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A '''Viterbi decoder''' uses the [[Viterbi algorithm]] for decoding a bitstream that has been
In [[mathematics]], the '''Iverson bracket''', named after [[Kenneth E. Iverson]]<!-- mentioned here so it's clear that it's an upper-case 'i' not an 'l'. -->, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,
encoded using a [[convolutional code]].
:<math>[P] = \begin{cases} 1 & \text{if } P \text{ is true;} \\ 0 & \text{otherwise.} \end{cases}</math>
where {{math|''P''}} is a [[statement (logic)|statement]] that can be true or false. This notation was introduced by [[Kenneth E. Iverson]] in his programming language [[APL (programming language)|APL]],<ref>[[Ronald Graham]], [[Donald Knuth]], and [[Oren Patashnik]]. ''[[Concrete Mathematics]]'', Section 2.2: Sums and Recurrences.</ref> while the specific restriction to square brackets was advocated by [[Donald Knuth]] to avoid ambiguity in parenthesized logical expressions.<ref>Knuth 1992.</ref>


There are other algorithms for decoding a convolutionally encoded stream (for example, the [[Robert Fano|Fano algorithm]]). The Viterbi algorithm is the most resource-consuming, but it does the [[maximum likelihood]] decoding. It is most often used for decoding convolutional codes with constraint lengths k<=10, but values up to k=15 are used in practice.
==Uses==
The Iverson bracket converts a [[Boolean data type|Boolean value]] to an integer value through the natural map <math>\textbf{false}\mapsto 0;  \textbf{true}\mapsto1</math>, which allows counting to be represented as summation. For instance, the [[Euler phi function]] that counts the number of integers less than ''n'' and [[coprime]] to it can be expressed by
: <math> \phi(n)=\sum_{i=0}^{n-1}[\gcd(i,n)=1],\qquad\text{for }n\in\mathbb N.</math>
More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example
: <math>\sum_{1\le i \le 10} i^2 = \sum_{i} i^2[1 \le i \le 10].</math>
In the first sum, the index <math>i</math> is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where ''i'' is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions.  
<!-- add an example of such a manipulation -->


Viterbi decoding was developed by [[Andrew J. Viterbi]] and published in the paper "Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm", [[IEEE Transactions on Information Theory]], Volume IT-13, pages 260-269, in April, 1967.
Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula
:<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)</math>


There are both hardware (in modems) and software implementations of a Viterbi decoder.
which is valid for {{math|''n'' &gt; 1}} but which is off by {{sfrac|1|2}} for {{math|''n'' {{=}} 1}}. To get an identity valid for all positive integers ''n'' (i.e., all values for which <math>\phi(n)</math> is defined), a correction term involving the Iverson bracket may be added:


== Hardware implementation ==
:<math>\sum_{1\le k\le n \atop \gcd(k,n)=1}\!\!k = \frac{1}{2}n(\varphi(n)+[n=1])</math>


[[Image:Viterbi decoder hardware implementation.png|right|thumb|350px|A common way to implement a hardware viterbi decoder]]
==Special cases==
The [[Kronecker delta]] notation is a specific case of Iverson notation when the condition is equality. That is,
: <math>\delta_{ij} = [i=j].\,</math>


A hardware Viterbi decoder for basic (not punctured) code usually consists of the following major blocks:
The [[indicator function]], another specific case, has set membership as its condition:
: <math>\mathbf{I}_A(x) = [x\in A].</math>


*Branch metric unit (BMU)
The [[sign function]] and [[Heaviside step function]] are also easily expressed in this notation:
*Path metric unit (PMU)
: <math> \sgn(x) = [x > 0] - [x < 0] \,</math>
*Traceback unit (TBU)
: <math> H(x) = [x > 0].</math>


=== Branch metric unit (BMU) ===
And the [[Trichotomy (mathematics)|trichotomy]] of the reals can be expressed:
:<math>[a < b] + [a = b] + [a > b] = 1. \, </math>


[[Image:Viterbi decoder hardware implementation BMU.png|right|thumb|350px|A sample implementation of a branch metric unit]]
== Notes ==
{{reflist}}


A branch metric unit's function is to calculate ''branch metrics'', which are normed distances between every possible symbol in the code alphabet, and the received symbol.
== References ==
*  Donald Knuth, "Two Notes on Notation", ''[[American Mathematical Monthly]]'', Volume 99, Number 5, May 1992, pp.&nbsp;403–422. ([http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz {{TeX}}], {{arxiv|math/9205211}})
* Kenneth E. Iverson, ''A Programming Language'', New York: Wiley, p.&nbsp;11, 1962.


There are hard decision and soft decision Viterbi decoders. A hard decision Viterbi decoder receives a simple bitstream on its input, and a [[Hamming distance]] is used as a metric. A soft decision Viterbi decoder receives a bitstream containing information about the ''reliability'' of each received symbol. For instance, in a 3-bit encoding, this ''reliability'' information is encoded as follows:
[[Category:Mathematical notation]]


{| border
[[de:Prädikatabbildung]]
| value
[[fr:Crochet_de_Iverson]]
| meaning
[[it:Parentesi di Iverson]]
|-
[[ja:アイバーソンの記法]]
| ''000''
[[pt:Colchetes de Iverson]]
| strongest '''0'''
[[ru:Нотация Айверсона]]
|-
[[sv:Iversonklammer]]
| ''001''
| relatively strong '''0'''
|-
| ''010''
| relatively weak '''0'''
|-
| ''011''
| weakest '''0'''
|-
| ''100''
| weakest '''1'''
|-
| ''101''
| relatively weak '''1'''
|-
| ''110''
| relatively strong '''1'''
|-
| ''111''
| strongest '''1'''
|}
 
Of course, it is not the only way to encode reliability data.
 
The ''squared'' [[Euclidean distance]] is used as a metric for soft decision decoders.
 
=== Path metric unit (PMU) ===
 
[[Image:Viterbi decoder hardware implementation PMU.png|right|thumb|350px|A sample implementation of a path metric unit for a specific K=4 decoder]]
 
A path metric unit summarizes branch metrics to get metrics for <math>2^{K-1}</math> paths, where K is the constraint length of the code, one of which can eventually be chosen as ''optimal''. Every clock it makes <math>2^{K-1}</math> decisions, throwing off wittingly nonoptimal paths. The results of these decisions are written to the memory of a traceback unit.
 
The core elements of a PMU are ''ACS (Add-Compare-Select)'' units. The way in which they are connected between themselves is defined by a specific code's [[trellis diagram]].
 
Since branch metrics are always <math>\ge 0</math>, there must be an additional circuit preventing metric counters from overflow (it isn't shown on the image).  An alternate method that eliminates the need to monitor the path metric growth is to allow the path metrics to "roll over", to use this method it is necessary to make sure the path metric accumulators contain enough bits to prevent the "best" and "worst" values from coming within 2<sup>(n-1)</sup> of each other.  The compare circuit is essentially unchanged. 
 
[[Image:Viterbi decoder hardware implementation ACS.png|right|thumb|350px|A sample implementation of an ACS unit]]
 
It is possible to monitor the noise level on the incoming bit stream by monitoring the rate of growth of the "best" path metric. A simpler way to do this is to monitor a single location or "state" and watch it pass "upward" through say four discrete levels within the range of the accumulator.  As it passes upward through each of these thresholds, a counter is incremented that reflects the "noise" present on the incoming signal.
 
=== Traceback unit (TBU) ===
 
[[Image:Viterbi decoder hardware implementation TBU.png|right|thumb|350px|A sample implementation of a traceback unit]]
 
Back-trace unit restores an (almost) maximum-likelihood path from the decisions made by PMU. Since it does it in inverse direction, a viterbi decoder comprises a FILO (first-in-last-out) buffer to reconstruct a correct order.
 
Note that the implementation shown on the image requires double frequency. There are some tricks that eliminate this requirement.
 
== Implementation issues ==
=== Quantization for soft decision decoding ===
In order to fully exploit benefits of soft decision decoding, one needs to quantize the input signal properly. The optimal quantization zone width is defined by the following formula:
 
<math>\,\! T = \sqrt{N_0/2^k},</math>
 
where <math>N_0</math> is a noise power spectral density, and ''k'' is a number of bits for soft decision.
 
=== Euclidean metric computation ===
 
The squared [[norm (mathematics)|norm]] (''<math>\ell</math><sub>2</sub>'') distance between the received and the actual symbols in the code alphabet may be further simplified into a linear sum/difference form, which makes it less computationally intensive.
 
Consider a 1/2 [[convolutional code]]r, which generates 2 bits (''00'', ''01'', ''10'' or ''11'') for every input bit (''1'' or ''0''). These ''Return-to-Zero'' signals are translated into a ''Non-Return-to-Zero'' form shown alongside.
 
{| border
| code alphabet
| vector mapping
|-
| ''00''
| ''1, 1''
|-
| ''01''
| ''1, -1''
|-
| ''10''
| ''-1, 1''
|-
| ''11''
| ''-1, -1''
|}
 
Each received symbol may be represented in vector form as '''v<sub>r</sub>''' = {r<sub>0</sub>, r<sub>1</sub>}, where r<sub>0</sub> and r<sub>1</sub> are soft decision values, whose magnitudes signify the ''joint reliability'' of the received vector, '''v<sub>r</sub>'''.
 
Every symbol in the code alphabet may, likewise, be represented by the vector '''v<sub>i</sub>''' = {±1, ±1}.
 
The actual computation of the Euclidean distance metric is:
 
<math>\,\!D = (\overrightarrow{v_r} - \overrightarrow{v_i})^2 = \overrightarrow{v_r}^2 - 2 \overrightarrow{v_r} \overrightarrow{v_i} + \overrightarrow{v_i}^2</math>
 
Each square term is a normed distance, depicting the ''energy'' of the symbol. For ex., the ''energy'' of the symbol '''v<sub>i</sub>''' = {±1, ±1} may be computed as
 
<math>\,\!\overrightarrow{v_i}^2 = (\pm 1)^2 + (\pm 1)^2 = 2</math>
 
Thus, the energy term of all symbols in the code alphabet is constant (at (''normalized'') value 2).
 
The ''Add-Compare-Select'' (''ACS'') operation compares the metric distance between the received symbol '''||v<sub>r</sub>||''' and any 2 symbols in the code alphabet whose paths merge at a node in the corresponding trellis, '''||v<sub>i</sub><sup>(0)</sup>||''' and '''||v<sub>i</sub><sup>(1)</sup>||'''. This is equivalent to comparing
 
<math>\,\!D_0 = \overrightarrow{v_r}^2 - 2 \overrightarrow{v_r} \overrightarrow{v_i^0} + \overrightarrow{v_i^0}^2</math>
 
and
 
<math>\,\!D_1 = \overrightarrow{v_r}^2 - 2 \overrightarrow{v_r} \overrightarrow{v_i^1} + \overrightarrow{v_i^1}^2</math>
 
But, from above we know that the ''energy'' of '''v<sub>i</sub>''' is constant (equal to (normalized) value of 2), and the ''energy'' of '''v<sub>r</sub>''' is the same in both cases. This reduces the comparison to a minima function between the 2 (middle) ''[[dot product]]'' terms,
 
<math>\,\!min(-2 \overrightarrow{v_r} \overrightarrow{v_i^0},-2 \overrightarrow{v_r} \overrightarrow{v_i^1}) = max(\overrightarrow{v_r} \overrightarrow{v_i^0}, \overrightarrow{v_r} \overrightarrow{v_i^1})</math>
 
since a ''min'' operation on negative numbers may be interpreted as an equivalent ''max'' operation on positive quantities.
 
Each ''[[dot product]]'' term may be expanded as
 
<math>\,\! max(\pm r_0 \pm r_1, \pm r_0 \pm r_1)</math>
 
where, the signs of each term depend on symbols, '''v<sub>i</sub><sup>(0)</sup>''' and '''v<sub>i</sub><sup>(1)</sup>''', being compared. Thus, the ''squared'' Euclidean metric distance calculation to compute the ''branch metric'' may be performed with a simple add/subtract operation.
 
=== Traceback ===
 
The general approach to traceback is to accumulate path metrics for up to five times the constraint length (''5 * (K - 1)''), find the node with the largest accumulated cost, and begin traceback from this node.
 
However, computing the node which has accumulated the largest cost (either the largest or smallest integral path metric) involves finding the ''maxima'' or ''minima'' of several (usually 2<sup>''K-1''</sup>) numbers, which may be time consuming when implemented on embedded hardware systems.
 
Most communication systems employ Viterbi decoding involving data packets of fixed sizes, with a fixed [[bit]]/[[byte]] pattern either at the beginning or/and at the end of the data packet. By using the known [[bit]]/[[byte]] pattern as reference, the start node may be set to a fixed value, thereby obtaining a perfect Maximum Likelihood Path during traceback.
 
== Limitations ==
 
A physical implementation of a viterbi decoder will not yield an ''exact'' maximum-likelihood stream due to [[Quantization (signal processing)|quantization]] of the input signal, branch and path metrics, and finite ''traceback length''. Practical implementations do approach within 1dB of the ideal.
 
== Punctured codes ==
 
A hardware viterbi decoder of ''[[Puncturing|punctured]] codes'' is commonly implemented in such a way:
 
* A depuncturer, which transforms the input stream into the stream which looks like an original (non punctured) stream with ERASE marks at the places where bits were erased.
* A basic viterbi decoder understanding these ERASE marks (that is, not using them for branch metric calculation).
 
== Software implementation ==
 
See [[Viterbi algorithm]].
 
One of the most time-consuming operations is an ACS butterfly, which is usually implemented using an [[assembly language]] and appropriate instruction set extensions (such as [[SSE2]]) to speed up the decoding time.
 
== Applications ==
 
The Viterbi decoding algorithm is widely used in the following areas:
*Radio communication: digital TV ([[ATSC]], [[QAM (television)|QAM]], [[DVB-T]], etc.), [[Radio relay link|radio relay]], [[satellite communications]], [[PSK31]] digital mode for [[amateur radio]].
*Decoding [[trellis-coded modulation]] (TCM), the technique used in telephone-line modems to squeeze high [[spectral efficiency]] out of 3&nbsp;kHz-bandwidth analog telephone lines.
*Computer storage devices such as [[hard disk drive]]s.
*[[Automatic speech recognition]]
 
==External links==
*[http://arxiv.org/abs/cs/0504020v2 David Forney's take on the history of the Viterbi algorithm]
*[http://home.netcom.com/~chip.f/viterbi/tutorial.html Details on Viterbi decoding, as well as a bibliography].
*[http://www.1-core.com/library/comm/viterbi/ Viterbi algorithm explanation with the focus on hardware implementation issues].
*[http://quest.nasa.gov/saturn/qa/cassini/Error_correction.txt r=1/6 k=15 coding for the Cassini mission to Saturn].
*[http://www.spiral.net/software/viterbi.html Online Generator of optimized software Viterbi decoders (GPL)].
*[http://www.ka9q.net/code/fec/ GPL Viterbi decoder software for four standard codes].
*[http://opencores.org/project,viterbi_decoder_axi4s  Generic Viterbi decoder hardware (GPL)].
 
[[Category:Data transmission]]
[[Category:Error detection and correction]]
 
[[ru:Алгоритм свёрточного декодирования Витерби]]

Revision as of 23:39, 13 August 2014

In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that denotes a number that is 1 if the condition in square brackets is satisfied, and 0 otherwise. More exactly,

[P]={1if P is true;0otherwise.

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Uses

The Iverson bracket converts a Boolean value to an integer value through the natural map false↦0;true↦1, which allows counting to be represented as summation. For instance, the Euler phi function that counts the number of integers less than n and coprime to it can be expressed by

ϕ(n)=∑i=0n−1[gcd⁡(i,n)=1],for n∈ℕ.

More generally the notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically. For example

∑1≤i≤10i2=∑ii2[1≤i≤10].

In the first sum, the index i is limited to be in the range 1 to 10. The second sum is allowed to range over all integers, but where i is strictly less than 1 or strictly greater than 10, the summand is 0, contributing nothing to the sum. Such use of the Iverson bracket can permit easier manipulation of these expressions.

Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula

∑1≤k≤ngcd⁡(k,n)=1k=12nφ(n)

which is valid for Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. To get an identity valid for all positive integers n (i.e., all values for which ϕ(n) is defined), a correction term involving the Iverson bracket may be added:

∑1≤k≤ngcd⁡(k,n)=1k=12n(φ(n)+[n=1])

Special cases

The Kronecker delta notation is a specific case of Iverson notation when the condition is equality. That is,

δij=[i=j].

The indicator function, another specific case, has set membership as its condition:

IA(x)=[x∈A].

The sign function and Heaviside step function are also easily expressed in this notation:

sgn⁡(x)=[x>0]−[x<0]
H(x)=[x>0].

And the trichotomy of the reals can be expressed:

[a<b]+[a=b]+[a>b]=1.

Notes

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References

de:Prädikatabbildung fr:Crochet_de_Iverson it:Parentesi di Iverson ja:アイバーソンの記法 pt:Colchetes de Iverson ru:Нотация Айверсона sv:Iversonklammer

  1. ↑ Ronald Graham, Donald Knuth, and Oren Patashnik. Concrete Mathematics, Section 2.2: Sums and Recurrences.
  2. ↑ Knuth 1992.
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