an over-center spring or permanent magnet to hold the armature and contacts in position while the coil is relaxed, or with a remnant core. In the ratchet and cam example, the first pulse to the coil turns the relay on and the second pulse turns it off. In the two coil example, a pulse to one coil turns the relay on and a pulse to the opposite coil turns the relay off. This type of relay has the advantage that it consumes power only for an instant, while it is being switched, and it retains its last setting across a power outage.

===Reed relay===
A '''[[reed relay]]''' has a set of contacts inside a [[vacuum]] or [[inert gas]] filled glass tube, which protects the contacts against atmospheric [[corrosion]]. The contacts are closed by a magnetic field generated when current passes through a [[coil]] around the glass tube. Reed relays are capable of faster switching speeds than conventional relays. See also [[reed switch]].

====Mercury-wetted relay====
A '''mercury-wetted relay''' is a form of reed relay in which the contacts are wetted with [[mercury (element)|mercury]]. Such relays are used to switch low-voltage signals (one volt or less) because of its low contact resistance, or for high-speed counting and timing applications where the mercury eliminates contact bounce. Mercury wetted relays are position-sensitive and must be mounted vertically to work properly. Because of the toxicity and expense of liquid mercury, these relays are rarely specified for new equipment. See also [[mercury switch]].

===Polarized relay===
A '''Polarized Relay''' placed the armature between the poles of a permanent magnet to increase sensitivity. Polarized relays were used in middle 20th Century [[Crossbar switch|telephone exchange]]s to detect faint pulses and correct telegraphic distortion. The poles were on screws, so a technician could first adjust them for maximum sensitivity and then apply a bias spring to set the critical current that would operate the relay.

===Machine tool relay===
A '''machine tool relay''' is a type standardized for industrial control of machine tools, transfer machines, and other sequential control. They are characterized by a large number of contacts (sometimes extendable in the field) which are easily converted from normally-open to normally-closed status, easily replaceable coils, and a form factor that allows compactly installing many relays in a control panel. Although such relays once were the backbone of automation in such industries as automobile assembly, the [[programmable logic controller]] mostly displaced the machine tool relay from sequential control applications.

===Contactor relay===
A '''[[contactor]]''' is a very heavy-duty relay used for switching [[electric motor]]s and lighting loads. With high current, the contacts are made with pure silver. The unavoidable arcing causes the contacts to oxidize and silver oxide is still a good conductor. Such devices are often used for motor starters. A motor starter is a contactor with an overload protection devices attached. The overload sensing devices are a form of heat operated relay where a coil heats a bi-metal strip, or where a solder pot melts, releasing a spring to operate auxiliary contacts. These auxiliary contacts are in series with the coil. If the overload senses excess current in the load, the coil is de-energized. Contactor relays can be extremely loud to operate, making them unfit for use where noise is a chief concern.

===Solid state contactor relay===
[[Image:Solid-state-contactor.jpg|thumb|right|25 amp or 40 amp solid state contactors]]
A '''solid state contactor''' is a very heavy-duty [[solid state relay]], including the necessary heat sink, used for switching electric heaters, small [[electric motor]]s and lighting loads; where frequent on/off cycles are required. There are no moving parts to wear out and there is no contact bounce due to vibration. They are activated by AC control signals or DC control signals from [[Programmable logic controller]] (PLCs), PCs, [[Transistor-transistor logic]] (TTL) sources, or other microprocessor controls.

===Buchholz relay===
A '''[[Buchholz relay]]''' is a safety device sensing the accumulation of gas in large oil-filled [[transformer]]s, which will alarm on slow accumulation of gas or shut down the transformer if gas is produced rapidly in the transformer oil.

===Forced-guided contacts relay===
A '''forced-guided contacts relay''' has relay contacts that are mechanically linked together, so that when the relay coil is energized or de-energized, all of the linked contacts move together. If one set of contacts in the relay becomes immobilized, no other contact of the same relay will be able to move. The function of forced-guided contacts is to enable the safety circuit to check the status of the relay. Forced-guided contacts are also known as "positive-guided contacts", "captive contacts", "locked contacts", or "safety relays".

[[Image:Solid_state_relay.jpg|thumb|right|A [[solid state (electronics)|solid state]]
relay, which has no moving parts]]

===Solid-state relay===

A '''[[solid state relay]] (SSR)''' is a [[solid state (electronics)|solid state]] electronic component that provides a similar function to an [[electromechanical]] relay but does not have any moving components, increasing long-term reliability. With early SSR's, the tradeoff came from the fact that every transistor has a small voltage drop across it. This collective voltage drop limited the amount of current a given SSR could handle. As transistors improved, higher current SSR's, able to handle 100 to 1,200 [[amps]], have become commercially available.
Compared to electromagnetic relays, they may be falsely triggered by transients.

===Overload protection relay===
One type of [[electric motor]] overload protection relay is operated by a heating element in series with the [[electric motor]] . The heat generated by the motor current operates a bi-metal strip or melts [[solder]], releasing a spring to operate contacts. Where the overload relay is exposed to the same environment as the motor, a useful though crude compensation for motor ambient temperature is provided.

==Pole & Throw==
[[Image:Relay_symbols.gif|thumb|left|100px|Circuit symbols of relays. ''"C" denotes the common terminal in SPDT and DPDT types.'']]
Since relays are [[switch]]es, the terminology applied to switches is also applied to relays. A relay will switch one or more ''poles'', each of whose contacts can be ''thrown'' by energizing the coil in one of three ways:
* Normally-open ('''NO''') contacts connect the circuit when the relay is activated; the circuit is disconnected when the relay is inactive. It is also called a '''Form A''' contact or "make" contact.
* Normally-closed ('''NC''') contacts disconnect the circuit when the relay is activated; the circuit is connected when the relay is inactive. It is also called a '''Form B''' contact or "break" contact.
* Change-over, or double-throw, contacts control two circuits: one normally-open contact and one normally-closed contact with a common terminal. It is also called a '''Form C''' contact or "transfer" contact.

The following types of relays are commonly encountered:
* '''SPST''' - '''S'''ingle '''P'''ole '''S'''ingle '''T'''hrow. These have two terminals which can be connected or disconnected. Including two for the coil, such a relay has four terminals in total. It is ambiguous whether the pole is normally open or normally closed. The terminology "SPNO" and "SPNC" is sometimes used to resolve the ambiguity.
* '''SPDT''' - '''S'''ingle '''P'''ole '''D'''ouble '''T'''hrow. A common terminal connects to either of two others. Including two for the coil, such a relay has five terminals in total.
* '''DPST''' - '''D'''ouble '''P'''ole '''S'''ingle '''T'''hrow. These have two pairs of terminals. Equivalent to two SPST switches or relays actuated by a single coil. Including two for the coil, such a relay has six terminals in total. It is ambiguous whether the poles are normally open, normally closed, or one of each.
[[Image:Relaycov.jpg|thumb|right|100px|The diagram on the package of a DPDT AC coil relay]]
* '''DPDT''' - '''D'''ouble '''P'''ole '''D'''ouble '''T'''hrow. These have two rows of change-over terminals. Equivalent to two SPDT switches or relays actuated by a single coil. Such a relay has eight terminals, including the coil.
* '''QPDT''' - '''Q'''uadruple '''P'''ole '''D'''ouble '''T'''hrow. Often referred to as Quad Pole Double Throw, or 4PDT. These have four rows of change-over terminals. Equivalent to four SPDT switches or relays actuated by a single coil, or two DPDT relays. In total, fourteen terminals including the coil.


== Applications ==
[[Image:ACRelay.jpg|thumb|right|150px|A DPDT AC coil relay with "ice cube" packaging]]
Relays are used:
*to control a high-voltage circuit with a low-voltage signal, as in some types of [[modem]]s,
*to control a high-[[Current (electricity)|current]] circuit with a low-current signal, as in the [[starter motor|starter]] [[solenoid]] of an [[automobile]],
*to detect and isolate faults on transmission and distribution lines by opening and closing [[circuit breakers]] (protection relays),
*to isolate the controlling circuit from the controlled circuit when the two are at different potentials, for example when controlling a mains-powered device from a low-voltage switch. The latter is often applied to control office lighting as the low voltage wires are easily installed in partitions, which may be often moved as needs change. They may also be controlled by room occupancy detectors in an effort to conserve energy,
*to perform logic functions. For example, the boolean AND function is realised by connecting NO relay contacts in series, the OR function by connecting NO contacts in parallel. The change-over or Form C contacts perform the XOR (exclusive or) function. Similar functions for NAND and NOR are accomplished using NC contacts. Due to the failure modes of a relay compared with a semiconductor, they are widely used in safety critical logic, such as the control panels of radioactive waste handling machinery.
*to perform time delay functions. Relays can be modified to delay opening or delay closing a set of contacts. A very short (a fraction of a second) delay would use a copper disk between the armature and moving blade assembly. Current flowing in the disk maintains magnetic field for a short time, lengthening release time. For a slightly longer (up to a minute) delay, a dashpot is used. A dashpot is a piston filled with fluid that is allowed to escape slowly. The time period can be varied by increasing or decreasing the flow rate. For longer time periods, a mechanical clockwork timer is installed.

this is not good

==Relay application considerations==
[[Image:Phonerelay.png|thumb|right|A large relay with two coils and many sets of contacts, used in an old telephone switching system.]]
Selection of an appropriate relay for a particular application requires evaluation of many different factors:
* Number and type of contacts - normally open, normally closed, changeover (double-throw)
* In the case of changeover, there are two types. This style of relay can be manufactured two different ways. "Make before Break" and "Break before Make". The old style telephone switch required Make-before-break so that the connection didn't get dropped while dialing the number. The railroad still uses them to control railroad crossings.
* Rating of contacts - small relays switch a few amperes, large contactors are rated for up to 3000 amperes, alternating or direct current
* Voltage rating of contacts - typical control relays rated 300 VAC or 600 VAC, automotive types to 50 VDC, special high-voltage relays to about 15,000 V
* Coil voltage - machine-tool relays usually 24 VAC or 120 VAC, relays for switchgear may have 125 V or 250 VDC coils, "sensitive" relays operate on a few milliamperes
* Package/enclosure - open, touch-safe, double-voltage for isolation between circuits, explosion proof, outdoor, oil-splashresistant
* Mounting - sockets, plug board, rail mount, panel mount, through-panel mount, enclosure for mounting on walls or equipment
* Switching time - where high speed is required
* "Dry" contacts - when switching very low level signals, special contact materials may be needed such as gold-plated contacts
* Contact protection - suppress arcing in very inductive circuits
* Coil protection - suppress the surge voltage produced when switching the coil current
* Isolation between coil circuit and contacts
* Aerospace or radiation-resistant testing, special quality assurance
* Expected mechanical loads due to [[acceleration]] - some relays used in [[aerospace]] applications are designed to function in [[Shock (mechanics)|shock]] loads of 50 [[G-force|''g'']] or more
* Accessories such as timers, auxiliary contacts, pilot lamps, test buttons
* Regulatory approvals
* Stray magnetic linkage between coils of adjacent relays on a printed circuit board.

==Protective relay==
A '''[[protection and monitoring of the electrical energy transmission networks|protective relay]]''' is a complex electromechanical apparatus, often with more than one coil, designed to calculate operating conditions on an electrical circuit and trip circuit breakers when a fault was found. Unlike switching type relays with fixed and usually ill-defined operating voltage thresholds and operating times, protective relays had well-established, selectable, time/current (or other operating parameter) curves. Such relays were very elaborate, using arrays of induction disks, shaded-pole magnets, operating and restraint coils, solenoid-type operators, telephone-relay style contacts, and phase-shifting networks to allow the relay to respond to such conditions as over-current, over-voltage, reverse [[Electric power|power]] flow, over- and under- frequency, and even distance relays that would trip for faults up to a certain distance away from a substation but not beyond that point. An important transmission line or generator unit would have had cubicles dedicated to protection, with a score of individual electromechanical devices. Each of the protective functions available on a given relay are denoted by standard [[ANSI Device Numbers]]. For example, a relay including function 51 would be a timed overcurrent protective relay.

Design and theory of these protective devices is an important part of the education of an [[power engineering|electrical engineer]] who specializes in power systems. Today these devices are nearly entirely replaced (in new designs) with microprocessor-based instruments (numerical relays) that emulate their electromechanical ancestors with great precision and convenience in application. By combining several functions in one case, numerical relays also save capital cost and maintenance cost over electromechanical relays. However, due to their very long life span, tens of thousands of these "silent sentinels" are still protecting transmission lines and electrical apparatus all over the world.

[[Image:Reedrelay.jpg|thumb|right|Top, middle: reed switches, bottom: reed relay]]

==Overcurrent relay==
An "Overcurrent Relay" is a type of protective relay. The ANSI Device Designation Number is 50 for an Instantaneous OverCurrent (IOC), 51 for a Time OverCurrent (TOC). In a typical application the overcurrent relay is used for overcurrent protection, connected to a current transformer and calibrated to operate at or above a specific current level. When the relay operates, one or more contacts will operate and energize a trip coil in a Circuit Breaker and trip (open) the Circuit Breaker.

== See also ==

* [[Contactors]]
* [[Ladder programming language]]
* [[Wire spring relay]]

==References==
1. Vladimir Gurevich "Electrical Relays: Principles and Applications", CRC Press (Taylor & Francis group), London - New York, 2005, 704 pp.
2. Westinghouse Corporation, ''Applied Protective Relaying'', 1976, Westinghouse Corporation, no ISBN, Library of Congress card no. 76-8060 - a standard reference on electromechanical protection relays (out of print - current edition published by ABB)

3. Terrell Croft and Wilford Summers (ed), ''American Electricians' Handbook, Eleventh Edition'', McGraw Hill, New York (1987) ISBN 0-07-013932-6

<references />

==External links==
*[http://www.eleinmec.com/article.asp?24 Information about relays and the Latching Relay circuit]
*[http://www.uoguelph.ca/~antoon/circ/solidstate.html Example of the circuitry within a solid state relay]
*[http://www.asiarelay.com/products/relays-cross-reference.asp Relay Cross Reference]
*[http://www.mantatest.com/Protection%20Library.html Protective relay technical resource library] from Manta Test Systems
*[http://web.cecs.pdx.edu/~harry/Relay/index.html "Harry Porter's Relay Computer", a computer made out of relays.]

[[Category:BEV components]]
[[Category:Electromagnetic components]]
[[Category:Electrical components]]
[[Category:Switches]]
[[Category:Transducers]]
[[Category:Power engineering]]

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[[zh:继电器]]

Evoked potential

2007-05-14T04:40:37Z

200.55.139.25:

{{unreferenced||date=December 2005}}
In [[neurophysiology]], an '''evoked potential''' (or "evoked response") is an electrical potential recorded from a [[human]] or [[animal]] following presentation of a stimulus, as distinct from spontaneous potentials such as [[electroencephalogram]]s or [[electromyogram]]s.
Evoked potential [[amplitude]]s tend to be low, ranging from less than a [[microvolt]] to several microvolts, compared to tens of microvolts for EEG, millivolts for EMG, and often close to a volt for [[Electrocardiogram|EKG]]. To resolve these low-amplitude potentials against the background of ongoing EEG, EKG, EMG and other biological signals and ambient noise, signal averaging is usually required. The signal is time-locked to the stimulus and most of the [[noise]] occurs randomly, allowing the noise to be averaged out with averaging of repeated responses.

Signals can be recorded from [[cerebral cortex]], [[brain stem]], [[spinal cord]] and [[Peripheral nervous system|peripheral nerve]]s. Usually the term "evoked potential" is reserved for responses involving either recording from, or stimulation of, central nervous system structures. Thus evoked CMAP (compound motor action potentials) or SNAP (sensory nerve action potentials) as used in NCV (nerve conduction studies) are generally not thought of as evoked potentials, though they do meet the above definition.

== Sensory evoked potentials ==
Sensory evoked potentials (SEP or SSEP) are recorded from the central nervous system following stimulation of sense organs (for example, visual evoked potentials elicited by a flashing light or changing pattern on a monitor; auditory evoked potentials by a click or tone stimulus presented through earphones) or by electrical stimulation of a sensory or mixed nerve. They have been widely used in clinical diagnostic medicine since the 1970s, and also in intraoperative neurophysiology monitoring (IONM), also known as surgical neurophysiology.

There are three kinds of evoked potentials in widespread clinical use since the 1970s: auditory evoked potentials, usually recorded from the scalp but originating at brainstem level (ABR, BAER, BSER, BAEP, BSEP); visual evoked potentials, and [[somatosensory evoked potentials]], which are elicited by electrical stimulation of peripheral nerve. See the articles on each of these modalities.

=== Intraoperative monitoring ===
Somatosensory evoked potentials provide monitoring for the dorsal columns, MEP for the ventral cord, specifically the lateral corticospinal tract. Since the ventral and dorsal spinal cord have separate blood supply with very limited collateral flow, an anterior cord syndrome (paralysis or paresis with some preserved sensory function) is a possible surgical sequela, so it is important to have monitoring specific to the motor tracts as well as dorsal column monitoring.

Magnetic stimulation is generally regarded as unsuitable for intraoperative monitoring because it is more sensitive to anesthesia. Electrical stimulation is too painful for clinical use in awake patients. The two modalities are thus complementary, electrical stimulation being the choice for intraoperative monitoring, and magnetic for clinical applications. bindi.

== Motor evoked potentials ==
Motor evoked potentials (MEP) are recorded from muscles following direct stimulation of exposed motor cortex, or transcranial stimulation of motor cortex, either magnetic or electrical. Transcranial magnetic MEP (TCmMEP) potentially offer clinical diagnostic applications, but remain investigational as of 2005; transcranial electrical MEP (TCeMEP) has been in widespread use for several years for intraoperative monitoring of pyramidal tract functional integrity.

During the 1990s there were attempts to monitor "motor evoked potentials", including "neurogenic motor evoked potentials" recorded from peripheral nerves, following direct electrical stimulation of the spinal cord. It has become clear that these "motor" potentials were almost entirely elicited by antidromic stimulation of sensory tracts-- even when the recording was from muscles (antidromic sensory tract stimulation triggers myogenic responses through synapses at the root entry level). TCMEP, whether electrical or magnetic, is the most practical way to ensure pure motor responses, since stimulation of sensory cortex cannot result in descending impulses beyond the first synapse (synapses cannot be backfired).

== Evoked Potentials Procedure ==

Electrodes need to be attached to various points of on your scalp. Your head is measured using a standardized EEG measurement technique to determine the right spots (each spot corresponding to a type of EP that will be measured - e.g. the two locations on the back of the skull for the visual cortex, etc.), which are marked with a writing implement akin to a very thick pencil. Each of these spots is rubbed with an oil-removing scrub to get rid of the skin oil, then an electrode dipped in a liberal quantity of conductive gel (approximately the consistency of soft butter) is applied and pressed to each spot, and affixed with a strip of adhesive tape. 
For visual evoked potential (VEP), you are placed in front of a computer screen, which shows a pattern of white and black squares like a chessboard, and a red dot in the middle that you are supposed to focus your eyes on with minimal movement. The procedure is done one eye at a time, with the eye that is not being tested blocked off with an eye patch. During the actual procedure, these squares alternate (white ones become black, black ones become white) at a rate of several times a second, which produces responses in the visual cortex, which is picked up by your skull electrodes. Since the computer controls the exact timing of the changes of the square colors, and receives the exact timing of the electric response in the corresponding electrodes, it is able to determine precisely the amount of time it takes for the visual stimulus to reach the visual cortex.
For the somatosensory evoked potentials (SEP), additional electrodes are applied, in the same manner as described earlier. 
For the upper SEP (arms), two stimulus electrodes are attached on the inside wrist, closer to the thumb. These electrodes will receive timed electric pulses that will produce an involuntary twitch of the thumb. An additional sensor electrode is applied on the back of your shoulder, close to the attachment point of the clavicle. Similar to the VEP, the computer times the electric pulses (which come at a rate of several times a second) and gets the responses from the appropriate skull electrode, thus determining the exact time it takes for the stimulus to reach the intermediate point on your shoulder, and then the brain. The same is then repeated on the other arm.
For the lower SEP (legs), two stimulus electrodes are attached to the inside of your ankle, in such a way as to produce an involuntary twitch of the big toe. Additional sensor electrodes are placed at the back of the knee (closer to the outside), on the spine of the lower back, and on the spine of the upper back. Electric pulses are then sent at a rate of several times a second, and the responses are recorded in the same manner as above. 
For the brain auditory evoked potential (BAEP), the stimulus is supplied through headphones. The ear that is being tested receives a clicking sound, at a rate of several times a second, while the other ear receives static. Additional sensor electrodes are placed on the backs of your earlobes. The timing is determined as above. 

=== Evoked potential signal determination ===

There are many things going on at once in the brain, so it is difficult to determine when the evoked potential from a particular stimulus arrives from just one stimulus. The technique used to amplify the signal is called signal averaging. The stimulus in each evoked potential test is applied many times (one or two thousand times), and since everything else besides the evoked potential is not related to the signal, it happens at various random times relative to the stimulus, whereas the potential that is evoked by the stimulus always occurs at the same time relative to the stimulus. This allows the computer to pick out and amplify the one consistent peak or series of peaks, that are caused by the applied stimulus.

In order to improve the efficacy of this technique, you are advised to relax and not move, so as to reduce the noisiness of the signal and make the averaging technique more effective with fewer iterations of the stimulus.

== See also ==

* [[Steady-state evoked potential]]s
* [[Visual evoked potential]]s
* [[Auditory evoked potential]]s
* [[Somatosensory evoked potential]]s
* [[Event-related potential]]
* [[Evoked field]]
* [[Induced activity]]
* [[Ongoing brain activity|Ongoing activity]]

[[Category:Evoked potentials|*]]
[[Category:Electroencephalography]]
[[Category:Medical tests]]

[[de:Evozierte Potentiale]]
[[fr:Potentiel évoqué]]
[[it:Potenziali evocati]]
[[ru:Вызванный потенциал]]

Network simulation

2007-03-23T22:09:13Z

200.55.139.25: /* External links */

In [[computer network]] research, '''network simulation''' is a technique where a program simulates the behavior of a network. The program performs this simulation either by calculating the interaction between the different network entities (hosts/[[router]]s, [[data link]]s, [[packet]]s, etc) using mathematical formulas, or actually capturing and playing back network parameters from a production network. Using this input, the behavior of the network and the various applications and services it supports can be observed in a test lab. Various attributes of the environment can also be modified in a controlled manner to asses these behaviors under different conditions. When a simulation program is used in conjunction with live applications and services in order to observe end-to-end performance to the user desktop, this technique is also referred to as [[network emulation]].

==Uses==
Network simulators are used to predict the behavior of networks and applications under different situations. Researchers use network simulators to see how their protocols would behave if deployed. It is typical to use a network simulator to test [[routing protocols]], MAC ([[Medium Access Control]]) protocols, [[Transport protocol|transport protocols]], applications etc. Companies use simulators to design their networks and/or applications to get a feel for how they will perform under current or projected real-world conditions.

==Simulators==
The simulator (or network simulator) is the program in charge of calculating how the network would behave. They may be distributed in source form (software) or provided in the form of a hardware appliance. Users can then customize the simulator to fulfill their specific analysis needs. The simulators come with support for the most popular protocols implemented (such as IPv4, IPv6, UDP, and TCP).

===Open Source Simulators===
Free/[[open source]] network simulators in research include NS-2, OMNet++ and GloMoSim (the later no longer under active development), which run on [[Linux]], [[FreeBSD]], [[SunOS]], [[Solaris Operating Environment|Solaris]], [[Microsoft Windows]] and other operating systems. NS-2 and GloMoSim are widely used in universities.

===Proprietary Simulators===

Commercial network simulators include [http://www.opnet.com OPNET], [http://www.shunra.com Shunra], [http://www.qualnet.com QualNet] (formerly [http://pcl.cs.ucla.edu/projects/glomosim/ GloMoSim]), [http://www.tetcos.com NetSim], [http://www.itheon-networks.com Itheon Network Emulator],

==Simulations==
Some network simulators require input scripts or commands (network parameters) and produce trace files. The network parameters describe the state of the network (node placement, existing links) and the events (data transmissions, link failures, etc). Trace files can document every event that occurred in the simulation and are used for analysis. Some simulators incorporate or provide visualization tools for the trace files. [http://www.isi.edu/nsnam/nam NAM], for example, displays how a network looks and animates the packet transmissions. Network simulators can also capture this type of data directly from a functioning production environment. This data capture may be done at various times of the day, week, month, in order to reflect average, worst-case, and best-case conditions. Network simulators can also provide other tools to facilitate visual analysis of trends and potential trouble spots.

==Other==
Network simulation (or ''network modeling'') is also used to describe a [[computer simulation]] of a network of pipes or cables used in [[pipeline transport]] (e.g. of [[natural gas]], oil, water, etc.) [[electric power transmission]] and [[electricity distribution]].

==External links==
*[http://www.isi.edu/nsnam/ns NS2 - Free, Open Source ]
*[http://www.omnetpp.org/ OMNet++]
*[http://www.shunra.com/ Shunra]
*[http://www.opnet.com/ OPNET]
*[http://www.qualnet.com/ QualNet], formerly [http://pcl.cs.ucla.edu/projects/glomosim/ GloMoSim]
*[http://www.tetcos.com/ NetSim]
*[http://www.itheon-networks.com/ Itheon Networks]

[[Category:Networks]]

{{compsci-stub}}

Network simulation

2007-03-23T22:07:45Z

200.55.139.25: /* Open Source Simulators */

In [[computer network]] research, '''network simulation''' is a technique where a program simulates the behavior of a network. The program performs this simulation either by calculating the interaction between the different network entities (hosts/[[router]]s, [[data link]]s, [[packet]]s, etc) using mathematical formulas, or actually capturing and playing back network parameters from a production network. Using this input, the behavior of the network and the various applications and services it supports can be observed in a test lab. Various attributes of the environment can also be modified in a controlled manner to asses these behaviors under different conditions. When a simulation program is used in conjunction with live applications and services in order to observe end-to-end performance to the user desktop, this technique is also referred to as [[network emulation]].

==Uses==
Network simulators are used to predict the behavior of networks and applications under different situations. Researchers use network simulators to see how their protocols would behave if deployed. It is typical to use a network simulator to test [[routing protocols]], MAC ([[Medium Access Control]]) protocols, [[Transport protocol|transport protocols]], applications etc. Companies use simulators to design their networks and/or applications to get a feel for how they will perform under current or projected real-world conditions.

==Simulators==
The simulator (or network simulator) is the program in charge of calculating how the network would behave. They may be distributed in source form (software) or provided in the form of a hardware appliance. Users can then customize the simulator to fulfill their specific analysis needs. The simulators come with support for the most popular protocols implemented (such as IPv4, IPv6, UDP, and TCP).

===Open Source Simulators===
Free/[[open source]] network simulators in research include NS-2, OMNet++ and GloMoSim (the later no longer under active development), which run on [[Linux]], [[FreeBSD]], [[SunOS]], [[Solaris Operating Environment|Solaris]], [[Microsoft Windows]] and other operating systems. NS-2 and GloMoSim are widely used in universities.

===Proprietary Simulators===

Commercial network simulators include [http://www.opnet.com OPNET], [http://www.shunra.com Shunra], [http://www.qualnet.com QualNet] (formerly [http://pcl.cs.ucla.edu/projects/glomosim/ GloMoSim]), [http://www.tetcos.com NetSim], [http://www.itheon-networks.com Itheon Network Emulator],

==Simulations==
Some network simulators require input scripts or commands (network parameters) and produce trace files. The network parameters describe the state of the network (node placement, existing links) and the events (data transmissions, link failures, etc). Trace files can document every event that occurred in the simulation and are used for analysis. Some simulators incorporate or provide visualization tools for the trace files. [http://www.isi.edu/nsnam/nam NAM], for example, displays how a network looks and animates the packet transmissions. Network simulators can also capture this type of data directly from a functioning production environment. This data capture may be done at various times of the day, week, month, in order to reflect average, worst-case, and best-case conditions. Network simulators can also provide other tools to facilitate visual analysis of trends and potential trouble spots.

==Other==
Network simulation (or ''network modeling'') is also used to describe a [[computer simulation]] of a network of pipes or cables used in [[pipeline transport]] (e.g. of [[natural gas]], oil, water, etc.) [[electric power transmission]] and [[electricity distribution]].

==External links==
*[http://www.isi.edu/nsnam/ns NS2 - Free, Open Source ]
*[http://www.shunra.com/ Shunra]
*[http://www.opnet.com/ OPNET]
*[http://www.qualnet.com/ QualNet], formerly [http://pcl.cs.ucla.edu/projects/glomosim/ GloMoSim]
*[http://www.tetcos.com/ NetSim]
*[http://www.itheon-networks.com/ Itheon Networks]

[[Category:Networks]]

{{compsci-stub}}

Wisin & Yandel

2007-02-07T20:03:22Z

200.55.139.25: /* Singles */

{{Infobox musical artist 2
|Name = [[Image:w&y.jpg|100px]]
|Img = Wisin_y_Yandel1.jpg
|Img_capt = (left to right) Yandel & Wisin in Premios lo Nuestro at 2006
|Img_size = 200px
|Background = group_or_band
|Origin = [[Puerto Rico]]
|Genre = [[Reggaeton]]
|Years_active = [[1998]] – present
|Label = [[WY Records]]
|URL =
|Current_members =
}}
'''Wisin & Yandel''' (also known as "''¡W con Yandel!''") are a [[Puerto Rican]] [[reggaeton]] duo. They started their career together in [[1998]] and have been together since winning several awards during that time.

Some of their hits includes ''[["Mayor Que Yo"]]'', ''[["Noche de Sexo"]]'' featuring ''[[Aventura]]'', ''"Rakata"'', ''[[Llame Pa' Verte]] (Bailando Sexy)'', [[La Gitana]]'' and ''"[[Pam Pam]]"''. Wisin & Yandel were also featured on [[R. Kelly]]'s album, [[TP-3: Reloaded]].

==Duo History==
Wisin and Yandel were both born in [[Cayey, Puerto Rico]] where they met. Before beginning their musical career, Yandel worked as a [[barber]] while Wisin earned a degree in theater and acting. However, both shared the same musical interests.

In [[1998]], they worked with [[DJ Penis]] in his production ''No Fear 3''. The next year, they were signed to the ''Fresh Production'' label, appearing in ''La Misión, Vol. 1'' (''The Mission, Vol. 1''). That album reached gold status, and the label produced the duo's first solo [[album]] in [[2000]] titled ''Los Reyes del Nuevo Milenio'' (''The Kings of the New Millennium'').

From then on, they've released four albums together, while making appearances in the ''Misión'' compilations. All of those albums have reached gold status. They also won a [[Tu Musica]] Award for "Best Rap & Reggaeton Duo".<ref>{{cite web|author=|date=|url=http://www.solosabor.com/artist/wisin_y_yandel/|title=Wisin y Yandel Biography|accessdate=}}</ref>

In [[2004]], each of the members of the duo decided to release a solo effort each. Wisin released his album titled ''El Sobreviviente'' (''The Survivor'') while Yandel released his effort titled ''¿Quien Contra Mí?'' (''Who's Against Me?''). Both efforts did well on sales, but the duo returned together to release ''Pa'l Mundo'' (''For The World'') on [[2005]]. ''El Sobreviviente'' is now a reggaeton [[collector's item]].
That same year, the duo established their own record label, ''[[WY Records]]''. In [[2006]], they released a compilation album titled ''Los Vaqueros'' (''The Cowboys'')featuring several guest performers.

They appeared on [[Luny Tunes]]'s new album [[Mas Flow: Los Benjamins]].

==Duo Biography==
* '''Wisin''' (also known as ''W'' or ''El sobrevivente'' ([[Spanish language|Spanish]]: ''El Sobreviviente'')) was born Juan Luis Morera Luna on [[September 19]], [[1978]].
* '''Yandel''' (occasionally known as ''Y'' ([[Spanish language|Spanish]]: ''I Griega'')) was born Llandel Veguilla Malavé on [[January 14]], [[1977]].

==Discography==
===As Wisin & Yandel===
* ''[[Los Reyes del Nuevo Milenio]]'' ([[2000]])
* ''[[De Nuevos a Viejos]]'' ([[2000]])
* ''[[De Otra Manera]]'' ([[2001]])
* ''[[Mi Vida... My Life]]'' ([[2003]])
* ''[[Pa'l Mundo]]'' ([[2005]])
* ''[[Pa'l Mundo, Deluxe Edition]]'' ([[2006]])
* ''[[Los Vaqueros]]'' ([[2006]])
* ''[[Los Vaqueros Special Edition]]
(around 11 years)

==Singles==
*''Dembow'' (2003)
*''Rakata'' (2005)
*''[[Llame Pa' Verte]]'' (2006)
*''Mayor Que Yo Parte 2'' (2006)
*''Noche De Sexo Ft. Aventura'' (2006)
*''Mírala Bien'' (2006)
*''[[Pam Pam]]'' (2006)
*''El Teléfono ft. [[Hector El Father|Hector "El Father"]]'' (2006)
*''Pegao'' (2006)
*''Yo Te Quiero'' (2006)
Electrica (2006)

Nadie como tu (feat Don Omar)
===Solo Efforts===
* ''Wisin: El Sobreviviente'' ([[2004]])
* ''Yandel: Quien Contra Mi'' ([[2004]])
*wisin y yandel:yo quiero

==Notes==
<div class="references-small"><references/></div>

==External link==
* [http://www.wisinyandelpr.com/ Official website]
* [http://wisinyyandel.blogspot.com/ Wisin y Yandel]



[[Category:Puerto Rican reggaeton artists]]
[[Category:Roman Catholic rappers]]

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{{musician-stub}}

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