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Baseband

In telecommunications and signal processing, baseband is the range of frequencies occupied by a signal that has not been modulated to higher frequencies. Baseband signals typically originate from transducers, converting some other variable into an electrical signal. For example, the electronic output of a microphone is a baseband signal that is analogous to the applied voice audio. In conventional analog radio broadcasting, the baseband audio signal is used to modulate an RF carrier signal of a much higher frequency.

A baseband signal may have frequency components going all the way down to the DC bias, or at least it will have a high ratio bandwidth. A modulated baseband signal is called a passband signal. This occupies a higher range of frequencies and has a lower ratio and fractional bandwidth.

Baseband signal
A baseband signal or lowpass signal is a signal that can include frequencies that are very near zero, by comparison with its highest frequency (for example, a sound waveform can be considered as a baseband signal, whereas a radio signal or any other modulated signal is not).

A baseband bandwidth is equal to the highest frequency of a signal or system, or an upper bound on such frequencies, for example the upper cut-off frequency of a low-pass filter. By contrast, passband bandwidth is the difference between a highest frequency and a nonzero lowest frequency.

Baseband channel
A baseband channel or lowpass channel (or system, or network) is a communication channel that can transfer frequencies that are very near zero. Examples are serial cables and local area networks (LANs), as opposed to passband channels such as radio frequency channels and passband filtered wires of the analog telephone network. Frequency division multiplexing (FDM) allows an analog telephone wire to carry a baseband telephone call, concurrently as one or several carrier-modulated telephone calls.

Digital baseband transmission
Digital baseband transmission, also known as line coding, aims at transferring a digital bit stream over baseband channel, typically an unfiltered wire, contrary to passband transmission, also known as carrier-modulated transmission. Passband transmission makes communication possible over a bandpass filtered channel, such as the telephone network local-loop or a band-limited wireless channel.

Baseband transmission in Ethernet
The word "BASE" in Ethernet physical layer standards, for example 10BASE5, 100BASE-TX and 1000BASE-SX, implies baseband digital transmission (i.e. that a line code and an unfiltered wire are used).

Baseband processor
A baseband processor also known as BP or BBP is used to process the down-converted digital signal to retrieve essential data for a wireless digital system. The baseband processing block in GNSS receivers is responsible for providing observable data: that is, code pseudo-ranges and carrier phase measurements, as well as navigation data.

Equivalent baseband signal

An equivalent baseband signal or equivalent lowpass signal is a complex valued representation of the modulated physical signal (the so-called passband signal or RF signal). It is a concept within analog and digital modulation methods for (passband) signals with constant or varying carrier frequency (for example ASK, PSK QAM, and FSK). The equivalent baseband signal is $$Z(t)=I(t)+jQ(t)\,$$ where $$I(t)$$ is the inphase signal, $$Q(t)$$ the quadrature phase signal, and $$j$$ the imaginary unit. This signal is sometimes called IQ data. In a digital modulation method, the $$I(t)$$ and $$Q(t)$$ signals of each modulation symbol are evident from the constellation diagram. The frequency spectrum of this signal includes negative as well as positive frequencies. The physical passband signal corresponds to
 * $$I(t)\cos(\omega t) - Q(t)\sin(\omega t) = \mathrm{Re}\{Z(t)e^{j\omega t}\}\,$$

where $$\omega$$ is the carrier angular frequency in rad/s.

Modulation
A signal at baseband is often used to modulate a higher frequency carrier signal in order that it may be transmitted via radio. Modulation results in shifting the signal up to much higher frequencies (radio frequencies, or RF) than it originally spanned. A key consequence of the usual double-sideband amplitude modulation (AM) is that the range of frequencies the signal spans (its spectral bandwidth) is doubled. Thus, the RF bandwidth of a signal (measured from the lowest frequency as opposed to 0 Hz) is twice its baseband bandwidth. Steps may be taken to reduce this effect, such as single-sideband modulation. Conversely, some transmission schemes such as frequency modulation use even more bandwidth.

The figure below shows AM modulation:

title: Baseband


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Basic exchange telecommunications radio service

In telecommunication, a basic exchange telecommunications radio service (BETRS) is a commercial service that can extend telephone service to rural areas by replacing the local loop with radio communications. In the BETRS, non-government ultra high frequency (UHF) and very high frequency (VHF) common carrier and the private radio service frequencies are shared.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Basic service element

In telecommunication, a basic service element (BSE) is:

 * 1) An optional unbundled feature, generally associated with the basic serving arrangement (BSA), that an enhanced-service provider (ESP) may require or find useful in configuring an enhanced service.
 * 2) A fundamental (basic) communication network service; an optional network capability associated with a BSA.

BSEs constitute optional capabilities to which the customer may subscribe or decline to subscribe.

title: Basic service element


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Basic serving arrangement

In telecommunication, the term basic serving arrangement (BSA) has the following meanings:

 * 1)  The fundamental tariffed switching and transmission (and other) services that an operating company must provide to an enhanced service provider (ESP) to connect with its customers through the company network.
 * 2) In an open-network-architecture context, the fundamental underlying connection of an enhanced service provider (ESP) to and through the operating company's network including an ESP access link, the features and functions associated with that access link at the central office serving the ESP and/or other offices, and the transport (dedicated or switched) within the network that completes the connection from the ESP to the central office serving its customers or to capabilities associated with the customer's complementary network services. Each component may have a number of categories of network characteristics. Within these categories of network characteristics are alternatives from among which the customer must choose. Examples of BSA components are ESP access link, transport and/or usage.

title: Basic serving arrangement


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BCH code

In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called a Galois field). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D.K. Ray-Chaudhuri. The name Bose–Chaudhuri–Hocquenghem (and the acronym BCH) arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri).

One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an algebraic method known as syndrome decoding. This simplifies the design of the decoder for these codes, using small low-power electronic hardware.

BCH codes are used in applications such as satellite communications, compact disc players, DVDs, disk drives, USB flash drives, solid-state drives, and two-dimensional bar codes.

Primitive narrow-sense BCH codes
Given a prime number $q$ and prime power $q^{m}$ with positive integers $m$ and $d$ such that $d ≤ q^{m} − 1$, a primitive narrow-sense BCH code over the finite field (or Galois field) $GF(q)$ with code length $n = q^{m} − 1$ and distance at least $d$ is constructed by the following method.

Let $α$ be a primitive element of $GF(q^{m})$. For any positive integer $i$, let $m_{i}(x)$ be the minimal polynomial with coefficients in $GF(q)$ of $α^{i}$. The generator polynomial of the BCH code is defined as the least common multiple $g(x) = lcm(m_{1}(x),…,m_{d − 1}(x))$. It can be seen that $g(x)$ is a polynomial with coefficients in $GF(q)$ and divides $x^{n} − 1$. Therefore, the polynomial code defined by $g(x)$ is a cyclic code.

Example
Let $q = 2$ and $m = 4$ (therefore $n = 15$). We will consider different values of $d$ for $GF(16) = GF(2^{4})$ based on the reducing polynomial $z^{4} + z + 1$, using primitive element $α(z) = z$. There are fourteen minimum polynomials $m_{i}(x)$ with coefficients in $GF(2)$ satisfying
 * $$m_i\left(\alpha^i\right) \bmod \left(z^4 + z + 1\right) = 0.$$

The minimal polynomials are
 * $$\begin{align}

m_1(x) &= m_2(x) = m_4(x) = m_8(x) = x^4 + x + 1, \\ m_3(x) &= m_6(x) = m_9(x) = m_{12}(x) = x^4 + x^3 + x^2 + x + 1, \\ m_5(x) &= m_{10}(x) = x^2 + x + 1, \\ m_7(x) &= m_{11}(x) = m_{13}(x) = m_{14}(x) = x^4 + x^3 + 1. \end{align}$$

The BCH code with $$d = 2, 3$$ has the generator polynomial
 * $$g(x) = {\rm lcm}(m_1(x), m_2(x)) = m_1(x) = x^4 + x + 1.\,$$

It has minimal Hamming distance at least 3 and corrects up to one error. Since the generator polynomial is of degree 4, this code has 11 data bits and 4 checksum bits. It is also denoted as: (15, 11) BCH code.

The BCH code with $$d=4,5$$ has the generator polynomial
 * $$\begin{align}

g(x) &= {\rm lcm}(m_1(x),m_2(x),m_3(x),m_4(x)) = m_1(x) m_3(x) \\ &= \left(x^4 + x + 1\right)\left(x^4 + x^3 + x^2 + x + 1\right) = x^8 + x^7 + x^6 + x^4 + 1. \end{align}$$

It has minimal Hamming distance at least 5 and corrects up to two errors. Since the generator polynomial is of degree 8, this code has 7 data bits and 8 checksum bits. It is also denoted as: (15, 7) BCH code.

The BCH code with $$d=6,7$$ has the generator polynomial
 * $$\begin{align}

g(x) &= {\rm lcm}(m_1(x),m_2(x),m_3(x),m_4(x),m_5(x),m_6(x)) = m_1(x) m_3(x) m_5(x) \\ &= \left(x^4 + x + 1\right)\left(x^4 + x^3 + x^2 + x + 1\right)\left(x^2 + x + 1\right) = x^{10} + x^8 + x^5 + x^4 + x^2 + x + 1. \end{align}$$

It has minimal Hamming distance at least 7 and corrects up to three errors. Since the generator polynomial is of degree 10, this code has 5 data bits and 10 checksum bits. It is also denoted as: (15, 5) BCH code. (This particular generator polynomial has a real-world application, in the "format information" of the QR code.)

The BCH code with $$d=8$$ and higher has the generator polynomial
 * $$\begin{align}

g(x) &= {\rm lcm}(m_1(x),m_2(x),...,m_{14}(x)) = m_1(x) m_3(x) m_5(x) m_7(x)\\ &= \left(x^4 + x + 1\right)\left(x^4 + x^3 + x^2 + x + 1\right)\left(x^2 + x + 1\right)\left(x^4 + x^3 + 1\right) = x^{14} + x^{13} + x^{12} + \cdots + x^2 + x + 1. \end{align}$$

This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: (15, 1) BCH code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial repetition code).

General BCH codes
General BCH codes differ from primitive narrow-sense BCH codes in two respects.

First, the requirement that $$\alpha$$ be a primitive element of $$\mathrm{GF}(q^m)$$ can be relaxed. By relaxing this requirement, the code length changes from $$q^m - 1$$ to $$\mathrm{ord}(\alpha),$$ the order of the element $$\alpha.$$

Second, the consecutive roots of the generator polynomial may run from $$\alpha^c,\ldots,\alpha^{c+d-2}$$ instead of $$\alpha,\ldots,\alpha^{d-1}.$$

Definition. Fix a finite field $$GF(q),$$ where $$q$$ is a prime power. Choose positive integers $$m,n,d,c$$ such that $$2\leq d\leq n,$$ $${\rm gcd}(n,q)=1,$$ and $$m$$ is the multiplicative order of $$q$$ modulo $$n.$$

As before, let $$\alpha$$ be a primitive $n$th root of unity in $$GF(q^m),$$ and let $$m_i(x)$$ be the minimal polynomial over $$GF(q)$$ of $$\alpha^i$$ for all $$i.$$ The generator polynomial of the BCH code is defined as the least common multiple $$g(x) = {\rm lcm}(m_c(x),\ldots,m_{c+d-2}(x)).$$

Note: if $$n=q^m-1$$ as in the simplified definition, then $${\rm gcd}(n,q)$$ is 1, and the order of $$q$$ modulo $$n$$ is $$m.$$ Therefore, the simplified definition is indeed a special case of the general one.

Special cases

 * A BCH code with $$c=1$$ is called a narrow-sense BCH code.
 * A BCH code with $$n=q^m-1$$ is called primitive.

The generator polynomial $$g(x)$$ of a BCH code has coefficients from $$\mathrm{GF}(q).$$ In general, a cyclic code over $$\mathrm{GF}(q^p)$$ with $$g(x)$$ as the generator polynomial is called a BCH code over $$\mathrm{GF}(q^p).$$ The BCH code over $$\mathrm{GF}(q^m)$$ and generator polynomial $$g(x)$$ with successive powers of $$\alpha$$ as roots is one type of Reed–Solomon code where the decoder (syndromes) alphabet is the same as the channel (data and generator polynomial) alphabet, all elements of $$\mathrm{GF}(q^m)$$. The other type of Reed Solomon code is an original view Reed Solomon code which is not a BCH code.

Properties
The generator polynomial of a BCH code has degree at most $$(d-1)m$$. Moreover, if $$q=2$$ and $$c=1$$, the generator polynomial has degree at most $$dm/2$$.

Each minimal polynomial $$m_i(x)$$ has degree at most $$m$$. Therefore, the least common multiple of $$d-1$$ of them has degree at most $$(d-1)m$$. Moreover, if $$q=2,$$ then $$m_i(x) = m_{2i}(x)$$ for all $$i$$. Therefore, $$g(x)$$ is the least common multiple of at most $$d/2$$ minimal polynomials $$m_i(x)$$ for odd indices $$i,$$ each of degree at most $$m$$.

A BCH code has minimal Hamming distance at least $$d$$.

Suppose that $$p(x)$$ is a code word with fewer than $$d$$ non-zero terms. Then

 * $$p(x) = b_1x^{k_1} + \cdots + b_{d-1}x^{k_{d-1}},\text{ where }k_1<k_2<\cdots<k_{d-1}.$$

Recall that $$\alpha^c,\ldots,\alpha^{c+d-2}$$ are roots of $$g(x),$$ hence of $$p(x)$$. This implies that $$b_1,\ldots,b_{d-1}$$ satisfy the following equations, for each $$i \in \{c, \dotsc, c+d-2\}$$:

 * $$p(\alpha^i) = b_1\alpha^{ik_1} + b_2\alpha^{ik_2} + \cdots + b_{d-1}\alpha^{ik_{d-1}} = 0.$$

In matrix form, we have
 * $$\begin{bmatrix}

\alpha^{ck_1} &      \alpha^{ck_2} & \cdots &       \alpha^{ck_{d-1}} \\ \alpha^{(c+1)k_1} &  \alpha^{(c+1)k_2} & \cdots &   \alpha^{(c+1)k_{d-1}} \\ \vdots &             \vdots &        &                  \vdots \\ \alpha^{(c+d-2)k_1} & \alpha^{(c+d-2)k_2} & \cdots & \alpha^{(c+d-2)k_{d-1}} \\ \end{bmatrix}\begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_{d-1} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}. $$

The determinant of this matrix equals
 * $$\left(\prod_{i=1}^{d-1}\alpha^{ck_i}\right)\det\begin{pmatrix}

1 &                1 & \cdots &                     1 \\ \alpha^{k_1} &     \alpha^{k_2} & \cdots &      \alpha^{k_{d-1}} \\ \vdots &           \vdots &        &                \vdots \\ \alpha^{(d-2)k_1} & \alpha^{(d-2)k_2} & \cdots & \alpha^{(d-2)k_{d-1}} \\ \end{pmatrix} = \left(\prod_{i=1}^{d-1}\alpha^{ck_i}\right) \det(V).$$

The matrix $$V$$ is seen to be a Vandermonde matrix, and its determinant is
 * $$\det(V) = \prod_{1\le i<j\le d-1} \left(\alpha^{k_j} - \alpha^{k_i}\right),$$

which is non-zero. It therefore follows that $$b_1,\ldots,b_{d-1}=0,$$ hence $$p(x) = 0.$$

A BCH code is cyclic.

A polynomial code of length $$n$$ is cyclic if and only if its generator polynomial divides $$x^n-1.$$ Since $$g(x)$$ is the minimal polynomial with roots $$\alpha^c,\ldots,\alpha^{c+d-2},$$ it suffices to check that each of $$\alpha^c,\ldots,\alpha^{c+d-2}$$ is a root of $$x^n-1.$$ This follows immediately from the fact that $$\alpha$$ is, by definition, an $$n$$th root of unity.

Encoding
Because any polynomial that is a multiple of the generator polynomial is a valid BCH codeword, BCH encoding is merely the process of finding some polynomial that has the generator as a factor.

The BCH code itself is not prescriptive about the meaning of the coefficients of the polynomial; conceptually, a BCH decoding algorithm's sole concern is to find the valid codeword with the minimal Hamming distance to the received codeword. Therefore, the BCH code may be implemented either as a systematic code or not, depending on how the implementor chooses to embed the message in the encoded polynomial.

Non-systematic encoding: The message as a factor
The most straightforward way to find a polynomial that is a multiple of the generator is to compute the product of some arbitrary polynomial and the generator. In this case, the arbitrary polynomial can be chosen using the symbols of the message as coefficients.

 * $$s(x) = p(x)g(x)$$

As an example, consider the generator polynomial $$g(x)=x^{10}+x^9+x^8+x^6+x^5+x^3+1$$, chosen for use in the (31, 21) binary BCH code used by POCSAG and others. To encode the 21-bit message {101101110111101111101}, we first represent it as a polynomial over $$GF(2)$$:
 * $$p(x) = x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^3+x^2+1$$

Then, compute (also over $$GF(2)$$):
 * $$\begin{align}

s(x) &= p(x)g(x)\\ &= \left(x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^3+x^2+1\right)\left(x^{10}+x^9+x^8+x^6+x^5+x^3+1\right)\\ &= x^{30}+x^{29}+x^{26}+x^{25}+x^{24}+x^{22}+x^{19}+x^{17}+x^{16}+x^{15}+x^{14}+x^{12}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^2+1 \end{align}$$ Thus, the transmitted codeword is {1100111010010111101011101110101}.

The receiver can use these bits as coefficients in $$s(x)$$ and, after error-correction to ensure a valid codeword, can recompute $$p(x) = s(x)/g(x)$$

Systematic encoding: The message as a prefix
A systematic code is one in which the message appears verbatim somewhere within the codeword. Therefore, systematic BCH encoding involves first embedding the message polynomial within the codeword polynomial, and then adjusting the coefficients of the remaining (non-message) terms to ensure that $$s(x)$$ is divisible by $$g(x)$$.

This encoding method leverages the fact that subtracting the remainder from a dividend results in a multiple of the divisor. Hence, if we take our message polynomial $$p(x)$$ as before and multiply it by $$x^{n-k}$$ (to "shift" the message out of the way of the remainder), we can then use Euclidean division of polynomials to yield:
 * $$p(x)x^{n-k} = q(x)g(x) + r(x)$$

Here, we see that $$q(x)g(x)$$ is a valid codeword. As $$r(x)$$ is always of degree less than $$n-k$$ (which is the degree of $$g(x)$$), we can safely subtract it from $$p(x)x^{n-k}$$ without altering any of the message coefficients, hence we have our $$s(x)$$ as
 * $$s(x) = q(x)g(x) = p(x)x^{n-k} - r(x)$$

Over $$GF(2)$$ (i.e. with binary BCH codes), this process is indistinguishable from appending a cyclic redundancy check, and if a systematic binary BCH code is used only for error-detection purposes, we see that BCH codes are just a generalization of the mathematics of cyclic redundancy checks.

The advantage to the systematic coding is that the receiver can recover the original message by discarding everything after the first $$k$$ coefficients, after performing error correction.

Decoding
There are many algorithms for decoding BCH codes. The most common ones follow this general outline:
 * 1) Calculate the syndromes sj for the received vector
 * 2) Determine the number of errors t and the error locator polynomial Λ(x) from the syndromes
 * 3) Calculate the roots of the error location polynomial to find the error locations Xi
 * 4) Calculate the error values Yi at those error locations
 * 5) Correct the errors

During some of these steps, the decoding algorithm may determine that the received vector has too many errors and cannot be corrected. For example, if an appropriate value of t is not found, then the correction would fail. In a truncated (not primitive) code, an error location may be out of range. If the received vector has more errors than the code can correct, the decoder may unknowingly produce an apparently valid message that is not the one that was sent.

Calculate the syndromes
The received vector $$R$$ is the sum of the correct codeword $$C$$ and an unknown error vector $$E.$$ The syndrome values are formed by considering $$R$$ as a polynomial and evaluating it at $$\alpha^c, \ldots, \alpha^{c+d-2}.$$ Thus the syndromes are
 * $$s_j = R\left(\alpha^j\right) = C\left(\alpha^j\right) + E\left(\alpha^j\right)$$

for $$j = c$$ to $$c + d - 2.$$

Since $$\alpha^{j}$$ are the zeros of $$g(x),$$ of which $$C(x)$$ is a multiple, $$C\left(\alpha^j\right) = 0.$$ Examining the syndrome values thus isolates the error vector so one can begin to solve for it.

If there is no error, $$s_j = 0$$ for all $$j.$$ If the syndromes are all zero, then the decoding is done.

Calculate the error location polynomial
If there are nonzero syndromes, then there are errors. The decoder needs to figure out how many errors and the location of those errors.

If there is a single error, write this as $$E(x) = e\,x^i,$$ where $$i$$ is the location of the error and $$e$$ is its magnitude. Then the first two syndromes are
 * $$\begin{align}

s_c &= e\,\alpha^{c\,i} \\ s_{c+1} &= e\,\alpha^{(c+1)\,i} = \alpha^i s_c \end{align}$$ so together they allow us to calculate $$e$$ and provide some information about $$i$$ (completely determining it in the case of Reed–Solomon codes).

If there are two or more errors,
 * $$E(x) = e_1 x^{i_1} + e_2 x^{i_2} + \cdots \, $$

It is not immediately obvious how to begin solving the resulting syndromes for the unknowns $$e_k$$ and $$i_k.$$

The first step is finding, compatible with computed syndromes and with minimal possible $$t,$$ locator polynomial:
 * $$\Lambda(x) = \prod_{j=1}^t \left(x\alpha^{i_j} - 1\right)$$

Three popular algorithms for this task are:
 * 1) Peterson–Gorenstein–Zierler algorithm
 * 2) Berlekamp–Massey algorithm
 * 3) Sugiyama Euclidean algorithm

Peterson–Gorenstein–Zierler algorithm
Peterson's algorithm is the step 2 of the generalized BCH decoding procedure. Peterson's algorithm is used to calculate the error locator polynomial coefficients $$ \lambda_1, \lambda_2, \dots, \lambda_{v} $$ of a polynomial

 * $$ \Lambda(x) = 1 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_v x^v .$$

Now the procedure of the Peterson–Gorenstein–Zierler algorithm. Expect we have at least 2t syndromes sc, …, sc+2t−1. Let v = t. 1. Start by generating the $S_{v\times v}$ matrix with elements that are syndrome values
 * $S_{v \times v}=\begin{bmatrix}s_c&s_{c+1}&\dots&s_{c+v-1}\\

s_{c+1}&s_{c+2}&\dots&s_{c+v}\\ \vdots&\vdots&\ddots&\vdots\\ s_{c+v-1}&s_{c+v}&\dots&s_{c+2v-2}\end{bmatrix}. $

2. Generate a $c_{v \times 1}$ vector with elements
 * $C_{v \times 1}=\begin{bmatrix}s_{c+v}\\

s_{c+v+1}\\ \vdots\\ s_{c+2v-1}\end{bmatrix}. $

3. Let $\Lambda$ denote the unknown polynomial coefficients, which are given by
 * $\Lambda_{v \times 1} = \begin{bmatrix}\lambda_{v}\\

\lambda_{v-1}\\ \vdots\\ \lambda_{1}\end{bmatrix}. $

4. Form the matrix equation
 * $S_{v \times v} \Lambda_{v \times 1} = -C_{v \times 1\,} .$

5. If the determinant of matrix $S_{v \times v}$ is nonzero, then we can actually find an inverse of this matrix and solve for the values of unknown $\Lambda$ values.

6. If $\det\left(S_{v \times v}\right) = 0,$ then follow if $v = 0$ then declare an empty error locator polynomial stop Peterson procedure. end set $v \leftarrow v -1$ continue from the beginning of Peterson's decoding by making smaller $S_{v \times v}$

7. After you have values of $\Lambda$, you have the error locator polynomial.

8. Stop Peterson procedure.

Factor error locator polynomial
Now that you have the $$\Lambda(x)$$ polynomial, its roots can be found in the form $$\Lambda(x) = \left(\alpha^{i_1} x - 1\right)\left(\alpha^{i_2} x - 1\right) \cdots \left(\alpha^{i_v} x - 1\right)$$ by brute force for example using the Chien search algorithm. The exponential powers of the primitive element $$\alpha$$ will yield the positions where errors occur in the received word; hence the name 'error locator' polynomial.

The zeros of Λ(x) are α−i1, …, α−iv.

Calculate error values
Once the error locations are known, the next step is to determine the error values at those locations. The error values are then used to correct the received values at those locations to recover the original codeword.

For the case of binary BCH, (with all characters readable) this is trivial; just flip the bits for the received word at these positions, and we have the corrected code word. In the more general case, the error weights $$e_j$$ can be determined by solving the linear system
 * $$\begin{align}

s_c & = e_1 \alpha^{c\,i_1} + e_2 \alpha^{c\,i_2} + \cdots \\ s_{c+1} & = e_1 \alpha^{(c + 1)\,i_1} + e_2 \alpha^{(c + 1)\,i_2} + \cdots \\ & {}\ \vdots \end{align}$$

Forney algorithm
However, there is a more efficient method known as the Forney algorithm.

Let
 * $$S(x) = s_c + s_{c+1}x + s_{c+2}x^2 + \cdots + s_{c+d-2}x^{d-2}.$$
 * $$v \leqslant d-1, \lambda_0 \neq 0 \qquad \Lambda(x) = \sum_{i=0}^v\lambda_i x^i = \lambda_0 \prod_{k=0}^{v} \left(\alpha^{-i_k}x - 1\right).$$

And the error evaluator polynomial
 * $$\Omega(x) \equiv S(x) \Lambda(x) \bmod{x^{d-1}}$$

Finally:
 * $$\Lambda'(x) = \sum_{i=1}^v i \cdot \lambda_i x^{i-1},$$

where
 * $$i \cdot x := \sum_{k=1}^i x.$$

Than if syndromes could be explained by an error word, which could be nonzero only on positions $$i_k$$, then error values are
 * $$e_k = -{\alpha^{i_k}\Omega\left(\alpha^{-i_k}\right) \over \alpha^{c\cdot i_k}\Lambda'\left(\alpha^{-i_k}\right)}.$$

For narrow-sense BCH codes, c = 1, so the expression simplifies to:
 * $$e_k = -{\Omega\left(\alpha^{-i_k}\right) \over \Lambda'\left(\alpha^{-i_k}\right)}.$$

Explanation of Forney algorithm computation
It is based on Lagrange interpolation and techniques of generating functions.

Consider $$S(x)\Lambda(x),$$ and for the sake of simplicity suppose $$\lambda_k = 0$$ for $$k > v,$$ and $$s_k = 0$$ for $$k > c + d - 2.$$ Then

 * $$S(x)\Lambda(x) = \sum_{j=0}^{\infty}\sum_{i=0}^j s_{j-i+1}\lambda_i x^j.$$

 * $$\begin{align}

S(x)\Lambda(x) &= S(x) \left \{ \lambda_0\prod_{\ell=1}^v \left (\alpha^{i_\ell}x-1 \right ) \right \} \\ &= \left \{ \sum_{i=0}^{d-2}\sum_{j=1}^v e_j\alpha^{(c+i)\cdot i_j} x^i \right \} \left \{ \lambda_0\prod_{\ell=1}^v \left (\alpha^{i_\ell}x-1 \right ) \right \} \\ &= \left \{ \sum_{j=1}^v e_j \alpha^{c i_j}\sum_{i=0}^{d-2} \left (\alpha^{i_j} \right )^i x^i \right \} \left \{ \lambda_0\prod_{\ell=1}^v \left (\alpha^{i_\ell}x-1 \right ) \right \} \\ &= \left \{ \sum_{j=1}^v e_j \alpha^{c i_j} \frac{\left (x \alpha^{i_j} \right )^{d-1}-1}{x \alpha^{i_j}-1} \right \} \left \{ \lambda_0 \prod_{\ell=1}^v \left (\alpha^{i_\ell}x-1 \right ) \right \} \\ &= \lambda_0 \sum_{j=1}^v e_j\alpha^{c i_j} \frac{ \left (x\alpha^{i_j} \right)^{d-1}-1}{x\alpha^{i_j}-1} \prod_{\ell=1}^v \left (\alpha^{i_\ell}x-1 \right ) \\ &= \lambda_0 \sum_{j=1}^v e_j\alpha^{c i_j} \left ( \left (x\alpha^{i_j} \right)^{d-1}-1 \right ) \prod_{\ell\in\{1,\cdots,v\}\setminus\{j\}} \left (\alpha^{i_\ell}x-1 \right ) \end{align}$$

We want to compute unknowns $$e_j,$$ and we could simplify the context by removing the $$\left(x\alpha^{i_j}\right)^{d-1}$$ terms. This leads to the error evaluator polynomial
 * $$\Omega(x) \equiv S(x) \Lambda(x) \bmod{x^{d-1}}.$$

Thanks to $$v\leqslant d-1$$ we have
 * $$\Omega(x) = -\lambda_0\sum_{j=1}^v e_j\alpha^{c i_j} \prod_{\ell\in\{1,\cdots,v\}\setminus\{j\}} \left(\alpha^{i_\ell}x - 1\right).$$

Thanks to $$\Lambda$$ (the Lagrange interpolation trick) the sum degenerates to only one summand for $$x = \alpha^{-i_k}$$

 * $$\Omega \left(\alpha^{-i_k}\right) = -\lambda_0 e_k\alpha^{c\cdot i_k}\prod_{\ell\in\{1,\cdots,v\}\setminus\{k\}} \left(\alpha^{i_\ell}\alpha^{-i_k} - 1\right).$$

To get $$e_k$$ we just should get rid of the product. We could compute the product directly from already computed roots $$\alpha^{-i_j}$$ of $$\Lambda,$$ but we could use simpler form.

As formal derivative
 * $$\Lambda'(x) = \lambda_0\sum_{j=1}^v \alpha^{i_j}\prod_{\ell\in\{1,\cdots,v\}\setminus\{j\}} \left(\alpha^{i_\ell}x - 1\right),$$

we get again only one summand in
 * $$\Lambda'\left(\alpha^{-i_k}\right) = \lambda_0\alpha^{i_k}\prod_{\ell\in\{1,\cdots,v\}\setminus\{k\}} \left(\alpha^{i_\ell}\alpha^{-i_k} - 1\right).$$

So finally
 * $$e_k = -\frac{\alpha^{i_k}\Omega \left(\alpha^{-i_k}\right)}{\alpha^{c\cdot i_k}\Lambda' \left(\alpha^{-i_k}\right)}.$$

This formula is advantageous when one computes the formal derivative of $$\Lambda$$ form
 * $$\Lambda(x) = \sum_{i=1}^v \lambda_i x^i$$

yielding:
 * $$\Lambda'(x) = \sum_{i=1}^v i \cdot \lambda_i x^{i-1},$$

where
 * $$i\cdot x := \sum_{k=1}^i x.$$

Decoding based on extended Euclidean algorithm
An alternate process of finding both the polynomial Λ and the error locator polynomial is based on Yasuo Sugiyama's adaptation of the Extended Euclidean algorithm. Correction of unreadable characters could be incorporated to the algorithm easily as well.

Let $$k_1, ..., k_k$$ be positions of unreadable characters. One creates polynomial localising these positions $$\Gamma(x) = \prod_{i=1}^k\left(x\alpha^{k_i} - 1\right).$$ Set values on unreadable positions to 0 and compute the syndromes.

As we have already defined for the Forney formula let $$S(x)=\sum_{i=0}^{d-2}s_{c+i}x^i.$$

Let us run extended Euclidean algorithm for locating least common divisor of polynomials $$S(x)\Gamma(x)$$ and $$x^{d-1}.$$ The goal is not to find the least common divisor, but a polynomial $$r(x)$$ of degree at most $$\lfloor (d+k-3)/2\rfloor$$ and polynomials $$a(x), b(x)$$ such that $$r(x)=a(x)S(x)\Gamma(x)+b(x)x^{d-1}.$$ Low degree of $$r(x)$$ guarantees, that $$a(x)$$ would satisfy extended (by $$\Gamma$$) defining conditions for $$\Lambda.$$

Defining $$\Xi(x)=a(x)\Gamma(x)$$ and using $$\Xi$$ on the place of $$\Lambda(x)$$ in the Fourney formula will give us error values.

The main advantage of the algorithm is that it meanwhile computes $$\Omega(x)=S(x)\Xi(x)\bmod x^{d-1}=r(x)$$ required in the Forney formula.

Explanation of the decoding process
The goal is to find a codeword which differs from the received word minimally as possible on readable positions. When expressing the received word as a sum of nearest codeword and error word, we are trying to find error word with minimal number of non-zeros on readable positions. Syndrom $$s_i$$ restricts error word by condition

 * $$s_i=\sum_{j=0}^{n-1}e_j\alpha^{ij}.$$

We could write these conditions separately or we could create polynomial

 * $$S(x)=\sum_{i=0}^{d-2}s_{c+i}x^i$$

and compare coefficients near powers $$0$$ to $$d-2.$$

 * $$S(x) \stackrel{\{0,\cdots,\,d-2\}}{=} E(x)=\sum_{i=0}^{d-2}\sum_{j=0}^{n-1}e_j\alpha^{ij}\alpha^{cj}x^i.$$

Suppose there is unreadable letter on position $$k_1,$$ we could replace set of syndromes $$\{s_c,\cdots,s_{c+d-2}\}$$ by set of syndromes $$\{t_c,\cdots,t_{c+d-3}\}$$ defined by equation $$t_i=\alpha^{k_1}s_i-s_{i+1}.$$ Suppose for an error word all restrictions by original set $$\{s_c,\cdots,s_{c+d-2}\}$$ of syndromes hold, than

 * $$t_i=\alpha^{k_1}s_i-s_{i+1}=\alpha^{k_1}\sum_{j=0}^{n-1}e_j\alpha^{ij}-\sum_{j=0}^{n-1}e_j\alpha^j\alpha^{ij}=\sum_{j=0}^{n-1}e_j\left(\alpha^{k_1} - \alpha^j\right)\alpha^{ij}.$$

New set of syndromes restricts error vector

 * $$f_j=e_j\left(\alpha^{k_1} - \alpha^j\right)$$

the same way the original set of syndromes restricted the error vector $$e_j.$$ Except the coordinate $$k_1,$$ where we have $$f_{k_1}=0,$$ an $$f_j$$ is zero, if $$e_j = 0.$$ For the goal of locating error positions we could change the set of syndromes in the similar way to reflect all unreadable characters. This shortens the set of syndromes by $$k.$$

In polynomial formulation, the replacement of syndromes set $$\{s_c,\cdots,s_{c+d-2}\}$$ by syndromes set $$\{t_c,\cdots,t_{c+d-3}\}$$ leads to

 * $$T(x) = \sum_{i=0}^{d-3}t_{c+i}x^i=\alpha^{k_1}\sum_{i=0}^{d-3}s_{c+i}x^i-\sum_{i=1}^{d-2}s_{c+i}x^{i-1}.$$

Therefore,

 * $$xT(x) \stackrel{\{1,\cdots,\,d-2\}}{=} \left(x\alpha^{k_1} - 1\right)S(x).$$

After replacement of $$S(x)$$ by $$S(x)\Gamma(x)$$, one would require equation for coefficients near powers $$k,\cdots,d-2.$$

One could consider looking for error positions from the point of view of eliminating influence of given positions similarly as for unreadable characters. If we found $$v$$ positions such that eliminating their influence leads to obtaining set of syndromes consisting of all zeros, than there exists error vector with errors only on these coordinates. If $$\Lambda(x)$$ denotes the polynomial eliminating the influence of these coordinates, we obtain

 * $$S(x)\Gamma(x)\Lambda(x) \stackrel{\{k+v, \cdots, d-2\}}{=} 0.$$

In Euclidean algorithm, we try to correct at most $$\tfrac{1}{2}(d-1-k)$$ errors (on readable positions), because with bigger error count there could be more codewords in the same distance from the received word. Therefore, for $$\Lambda(x)$$ we are looking for, the equation must hold for coefficients near powers starting from

 * $$k + \left\lfloor \frac{1}{2} (d-1-k) \right\rfloor.$$

In Forney formula, $$\Lambda(x)$$ could be multiplied by a scalar giving the same result.

It could happen that the Euclidean algorithm finds $$\Lambda(x)$$ of degree higher than $$\tfrac{1}{2}(d-1-k)$$ having number of different roots equal to its degree, where the Fourney formula would be able to correct errors in all its roots, anyway correcting such many errors could be risky (especially with no other restrictions on received word). Usually after getting $$\Lambda(x)$$ of higher degree, we decide not to correct the errors. Correction could fail in the case $$\Lambda(x)$$ has roots with higher multiplicity or the number of roots is smaller than its degree. Fail could be detected as well by Forney formula returning error outside the transmitted alphabet.

Correct the errors
Using the error values and error location, correct the errors and form a corrected code vector by subtracting error values at error locations.

Decoding of binary code without unreadable characters
Consider a BCH code in GF(24) with $$d=7$$ and $$g(x) = x^{10} + x^8 + x^5 + x^4 + x^2 + x + 1$$. (This is used in QR codes.) Let the message to be transmitted be [1 1 0 1 1], or in polynomial notation, $$M(x) = x^4 + x^3 + x + 1.$$ The "checksum" symbols are calculated by dividing $$x^{10} M(x)$$ by $$g(x)$$ and taking the remainder, resulting in $$x^9 + x^4 + x^2$$ or [ 1 0 0 0 0 1 0 1 0 0 ]. These are appended to the message, so the transmitted codeword is [ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 ].

Now, imagine that there are two bit-errors in the transmission, so the received codeword is [ 1 0 0 1 1 1 0 0 0 1 1 0 1 0 0 ]. In polynomial notation:
 * $$R(x) = C(x) + x^{13} + x^5 = x^{14} + x^{11} + x^{10} + x^9 + x^5 + x^4 + x^2$$

In order to correct the errors, first calculate the syndromes. Taking $$\alpha = 0010,$$ we have $$s_1 = R(\alpha^1) = 1011,$$ $$s_2 = 1001,$$ $$s_3 = 1011,$$ $$s_4 = 1101,$$ $$s_5 = 0001,$$ and $$s_6 = 1001.$$ Next, apply the Peterson procedure by row-reducing the following augmented matrix.
 * $$\left [ S_{3 \times 3} | C_{3 \times 1} \right ] =

\begin{bmatrix}s_1&s_2&s_3&s_4\\ s_2&s_3&s_4&s_5\\ s_3&s_4&s_5&s_6\end{bmatrix} = \begin{bmatrix}1011&1001&1011&1101\\ 1001&1011&1101&0001\\ 1011&1101&0001&1001\end{bmatrix} \Rightarrow \begin{bmatrix}0001&0000&1000&0111\\ 0000&0001&1011&0001\\ 0000&0000&0000&0000 \end{bmatrix}$$ Due to the zero row, $S_{3×3}$ is singular, which is no surprise since only two errors were introduced into the codeword. However, the upper-left corner of the matrix is identical to $[S_{2×2} | C_{2×1}]$, which gives rise to the solution $$\lambda_2 = 1000,$$ $$\lambda_1 = 1011.$$ The resulting error locator polynomial is $$\Lambda(x) = 1000 x^2 + 1011 x + 0001,$$ which has zeros at $$0100 = \alpha^{-13}$$ and $$0111 = \alpha^{-5}.$$ The exponents of $$\alpha$$ correspond to the error locations. There is no need to calculate the error values in this example, as the only possible value is 1.

Decoding with unreadable characters
Suppose the same scenario, but the received word has two unreadable characters [ 1 0 0 ? 1 1 ? 0 0 1 1 0 1 0 0 ]. We replace the unreadable characters by zeros while creating the polynomial reflecting their positions $$\Gamma(x) = \left(\alpha^8x - 1\right)\left(\alpha^{11}x - 1\right).$$ We compute the syndromes $$s_1=\alpha^{-7}, s_2=\alpha^{1}, s_3=\alpha^{4}, s_4=\alpha^{2}, s_5=\alpha^{5},$$ and $$s_6=\alpha^{-7}.$$ (Using log notation which is independent on GF(24) isomorphisms. For computation checking we can use the same representation for addition as was used in previous example. Hexadecimal description of the powers of $$\alpha$$ are consecutively 1,2,4,8,3,6,C,B,5,A,7,E,F,D,9 with the addition based on bitwise xor.)

Let us make syndrome polynomial

 * $$S(x)=\alpha^{-7}+\alpha^{1}x+\alpha^{4}x^2+\alpha^{2}x^3+\alpha^{5}x^4+\alpha^{-7}x^5,$$

compute

 * $$S(x)\Gamma(x)=\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^2+\alpha^{6}x^3+\alpha^{-1}x^4+\alpha^{5}x^5+\alpha^{7}x^6+\alpha^{-3}x^7.$$

Run the extended Euclidean algorithm:

 * $$\begin{align}

&\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix} \\ [6pt] ={} &\begin{pmatrix}\alpha^{-7} +\alpha^{4}x+ \alpha^{-1}x^2+ \alpha^{6}x^3+ \alpha^{-1}x^4+ \alpha^{5}x^5 +\alpha^{7}x^6+ \alpha^{-3}x^7 \\ x^6\end{pmatrix} \\ [6pt] ={} &\begin{pmatrix}\alpha^{7}+ \alpha^{-3}x & 1\\ 1 & 0\end{pmatrix} \begin{pmatrix}x^6\\ \alpha^{-7} +\alpha^{4}x +\alpha^{-1}x^2 +\alpha^{6}x^3 +\alpha^{-1}x^4 +\alpha^{5}x^5 +2\alpha^{7}x^6 +2\alpha^{-3}x^7\end{pmatrix} \\ [6pt] ={} &\begin{pmatrix}\alpha^{7}+ \alpha^{-3}x & 1\\ 1 & 0\end{pmatrix} \begin{pmatrix}\alpha^4 + \alpha^{-5}x & 1\\ 1 & 0\end{pmatrix} \\ &\qquad \begin{pmatrix}\alpha^{-7}+ \alpha^{4}x+ \alpha^{-1}x^2+ \alpha^{6}x^3+ \alpha^{-1}x^4+ \alpha^{5}x^5\\ \alpha^{-3} +\left(\alpha^{-7}+ \alpha^{3}\right)x+ \left(\alpha^{3}+ \alpha^{-1}\right)x^2+ \left(\alpha^{-5}+ \alpha^{-6}\right)x^3+ \left(\alpha^3+ \alpha^{1}\right)x^4+ 2\alpha^{-6}x^5+ 2x^6\end{pmatrix} \\ [6pt] ={} &\begin{pmatrix}\left(1+ \alpha^{-4}\right)+ \left(\alpha^{1}+ \alpha^{2}\right)x+ \alpha^{7}x^2 & \alpha^{7}+ \alpha^{-3}x \\ \alpha^4+ \alpha^{-5}x & 1\end{pmatrix} \begin{pmatrix}\alpha^{-7}+ \alpha^{4}x+ \alpha^{-1}x^2+ \alpha^{6}x^3+ \alpha^{-1}x^4+ \alpha^{5}x^5\\ \alpha^{-3}+ \alpha^{-2}x+ \alpha^{0}x^2+ \alpha^{-2}x^3+ \alpha^{-6}x^4\end{pmatrix} \\ [6pt] ={} &\begin{pmatrix}\alpha^{-3}+ \alpha^{5}x+ \alpha^{7}x^2 & \alpha^{7}+ \alpha^{-3}x \\ \alpha^4+ \alpha^{-5}x & 1\end{pmatrix} \begin{pmatrix}\alpha^{-5}+ \alpha^{-4}x & 1\\ 1 & 0 \end{pmatrix} \\ &\qquad \begin{pmatrix}\alpha^{-3}+ \alpha^{-2}x+ \alpha^{0}x^2+ \alpha^{-2}x^3+ \alpha^{-6}x^4\\ \left(\alpha^{7}+ \alpha^{-7}\right)+ \left(2\alpha^{-7}+ \alpha^{4}\right)x+ \left(\alpha^{-5}+ \alpha^{-6}+ \alpha^{-1}\right)x^2+ \left(\alpha^{-7}+ \alpha^{-4}+ \alpha^{6}\right)x^3+ \left(\alpha^{4}+ \alpha^{-6}+ \alpha^{-1}\right)x^4+ 2\alpha^{5}x^5\end{pmatrix} \\ [6pt] ={} &\begin{pmatrix}\alpha^{7}x+ \alpha^{5}x^2+ \alpha^{3}x^3 & \alpha^{-3}+ \alpha^{5}x+ \alpha^{7}x^2\\ \alpha^{3}+ \alpha^{-5}x+ \alpha^{6}x^2 & \alpha^4+ \alpha^{-5}x\end{pmatrix} \begin{pmatrix}\alpha^{-3}+ \alpha^{-2}x+ \alpha^{0}x^2+ \alpha^{-2}x^3+ \alpha^{-6}x^4\\ \alpha^{-4}+ \alpha^{4}x+ \alpha^{2}x^2+ \alpha^{-5}x^3\end{pmatrix}. \end{align}$$

We have reached polynomial of degree at most 3, and as

 * $$\begin{pmatrix}-\left(\alpha^4+ \alpha^{-5}x\right) & \alpha^{-3}+ \alpha^{5}x+ \alpha^{7}x^2\\ \alpha^{3}+ \alpha^{-5}x+ \alpha^{6}x^2 & -\left(\alpha^{7}x+ \alpha^{5}x^2+ \alpha^{3}x^3\right)\end{pmatrix} \begin{pmatrix}\alpha^{7}x+ \alpha^{5}x^2+ \alpha^{3}x^3 & \alpha^{-3} + \alpha^{5}x + \alpha^{7}x^2\\ \alpha^{3} + \alpha^{-5}x + \alpha^{6}x^2 & \alpha^4 + \alpha^{-5}x\end{pmatrix} = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix},$$

we get

 * $$\begin{pmatrix}-\left(\alpha^4+ \alpha^{-5}x\right) & \alpha^{-3}+ \alpha^{5}x+ \alpha^{7}x^2\\ \alpha^{3}+ \alpha^{-5}x+ \alpha^{6}x^2 & -\left(\alpha^{7}x+ \alpha^{5}x^2+ \alpha^{3}x^3\right)\end{pmatrix}

\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix} = \begin{pmatrix}\alpha^{-3}+ \alpha^{-2}x+ \alpha^{0}x^2+ \alpha^{-2}x^3+ \alpha^{-6}x^4\\ \alpha^{-4}+ \alpha^{4}x+ \alpha^{2}x^2+ \alpha^{-5}x^3\end{pmatrix}. $$

Therefore,

 * $$S(x)\Gamma(x)\left(\alpha^{3} + \alpha^{-5}x + \alpha^{6}x^2\right) - \left(\alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3\right)x^6 = \alpha^{-4} + \alpha^{4}x + \alpha^{2}x^2 + \alpha^{-5}x^3.$$

Let $$\Lambda(x) = \alpha^{3}+ \alpha^{-5}x+ \alpha^{6}x^2.$$ Don't worry that $$\lambda_0\neq 1.$$ Find by brute force a root of $$\Lambda.$$ The roots are $$\alpha^2,$$ and $$\alpha^{10}$$ (after finding for example $$\alpha^2$$ we can divide $$\Lambda$$ by corresponding monom $$\left(x - \alpha^2\right)$$ and the root of resulting monom could be found easily).

Let

 * $$\begin{align}

\Xi(x) &= \Gamma(x)\Lambda(x) = \alpha^3 + \alpha^4x^2 + \alpha^2x^3 + \alpha^{-5}x^4 \\ \Omega(x) &= S(x)\Xi(x) \equiv \alpha^{-4} + \alpha^4x + \alpha^2x^2 + \alpha^{-5}x^3 \bmod{x^6} \end{align}$$

Let us look for error values using formula

 * $$e_j = -\frac{\Omega \left(\alpha^{-i_j} \right)}{\Xi' \left(\alpha^{-i_j} \right)},$$

where $$\alpha^{-i_j}$$ are roots of $$\Xi(x).$$ $$\Xi'(x)=\alpha^{2}x^2.$$ We get

 * $$\begin{align}

e_1 &=-\frac{\Omega(\alpha^4)}{\Xi'(\alpha^{4})} = \frac{\alpha^{-4}+\alpha^{-7}+\alpha^{-5}+\alpha^{7}}{\alpha^{-5}} =\frac{\alpha^{-5}}{\alpha^{-5}}=1 \\ e_2 &=-\frac{\Omega(\alpha^7)}{\Xi'(\alpha^{7})} = \frac{\alpha^{-4}+\alpha^{-4}+\alpha^{1}+\alpha^{1}}{\alpha^{1}}=0 \\ e_3 &=-\frac{\Omega(\alpha^{10})}{\Xi'(\alpha^{10})} = \frac{\alpha^{-4}+\alpha^{-1}+\alpha^{7}+\alpha^{-5}}{\alpha^{7}}=\frac{\alpha^{7}}{\alpha^{7}}=1 \\ e_4 &=-\frac{\Omega(\alpha^{2})}{\Xi'(\alpha^{2})} = \frac{\alpha^{-4}+\alpha^{6}+\alpha^{6}+\alpha^{1}}{\alpha^{6}}=\frac{\alpha^{6}}{\alpha^{6}}=1 \end{align}$$

Fact, that $$e_3=e_4=1,$$ should not be surprising.

Corrected code is therefore [ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0].

Decoding with unreadable characters with a small number of errors
Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 0 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ].

Again, replace the unreadable characters by zeros while creating the polynomial reflecting their positions $$\Gamma(x) = \left(\alpha^{8}x - 1\right)\left(\alpha^{11}x - 1\right).$$ Compute the syndromes $$s_1 = \alpha^{4}, s_2 = \alpha^{-7}, s_3 = \alpha^{1}, s_4 = \alpha^{1}, s_5 = \alpha^{0},$$ and $$s_6 = \alpha^{2}.$$ Create syndrome polynomial

 * $$\begin{align}

S(x) &= \alpha^{4} + \alpha^{-7}x + \alpha^{1}x^2 + \alpha^{1}x^3 + \alpha^{0}x^4 + \alpha^{2}x^5, \\ S(x)\Gamma(x) &= \alpha^{4} + \alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3 + \alpha^{1}x^4 + \alpha^{-1}x^5 + \alpha^{-1}x^6 + \alpha^{6}x^7. \end{align}$$

Let us run the extended Euclidean algorithm:

 * $$\begin{align}

\begin{pmatrix} S(x)\Gamma(x) \\ x^6 \end{pmatrix} &= \begin{pmatrix} \alpha^{4} + \alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3 + \alpha^{1}x^4 + \alpha^{-1}x^5 + \alpha^{-1}x^6 + \alpha^{6}x^7 \\ x^6 \end{pmatrix} \\ &= \begin{pmatrix} \alpha^{-1} + \alpha^{6}x & 1 \\ 1 & 0      \end{pmatrix} \begin{pmatrix} x^6 \\ \alpha^{4} + \alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3 + \alpha^{1}x^4 + \alpha^{-1}x^5 + 2\alpha^{-1}x^6 + 2\alpha^{6}x^7 \end{pmatrix} \\ &= \begin{pmatrix} \alpha^{-1} + \alpha^{6}x & 1 \\ 1 & 0      \end{pmatrix} \begin{pmatrix} \alpha^{3} + \alpha^{1}x & 1 \\ 1 & 0      \end{pmatrix} \begin{pmatrix} \alpha^{4} + \alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3 + \alpha^{1}x^4 + \alpha^{-1}x^5 \\ \alpha^{7} + \left(\alpha^{-5} + \alpha^{5}\right)x + 2\alpha^{-7}x^2 + 2\alpha^{6}x^3 + 2\alpha^{4}x^4 + 2\alpha^{2}x^5 + 2x^6 \end{pmatrix} \\ &= \begin{pmatrix} \left(1 + \alpha^{2}\right) + \left(\alpha^{0} + \alpha^{-6}\right)x + \alpha^{7}x^2 & \alpha^{-1} + \alpha^{6}x \\ \alpha^{3} + \alpha^{1}x & 1 \end{pmatrix} \begin{pmatrix} \alpha^{4} + \alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3 + \alpha^{1}x^4 + \alpha^{-1}x^5 \\ \alpha^{7} + \alpha^{0}x \end{pmatrix} \end{align}$$

We have reached polynomial of degree at most 3, and as

\begin{pmatrix} -1 & \alpha^{-1} + \alpha^{6}x \\ \alpha^{3} + \alpha^{1}x & -\left(\alpha^{-7} + \alpha^{7}x + \alpha^{7}x^2\right) \end{pmatrix} \begin{pmatrix} \alpha^{-7} + \alpha^{7}x + \alpha^{7}x^2 & \alpha^{-1} + \alpha^{6}x \\ \alpha^{3} + \alpha^{1}x & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, $$ we get

\begin{pmatrix} -1 & \alpha^{-1} + \alpha^{6}x \\ \alpha^{3} + \alpha^{1}x & -\left(\alpha^{-7} + \alpha^{7}x + \alpha^{7}x^2\right) \end{pmatrix}\begin{pmatrix} S(x)\Gamma(x) \\ x^6 \end{pmatrix} = \begin{pmatrix} \alpha^{4} + \alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3 + \alpha^{1}x^4 + \alpha^{-1}x^5 \\ \alpha^{7} + \alpha^{0}x \end{pmatrix}. $$

Therefore,
 * $$S(x)\Gamma(x)\left(\alpha^{3} + \alpha^{1}x\right) - \left(\alpha^{-7} + \alpha^{7}x + \alpha^{7}x^2\right)x^6 = \alpha^{7} + \alpha^{0}x.$$

Let $$\Lambda(x) = \alpha^{3} + \alpha^{1}x.$$ Don't worry that $$\lambda_0 \neq 1.$$ The root of $$\Lambda(x)$$ is $$\alpha^{3-1}.$$

Let
 * $$\begin{align}

\Xi(x) &= \Gamma(x)\Lambda(x) = \alpha^{3} + \alpha^{-7}x + \alpha^{-4}x^2 + \alpha^{5}x^3, \\ \Omega(x) &= S(x)\Xi(x) \equiv \alpha^{7} + \alpha^{0}x \bmod{x^6} \end{align}$$

Let us look for error values using formula $$e_j = -\Omega\left(\alpha^{-i_j}\right)/\Xi'\left(\alpha^{-i_j}\right),$$ where $$\alpha^{-i_j}$$ are roots of polynomial $$\Xi(x).$$
 * $$\Xi'(x) = \alpha^{-7} + \alpha^{5}x^2.$$

We get
 * $$\begin{align}

e_1 &= -\frac{\Omega\left(\alpha^4\right)}{\Xi'\left(\alpha^{4}\right)} = \frac{\alpha^{7} + \alpha^{4}}{\alpha^{-7} + \alpha^{-2}} = \frac{\alpha^{3}}{\alpha^{3}} = 1 \\  e_2 &= -\frac{\Omega\left(\alpha^7\right)}{\Xi'\left(\alpha^{7}\right)} = \frac{\alpha^{7} + \alpha^{7}}{\alpha^{-7} + \alpha^{4}} = 0 \\  e_3 &= -\frac{\Omega\left(\alpha^2\right)}{\Xi'\left(\alpha^2\right)} = \frac{\alpha^{7} + \alpha^{2}}{\alpha^{-7} + \alpha^{-6}} = \frac{\alpha^{-3}}{\alpha^{-3}} = 1 \end{align}$$

The fact that $$e_3 = 1$$ should not be surprising.

Corrected code is therefore [ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0].

title: BCH code


text:
Beam diameter

The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. Since beams typically do not have sharp edges, the diameter can be defined in many different ways. Five definitions of the beam width are in common use: D4σ, 10/90 or 20/80 knife-edge, 1/e2, FWHM, and D86. The beam width can be measured in units of length at a particular plane perpendicular to the beam axis, but it can also refer to the angular width, which is the angle subtended by the beam at the source. The angular width is also called the beam divergence.

Beam diameter is usually used to characterize electromagnetic beams in the optical regime, and occasionally in the microwave regime, that is, cases in which the aperture from which the beam emerges is very large with respect to the wavelength.

Beam diameter usually refers to a beam of circular cross section, but not necessarily so. A beam may, for example, have an elliptical cross section, in which case the orientation of the beam diameter must be specified, for example with respect to the major or minor axis of the elliptical cross section. The term "beam width" may be preferred in applications where the beam does not have circular symmetry.

Rayleigh beamwidth
The angle between the maximum peak of radiated power and the first null (no power radiated in this direction) is called the Rayleigh beamwidth.

Full width at half maximum
The simplest way to define the width of a beam is to choose two diametrically opposite points at which the irradiance is a specified fraction of the beam's peak irradiance, and take the distance between them as a measure of the beam's width. An obvious choice for this fraction is $1⁄2$ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). This is also called the half-power beam width (HPBW).

1/e2 width
The 1/e2 width is equal to the distance between the two points on the marginal distribution that are 1/e2 = 0.135 times the maximum value. In many cases, it makes more sense to take the distance between points where the intensity falls to 1/e2 = 0.135 times the maximum value. If there are more than two points that are 1/e2 times the maximum value, then the two points closest to the maximum are chosen. The 1/e2 width is important in the mathematics of Gaussian beams, in which the intensity profile is described by $$I(r) = I_{0} \left( \frac{w_0}{w} \right)^2 \exp \! \left( \! -2 \frac{r^2}{w^2}\right ) $$.

The American National Standard Z136.1-2007 for Safe Use of Lasers (p. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. This is the beam diameter definition that is used for computing the maximum permissible exposure to a laser beam. In addition, the Federal Aviation Administration also uses the 1/e definition for laser safety calculations in FAA Order JO 7400.2, Para. 29-1-5d.

Measurements of the 1/e2 width only depend on three points on the marginal distribution, unlike D4σ and knife-edge widths that depend on the integral of the marginal distribution. 1/e2 width measurements are noisier than D4σ width measurements. For multimodal marginal distributions (a beam profile with multiple peaks), the 1/e2 width usually does not yield a meaningful value and can grossly underestimate the inherent width of the beam. For multimodal distributions, the D4σ width is a better choice. For an ideal single-mode Gaussian beam, the D4σ, D86 and 1/e2 width measurements would give the same value.

For a Gaussian beam, the relationship between the 1/e2 width and the full width at half maximum is $$2w = \frac{\sqrt 2\ \mathrm{FWHM}}{\sqrt{\ln 2}} = 1.699 \times \mathrm{FWHM}$$, where $$2w$$ is the full width of the beam at 1/e2.

D4σ or second-moment width
The D4σ width of a beam in the horizontal or vertical direction is 4 times σ, where σ is the standard deviation of the horizontal or vertical marginal distribution respectively. Mathematically, the D4σ beam width in the x dimension for the beam profile $$ I(x,y) $$ is expressed as

 * $$ D4\sigma = 4 \sigma = 4 \sqrt{\frac{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) (x - \bar{x})^2 \,dx \,dy} {\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y)\, dx \,dy}}, $$

where

 * $$ \bar{x} = \frac{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) x \,dx \,dy}{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) \,dx \,dy} $$

is the centroid of the beam profile in the x direction.

When a beam is measured with a laser beam profiler, the wings of the beam profile influence the D4σ value more than the center of the profile, since the wings are weighted by the square of its distance, x2, from the center of the beam. If the beam does not fill more than a third of the beam profiler's sensor area, then there will be a significant number of pixels at the edges of the sensor that register a small baseline value (the background value). If the baseline value is large or if it is not subtracted out of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by x2. Therefore, baseline subtraction is necessary for accurate D4σ measurements. The baseline is easily measured by recording the average value for each pixel when the sensor is not illuminated. The D4σ width, unlike the FWHM and 1/e2 widths, is meaningful for multimodal marginal distributions — that is, beam profiles with multiple peaks — but requires careful subtraction of the baseline for accurate results. The D4σ is the ISO international standard definition for beam width.

Knife-edge width
Before the advent of the CCD beam profiler, the beam width was estimated using the knife-edge technique: slice a laser beam with a razor and measure the power of the clipped beam as a function of the razor position. The measured curve is the integral of the marginal distribution, and starts at the total beam power and decreases monotonically to zero power. The width of the beam is defined as the distance between the points of the measured curve that are 10% and 90% (or 20% and 80%) of the maximum value. If the baseline value is small or subtracted out, the knife-edge beam width always corresponds to 60%, in the case of 20/80, or 80%, in the case of 10/90, of the total beam power no matter what the beam profile. On the other hand, the D4σ, 1/e2, and FWHM widths encompass fractions of power that are beam-shape dependent. Therefore, the 10/90 or 20/80 knife-edge width is a useful metric when the user wishes to be sure that the width encompasses a fixed fraction of total beam power. Most CCD beam profiler's software can compute the knife-edge width numerically.

Fusing knife-edge method with imaging
The main drawback of the knife-edge technique is that the measured value is displayed only on the scanning direction, minimizing the amount of relevant beam information. To overcome this drawback, an innovative technology offered commercially allows multiple directions beam scanning to create an image like beam representation.

By mechanically moving the knife edge across the beam, the amount of energy impinging the detector area is determined by the obstruction. The profile is then measured from the knife-edge velocity and its relation to the detector's energy reading. Unlike other systems, a unique scanning technique uses several different oriented knife-edges to sweep across the beam. By using tomographic reconstruction, mathematical processes reconstruct the laser beam size in different orientations to an image similar to the one produced by CCD cameras. The main advantage of this scanning method is that it is free from pixel size limitations (as in CCD cameras) and allows beam reconstructions with wavelengths not usable with existing CCD technology. Reconstruction is possible for beams in deep UV to far IR.

D86 width
The D86 width is defined as the diameter of the circle that is centered at the centroid of the beam profile and contains 86% of the beam power. The solution for D86 is found by computing the area of increasingly larger circles around the centroid until the area contains 0.86 of the total power. Unlike the previous beam width definitions, the D86 width is not derived from marginal distributions. The percentage of 86, rather than 50, 80, or 90, is chosen because a circular Gaussian beam profile integrated down to 1/e2 of its peak value contains 86% of its total power. The D86 width is often used in applications that are concerned with knowing exactly how much power is in a given area. For example, applications of high-energy laser weapons and lidars require precise knowledge of how much transmitted power actually illuminates the target.

ISO11146 beam width for elliptic beams
The definition given before holds for stigmatic (circular symmetric) beams only. For astigmatic beams, however, a more rigorous definition of the beam width has to be used:
 * $$ d_{\sigma x} = 2 \sqrt{2} \left( \langle x^2 \rangle + \langle y^2 \rangle + \gamma \left( \left( \langle x^2 \rangle - \langle y^2 \rangle \right)^2 + 4 \langle xy \rangle^2 \right)^{1/2} \right)^{1/2} $$

and
 * $$ d_{\sigma y} = 2 \sqrt{2} \left( \langle x^2 \rangle + \langle y^2 \rangle - \gamma \left( \left( \langle x^2 \rangle - \langle y^2 \rangle \right)^2 + 4 \langle xy \rangle^2 \right)^{1/2} \right)^{1/2}. $$

This definition also incorporates information about x–y correlation $$ \langle xy \rangle $$, but for circular symmetric beams, both definitions are the same.

Some new symbols appeared within the formulas, which are the first- and second-order moments:
 * $$ \langle x \rangle = \frac{1}{P} \int I(x,y) x \,dx \,dy, $$
 * $$ \langle y \rangle = \frac{1}{P} \int I(x,y) y \,dx \,dy, $$
 * $$ \langle x^2 \rangle = \frac{1}{P} \int I(x,y) (x - \langle x \rangle )^2 \,dx \,dy, $$
 * $$ \langle xy \rangle = \frac{1}{P} \int I(x,y) (x - \langle x \rangle ) (y - \langle y \rangle ) \,dx \,dy, $$
 * $$ \langle y^2 \rangle = \frac{1}{P} \int I(x,y) (y - \langle y \rangle )^2 \,dx \,dy, $$

the beam power
 * $$ P = \int I(x,y) \,dx \,dy, $$

and
 * $$ \gamma = \sgn \left( \langle x^2 \rangle - \langle y^2 \rangle \right) = \frac{\langle x^2 \rangle - \langle y^2 \rangle}{|\langle x^2 \rangle - \langle y^2 \rangle|}. $$

Using this general definition, also the beam azimuthal angle $$ \phi $$ can be expressed. It is the angle between the beam directions of minimal and maximal elongations, known as principal axes, and the laboratory system, being the $$x$$ and $$y$$ axes of the detector and given by
 * $$ \phi = \frac{1}{2} \arctan \frac{2 \langle xy \rangle}{\langle x^2 \rangle - \langle y^2 \rangle }.$$

Measurement
International standard ISO 11146-1:2005 specifies methods for measuring beam widths (diameters), divergence angles and beam propagation ratios of laser beams (if the beam is stigmatic) and for general astigmatic beams ISO 11146-2 is applicable. The D4σ beam width is the ISO standard definition and the measurement of the M2 beam quality parameter requires the measurement of the D4σ widths.

The other definitions provide complementary information to the D4σ. The D4σ and knife-edge widths are sensitive to the baseline value, whereas the 1/e2 and FWHM widths are not. The fraction of total beam power encompassed by the beam width depends on which definition is used.

The width of laser beams can be measured by capturing an image on a camera, or by using a laser beam profiler.

title: Beam diameter


text:
Beam divergence

In electromagnetics, especially in optics, beam divergence is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the beam emerges. The term is relevant only in the "far field", away from any focus of the beam. Practically speaking, however, the far field can commence physically close to the radiating aperture, depending on aperture diameter and the operating wavelength.

Beam divergence is often used to characterize electromagnetic beams in the optical regime, for cases in which the aperture from which the beam emerges is very large with respect to the wavelength. However, it is also used in the radio frequency (RF) band for cases in which the antenna is very large relative to a wavelength.

Beam divergence usually refers to a beam of circular cross section, but not necessarily so. A beam may, for example, have an elliptical cross section, in which case the orientation of the beam divergence must be specified, for example with respect to the major or minor axis of the elliptical cross section.

The divergence of a beam can be calculated if one knows the beam diameter at two separate points far from any focus (Di, Df), and the distance (l) between these points. The beam divergence, $$\Theta$$, is given by
 * $$\Theta=2 \arctan\left(\frac{D_f-D_i}{2l}\right).$$

If a collimated beam is focused with a lens, the diameter $$D_m$$ of the beam in the rear focal plane of the lens is related to the divergence of the initial beam by
 * $$\Theta=\frac{D_m}{f},\,$$

where f is the focal length of the lens. Note that this measurement is valid only when the beam size is measured at the rear focal plane of the lens, i.e. where the focus would lie for a truly collimated beam, and not at the actual focus of the beam, which would occur behind the rear focal plane for a divergent beam.

Like all electromagnetic beams, lasers are subject to divergence, which is measured in milliradians (mrad) or degrees. For many applications, a lower-divergence beam is preferable. Neglecting divergence due to poor beam quality, the divergence of a laser beam is proportional to its wavelength and inversely proportional to the diameter of the beam at its narrowest point. For example, an ultraviolet laser that emits at a wavelength of 308 nm will have a lower divergence than an infrared laser at 808 nm, if both have the same minimum beam diameter. The divergence of good-quality laser beams is modeled using the mathematics of Gaussian beams.

Gaussian laser beams are said to be diffraction limited when their radial beam divergence $$\theta=\Theta/2$$ is close to the minimum possible value, which is given by

 * $$ \theta = {\lambda \over\pi w_0},$$

where $$\lambda$$ is the laser wavelength and $$w_0$$ is the radius of the beam at its narrowest point, which is called the "beam waist". This type of beam divergence is observed from optimized laser cavities. Information on the diffraction-limited divergence of a coherent beam is inherently given by the N-slit interferometric equation.

title: Beam divergence


text:
Beam steering

Beam steering is a technique for changing the direction of the main lobe of a radiation pattern.

In radio and radar systems, beam steering may be accomplished by switching the antenna elements or by changing the relative phases of the RF signals driving the elements. As a result, this directs the transmit signal towards an intended receiver. In recent days, beam steering is playing a significant role in 5G communication because of quasi-optic nature of 5G frequencies.

In acoustics, beam steering is used to direct the audio from loudspeakers to a specific location in the listening area. This is done by changing the magnitude and phase of two or more loudspeakers installed in a column where the combined sound is added and cancelled at the required position. Commercially, this type of loudspeaker arrangement is known as a line array. This technique has been around for many years but since the emergence of modern digital signal processing (DSP) technology there are now many commercially available products on the market. Beam steering and directivity Control using DSP was pioneered in the early 1990s by Duran Audio who launched a technology called DDC (Digital Directivity Control).

In optical systems, beam steering may be accomplished by changing the refractive index of the medium through which the beam is transmitted or by the use of mirrors, prisms, lenses, or rotating diffraction gratings. Examples of optical beam steering approaches include mechanical mirror-based gimbals or beam-director units, galvanometer mechanisms that rotate mirrors, Risley prisms, phased-array optics, and microelectromechanical systems using micro-mirrors.

Source: from Federal Standard 1037C

title: Beam steering


text:
Bel

Bel can mean:

Mythology

 * Belenus or Bel, a Celtic deity
 * Bel (mythology), a title (meaning "lord" or "master") for various gods in Babylonian religion

People

 * Bel (name)
 * Annabel Linquist, known as Bel, American artist, musician, and entrepreneur

Places

 * Bél, the Hungarian name for Beliu Commune, Arad County, Romania
 * Bel Mountain, in the Zagros Mountains of western and southwestern Iran
 * Bel, Iran (disambiguation)
 * Bel, Osh, village in Osh Region, Kyrgyzstan
 * Bel, Syria, village in Aleppo Governorate
 * Temple of Bel in Palmyra, Syria

Languages

 * Bel languages, spoken in northern Papua New Guinea
 * ISO 639 code for the Belarusian language

Other uses

 * Groupe Bel, a France-based multinational cheese distributor
 * Bel (unit), a unit of gain or loss equal to ten decibels

title: Bel


text:
Bias

Bias is a disproportionate weight in favor of or against an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, or a belief. In science and engineering, a bias is a systematic error. Statistical bias results from an unfair sampling of a population, or from an estimation process that does not give accurate results on average.

Etymology
The word appears to derive from Old Provençal into Old French biais, "sideways, askance, against the grain". Whence comes French biais, "a slant, a slope, an oblique".

It seems to have entered English via the game of bowls, where it referred to balls made with a greater weight on one side. Which expanded to the figurative use, "a one-sided tendency of the mind", and, at first especially in law, "undue propensity or prejudice".

Cognitive biases
A cognitive bias is a repeating or basic misstep in thinking, assessing, recollecting, or other cognitive processes. That is, a pattern of deviation from standards in judgment, whereby inferences may be created unreasonably. People create their own "subjective social reality" from their own perceptions, their view of the world may dictate their behaviour. Thus, cognitive biases may sometimes lead to perceptual distortion, inaccurate judgment, illogical interpretation, or what is broadly called irrationality. However some cognitive biases are taken to be adaptive, and thus may lead to success in the appropriate situation. Furthermore, cognitive biases may allow speedier choices when speed is more valuable than precision. Other cognitive biases are a "by-product" of human processing limitations, coming about because of an absence of appropriate mental mechanisms, or just from human limitations in information processing.

Anchoring
Anchoring is a psychological heuristic that describes the propensity to rely on the first piece of information encountered when making decisions. According to this heuristic, individuals begin with an implicitly suggested reference point (the "anchor") and make adjustments to it to reach their estimate. For example, the initial price offered for a used car sets the standard for the rest of the negotiations, so that prices lower than the initial price seem more reasonable even if they are still higher than what the car is worth.

Apophenia
Apophenia, also known as patternicity, or agenticity, is the human tendency to perceive meaningful patterns within random data. Apophenia is well documented as a rationalization for gambling. Gamblers may imagine that they see patterns in the numbers which appear in lotteries, card games, or roulette wheels. One manifestation of this is known as the "gambler's fallacy".

Pareidolia is the visual or auditory form of apophenia. It has been suggested that pareidolia combined with hierophany may have helped ancient societies organize chaos and make the world intelligible.

Attribution bias
An attribution bias can happen when individuals assess or attempt to discover explanations behind their own and others' behaviors. People make attributions about the causes of their own and others' behaviors; but these attributions do not necessarily precisely reflect reality. Rather than operating as objective perceivers, individuals are inclined to perceptual slips that prompt biased understandings of their social world. When judging others we tend to assume their actions are the result of internal factors such as personality, whereas we tend to assume our own actions arise because of the necessity of external circumstances. There are a wide range of sorts of attribution biases, such as the ultimate attribution error, fundamental attribution error, actor-observer bias, and self-serving bias.

Examples of attribution bias:

Confirmation bias

Confirmation bias is the tendency to search for, interpret, favor, and recall information in a way that confirms one's beliefs or hypotheses while giving disproportionately less attention to information that contradicts it. The effect is stronger for emotionally charged issues and for deeply entrenched beliefs. People also tend to interpret ambiguous evidence as supporting their existing position. Biased search, interpretation and memory have been invoked to explain attitude polarization (when a disagreement becomes more extreme even though the different parties are exposed to the same evidence), belief perseverance (when beliefs persist after the evidence for them is shown to be false), the irrational primacy effect (a greater reliance on information encountered early in a series) and illusory correlation (when people falsely perceive an association between two events or situations). Confirmation biases contribute to overconfidence in personal beliefs and can maintain or strengthen beliefs in the face of contrary evidence. Poor decisions due to these biases have been found in political and organizational contexts.

Framing
Framing involves the social construction of social phenomena by mass media sources, political or social movements, political leaders, and so on. It is an influence over how people organize, perceive, and communicate about reality. It can be positive or negative, depending on the audience and what kind of information is being presented. For political purposes, framing often presents facts in such a way that implicates a problem that is in need of a solution. Members of political parties attempt to frame issues in a way that makes a solution favoring their own political leaning appear as the most appropriate course of action for the situation at hand. As understood in social theory, framing is a schema of interpretation, a collection of anecdotes and stereotypes, that individuals rely on to understand and respond to events. People use filters to make sense of the world, the choices they then make are influenced by their creation of a frame.

Cultural bias is the related phenomenon of interpreting and judging phenomena by standards inherent to one's own culture. Numerous such biases exist, concerning cultural norms for color, location of body parts, mate selection, concepts of justice, linguistic and logical validity, acceptability of evidence, and taboos. Ordinary people may tend to imagine other people as basically the same, not significantly more or less valuable, probably attached emotionally to different groups and different land.

Halo effect and horn effect
The halo effect and the horn effect are when an observer's overall impression of a person, organization, brand, or product influences their feelings about specifics of that entity's character or properties.

The name halo effect is based on the concept of the saint's halo, and is a specific type of confirmation bias, wherein positive sentiments in one area cause questionable or unknown characteristics to be seen positively. If the observer likes one aspect of something, they will have a positive predisposition toward everything about it. A person's appearance has been found to produce a halo effect. The halo effect is also present in the field of brand marketing, affecting perception of companies and non-governmental organizations (NGOs).

The opposite of the halo is the horn effect, when "individuals believe (that negative) traits are inter-connected." The term horn effect refers to Devil's horns. It works in a negative direction: if the observer dislikes one aspect of something, they will have a negative predisposition towards other aspects.

Both of these bias effects often clash with phrases such as "words mean something" and "Your words have a history."

Self-serving bias
Self-serving bias is the tendency for cognitive or perceptual processes to be distorted by the individual's need to maintain and enhance self-esteem. It is the propensity to credit accomplishment to our own capacities and endeavors, yet attribute failure to outside factors, to dismiss the legitimacy of negative criticism, concentrate on positive qualities and accomplishments yet disregard flaws and failures. Studies have demonstrated that this bias can affect behavior in the workplace, in interpersonal relationships, playing sports, and in consumer decisions.

Status quo bias
Status quo bias is an emotional bias; a preference for the current state of affairs. The current baseline (or status quo) is taken as a reference point, and any change from that baseline is perceived as a loss. Status quo bias should be distinguished from a rational preference for the status quo ante, as when the current state of affairs is objectively superior to the available alternatives, or when imperfect information is a significant problem. A large body of evidence, however, shows that status quo bias frequently affects human decision-making.

Conflicts of interest
A conflict of interest is when a person or association has intersecting interests (financial, personal, etc.) which could potentially corrupt. The potential conflict is autonomous of actual improper actions, it can be found and intentionally defused before corruption, or the appearance of corruption, happens. "A conflict of interest is a set of circumstances that creates a risk that professional judgement or actions regarding a primary interest will be unduly influenced by a secondary interest." It exists if the circumstances are sensibly accepted to present a hazard that choices made may be unduly affected by auxiliary interests.

Bribery
Bribery is giving of money, goods or other forms of recompense to in order to influence the recipient's behavior. Bribes can include money (including tips), goods, rights in action, property, privilege, emolument, gifts, perks, skimming, return favors, discounts, sweetheart deals, kickbacks, funding, donations, campaign contributions, sponsorships, stock options, secret commissions, or promotions. Expectations of when a monetary transaction is appropriate can differ from place to place. Political campaign contributions in the form of cash are considered criminal acts of bribery in some countries, while in the United States they are legal provided they adhere to election law. Tipping is considered bribery in some societies, but not others.

Favoritism
Favoritism, sometimes known as in-group favoritism, or in-group bias, refers to a pattern of favoring members of one's in-group over out-group members. This can be expressed in evaluation of others, in allocation of resources, and in many other ways. This has been researched by psychologists, especially social psychologists, and linked to group conflict and prejudice. Cronyism is favoritism of long-standing friends, especially by appointing them to positions of authority, regardless of their qualifications. Nepotism is favoritism granted to relatives.

Lobbying

Lobbying is the attempt to influence choices made by administrators, frequently lawmakers or individuals from administrative agencies. Lobbyists may be among a legislator's constituencies, or not; they may engage in lobbying as a business, or not. Lobbying is often spoken of with contempt, the implication is that people with inordinate socioeconomic power are corrupting the law in order to serve their own interests. When people who have a duty to act on behalf of others, such as elected officials with a duty to serve their constituents' interests or more broadly the common good, stand to benefit by shaping the law to serve the interests of some private parties, there is a conflict of interest. This can lead to all sides in a debate looking to sway the issue by means of lobbyists.

Regulatory issues
Self-regulation is the process whereby an organization monitors its own adherence to legal, ethical, or safety standards, rather than have an outside, independent agency such as a third party entity monitor and enforce those standards. Self-regulation of any group can create a conflict of interest. If any organization, such as a corporation or government bureaucracy, is asked to eliminate unethical behavior within their own group, it may be in their interest in the short run to eliminate the appearance of unethical behavior, rather than the behavior itself.

Regulatory capture is a form of political corruption that can occur when a regulatory agency, created to act in the public interest, instead advances the commercial or political concerns of special interest groups that dominate the industry or sector it is charged with regulating. Regulatory capture occurs because groups or individuals with a high-stakes interest in the outcome of policy or regulatory decisions can be expected to focus their resources and energies in attempting to gain the policy outcomes they prefer, while members of the public, each with only a tiny individual stake in the outcome, will ignore it altogether. Regulatory capture is a risk to which a regulatory agency is exposed by its very nature.

Shilling
Shilling is deliberately giving spectators the feeling that one is an energetic autonomous client of a vendor for whom one is working. The effectiveness of shilling relies on crowd psychology to encourage other onlookers or audience members to purchase the goods or services (or accept the ideas being marketed). Shilling is illegal in some places, but legal in others. An example of shilling is paid reviews that give the impression of being autonomous opinions.

Statistical biases
Statistical bias is a systematic tendency in the process of data collection, which results in lopsided, misleading results. This can occur in any of a number of ways, in the way the sample is selected, or in the way data are collected. It is a property of a statistical technique or of its results whereby the expected value of the results differs from the true underlying quantitative parameter being estimated.

Forecast bias
A forecast bias is when there are consistent differences between results and the forecasts of those quantities; that is: forecasts may have an overall tendency to be too high or too low.

Observer-expectancy effect
The observer-expectancy effect is when a researcher's expectations cause them to subconsciously influence the people participating in an experiment. It is usually controlled using a double-blind system, and was an important reason for the development of double-blind experiments.

Reporting bias and social desirability bias
In epidemiology and empirical research, reporting bias is defined as "selective revealing or suppression of information" of undesirable behavior by subjects or researchers. It refers to a tendency to under-report unexpected or undesirable experimental results, while being more trusting of expected or desirable results. This can propagate, as each instance reinforces the status quo, and later experimenters justify their own reporting bias by observing that previous experimenters reported different results.

Social desirability bias is a bias within social science research where survey respondents can tend to answer questions in a manner that will be viewed positively by others. It can take the form of over-reporting laudable behavior, or under-reporting undesirable behavior. This bias interferes with the interpretation of average tendencies as well as individual differences. The inclination represents a major issue with self-report questionnaires; of special concern are self-reports of abilities, personalities, sexual behavior, and drug use.

Selection bias
Selection bias is the conscious or unconscious bias introduced into a study by the way individuals, groups or data are selected for analysis, if such a way means that true randomization is not achieved, thereby ensuring that the sample obtained is not representative of the population intended to be analyzed. This results in a sample that may be significantly different from the overall population.

Prejudices
Bias and prejudice are usually considered to be closely related. Prejudice is prejudgment, or forming an opinion before becoming aware of the relevant facts of a case. The word is often used to refer to preconceived, usually unfavorable, judgments toward people or a person because of gender, political opinion, social class, age, disability, religion, sexuality, race/ethnicity, language, nationality, or other personal characteristics. Prejudice can also refer to unfounded beliefs and may include "any unreasonable attitude that is unusually resistant to rational influence".

Ageism
Ageism is the stereotyping and/or discrimination against individuals or groups on the basis of their age. It can be used in reference to prejudicial attitudes towards older people, or towards younger people.

Classism
Classism is discrimination on the basis of social class. It includes attitudes that benefit the upper class at the expense of the lower class, or vice versa.

Lookism
Lookism is stereotypes, prejudice, and discrimination on the basis of physical attractiveness, or more generally to people whose appearance matches cultural preferences. Many people make automatic judgments of others based on their physical appearance that influence how they respond to those people.

Racism
Racism consists of ideologies based on a desire to dominate or a belief in the inferiority of another race. It may also hold that members of different races should be treated differently.

Academic bias
Academic bias is the bias or perceived bias of scholars allowing their beliefs to shape their research and the scientific community. Claims of bias are often linked to claims by conservatives of pervasive bias against political conservatives and religious Christians. Some have argued that these claims are based upon anecdotal evidence which would not reliably indicate systematic bias, and have suggested that this divide is due to self-selection of conservatives choosing not to pursue academic careers. There is some evidence that perception of classroom bias may be rooted in issues of sexuality, race, class and sex as much or more than in religion.

Experimenter bias
In science research, experimenter bias occurs when experimenter expectancies regarding study results bias the research outcome. Examples of experimenter bias include conscious or unconscious influences on subject behavior including creation of demand characteristics that influence subjects, and altered or selective recording of experimental results themselves. It can also involve asking leading probes and not neutrally redirecting the subject back to the task when they ask for validation or questions.

Funding bias
Funding bias refers to the tendency of a scientific study to support the interests of the study's financial sponsor. This phenomenon is recognized sufficiently that researchers undertake studies to examine bias in past published studies. It can be caused by any or all of: a conscious or subconscious sense of obligation of researchers towards their employers, misconduct or malpractice, publication bias, or reporting bias.

Full text on net bias
Full text on net (or FUTON) bias is a tendency of scholars to cite academic journals with open access—that is, journals that make their full text available on the internet without charge—in their own writing as compared with toll access publications. Scholars can more easily discover and access articles that have their full text on the internet, which increases authors' likelihood of reading, quoting, and citing these articles, this may increase the impact factor of open access journals relative to journals without open access.

The related bias, no abstract available bias (NAA bias) is scholars' tendency to cite journal articles that have an abstract available online more readily than articles that do not.

Publication bias
Publication bias is a type of bias with regard to what academic research is likely to be published because of a tendency among researchers and journal editors to prefer some outcomes rather than others (e.g., results showing a significant finding), which leads to a problematic bias in the published literature. This can propagate further as literature reviews of claims about support for a hypothesis will themselves be biased if the original literature is contaminated by publication bias. Studies with significant results often do not appear to be superior to studies with a null result with respect to quality of design. However, statistically significant results have been shown to be three times more likely to be published compared to papers with null results.

Driving while black
Driving while black refers to the racial profiling of African American drivers. The phrase implies that a motorist might be pulled over by a police officer, questioned, and searched, because of a racial bias.

Racial profiling
Racial profiling, or ethnic profiling, is the act of suspecting or targeting a person of a certain race on the basis of racially observed characteristics or behavior, rather than on individual suspicion. Racial profiling is commonly referred to regarding its use by law enforcement, and its leading to discrimination against minorities.

Victim blaming
Victim blaming occurs when the victim of a wrongful act is held at fault for the harm that befell them. The study of victimology seeks to mitigate the perception of victims as responsible.

Biases in media
Media bias is the bias or perceived bias of journalists and news producers within the mass media in the selection of events, the stories that are reported, and how they are covered. The term generally implies a pervasive or widespread bias violating the standards of journalism, rather than the perspective of an individual journalist or article. The level of media bias in different nations is debated. There are also watchdog groups that report on media bias.

Practical limitations to media neutrality include the inability of journalists to report all available stories and facts, the requirement that selected facts be linked into a coherent narrative, government influence including overt and covert censorship, the influence of the owners of the news source, concentration of media ownership, the selection of staff, the preferences of an intended audience, and pressure from advertisers.

Bias has been a feature of the mass media since its birth with the invention of the printing press. The expense of early printing equipment restricted media production to a limited number of people. Historians have found that publishers often served the interests of powerful social groups.

Agenda setting
Agenda setting describes the capacity of the media to focus on particular stories, if a news item is covered frequently and prominently, the audience will regard the issue as more important. That is, its salience will increase.

Gatekeeping
Gatekeeping is the way in which information and news are filtered to the public, by each person or corporation along the way. It is the "process of culling and crafting countless bits of information into the limited number of messages that reach people every day, and it is the center of the media's role in modern public life. [...] This process determines not only which information is selected, but also what the content and nature of the messages, such as news, will be."

Sensationalism
Sensationalism is when events and topics in news stories and pieces are overhyped to present skewed impressions of events, which may cause a misrepresentation of the truth of a story. Sensationalism may involve reporting about insignificant matters and events, or the presentation of newsworthy topics in a trivial or tabloid manner contrary to the standards of professional journalism.

Educational bias
Bias in education refers to real or perceived bias in the educational system. The content of school textbooks is often the issue of debate, as their target audience is young people, and the term "whitewashing" is used to refer to selective removal of critical or damaging evidence or comment. Religious bias in textbooks is observed in countries where religion plays a dominant role. There can be many forms of educational bias. Some overlooked aspects, occurring especially with the pedagogical circles of public and private schools—sources that are unrelated to fiduciary or mercantile impoverishment which may be unduly magnified—include teacher bias as well as a general bias against women who are going into STEM research.

Inductive bias
Inductive bias occurs within the field of machine learning. In machine learning one seeks to develop algorithms that are able to learn to anticipate a particular output. To accomplish this, the learning algorithm is given training cases that show the expected connection. Then the learner is tested with new examples. Without further assumptions, this problem cannot be solved exactly as unknown situations may not be predictable. The inductive bias of the learning algorithm is the set of assumptions that the learner uses to predict outputs given inputs that it has not encountered. It may bias the learner towards the correct solution, the incorrect, or be correct some of the time. A classical example of an inductive bias is Occam's Razor, which assumes that the simplest consistent hypothesis is the best.

Insider trading
Insider trading is the trading of a public company's stock or other securities (such as bonds or stock options) by individuals with access to non-public information about the company. In various countries, trading based on insider information is illegal because it is seen as unfair to other investors who do not have access to the information as the investor with insider information could potentially make far larger profits that a typical investor could make.

Match fixing
In organized sports, match fixing occurs when a match is played to a completely or partially pre-determined result, violating the rules of the game and often the law. There is a variety of reasons for this, but the most common is in exchange for a payoff from gamblers. Players might also intentionally perform poorly to get an advantage in the future (such as a better draft pick, or an easier opponent in a playoff), or to rig a handicap system. Match-fixing generally refers to fixing the final result of the game. Another form of match-fixing, known as spot-fixing, involves fixing small events within a match which can be gambled upon, but which are unlikely to prove decisive in determining the final result of the game.

Implicit bias
An implicit bias, or implicit stereotype, is the unconscious attribution of particular qualities to a member of a certain social group.

Implicit stereotypes are shaped by experience and based on learned associations between particular qualities and social categories, including race and/or gender. Individuals' perceptions and behaviors can be influenced by the implicit stereotypes they hold, even if they are unaware/unintentionally hold such stereotypes. Implicit bias is an aspect of implicit social cognition: the phenomenon that perceptions, attitudes, and stereotypes operate without conscious intention. For example, researchers may have implicit bias when designing survey questions and as a result, the questions do not produce accurate results or fail to encourage survey participation. The existence of implicit bias is supported by a variety of scientific articles in psychological literature. Implicit stereotype was first defined by psychologists Mahzarin Banaji and Anthony Greenwald in 1995.

title: Bias


text:
User:Rgamble

I'm a Biological Laboratory technician at National Marine Fisheries in Woods Hole, Massachusetts. I work for the Fishery Oceanography Branch.

http://www.renaissoft.com/april/cgi-bin/wiki.pl is a link to a Science Hyperlink textbook that -- April is working on. I'm advertising it because I'm a firm believer that scientists need to work harder to make their fields understandable to those who are not in science.

For those curious why I'm back, just couldn't stand by and watch the defilement of that most holy of topics... fishes.. Will be modifying the horribly outdated and mostly useless entries as time permits.

I do want to make one rant... The main reason I left besides the vandalisms I noticed, is that I have a very low tolerance for intolerance... (I hope that sentence makes sense to people other than myself). I also have a very low tolerance for people who argue against science without understanding it. And one day I found myself defending someone against religious intolerance (I was a very devout Christian once myself, so have empathy) and then finding the person I'd defended threatening to pull scientifically valid information from articles without checking for the validity.

This latter point is something that is a huge pet peeve of mine. Especially on here where the desire is to have factual articles, do _not_ edit something unless you have more information than "Well, all my friends say this..." or "Well, I don't want to believe this..." Further advice. Do not tell others to do the research for you or else you will change the article. Look it up yourself, understand the science involved, otherwise you have no business copyediting it, just as I have no business copyediting Islam beliefs or articles on architecture.

title: User:Rgamble


text:
Bias distortion

In telecommunication, the term bias distortion has the following meanings:

 * 1) Signal distortion resulting from a shift in the bias.
 * 2) In digital signaling, distortion of the signal in which all the significant intervals have uniformly longer or shorter durations than their theoretical durations.

Bias distortion is expressed in percent of the system-specified unit interval.

title: Bias distortion


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Bilateral synchronization

In telecommunication, bilateral synchronization (or bilateral control) is a synchronization control system between exchanges A and B in which the clock at telephone exchange A controls the data received at exchange B and the clock at exchange B controls the data received at exchange A.

Bilateral synchronization is usually implemented by deriving the timing from the incoming bitstream.

Source: from Federal Standard 1037C in support of MIL-STD-188

title: Bilateral synchronization


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Bipolar signal

In telecommunication, a bipolar signal is a signal that may assume either of two polarities, neither of which is zero.

A bipolar signal may have a two-state non-return-to-zero (NRZ) or a three-state return-to-zero (RZ) binary coding scheme.

A bipolar signal is usually symmetrical with respect to zero amplitude, i.e., the absolute values of the positive and negative signal states are nominally equal.

title: Bipolar signal


text:
Bit-count integrity

In telecommunication, the term bit-count integrity (BCI) has the following meanings:

 * 1) In message communications, the preservation of the exact number of bits that are in the original message.
 * 2) In connection-oriented services, preservation of the number of bits per unit time.

Note: Bit-count integrity is not the same as bit integrity, which requires that the delivered bits correspond exactly with the original bits.

Source: from Federal Standard 1037C and from MIL-STD-188

title: Bit-count integrity