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Signal-to-noise ratio ( SNR or S/N ) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise . SNR is defined as the ratio of signal power to noise power , often expressed in decibels . A ratio higher than 1:1 (greater than 0 dB) indicates more signal than noise.

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84-523: (Redirected from S/N ) [REDACTED] Look up S/N  or s/n in Wiktionary, the free dictionary. S/n , S/N or s/n may refer to: Signal-to-noise ratio , a measure in science and engineering Screen name (computing) , of a computer user Serial number , a unique identifier See also [ edit ] SN (disambiguation) Signal-to-noise (disambiguation) Topics referred to by

168-521: A b x f ( x ) d x = ∫ a b x x 2 + π 2 d x = 1 2 ln ⁡ b 2 + π 2 a 2 + π 2 . {\displaystyle \int _{a}^{b}xf(x)\,dx=\int _{a}^{b}{\frac {x}{x^{2}+\pi ^{2}}}\,dx={\frac {1}{2}}\ln {\frac {b^{2}+\pi ^{2}}{a^{2}+\pi ^{2}}}.} The limit of this expression as

252-401: A weighted average of the x i values, with weights given by their probabilities p i . In the special case that all possible outcomes are equiprobable (that is, p 1 = ⋅⋅⋅ = p k ), the weighted average is given by the standard average . In the general case, the expected value takes into account the fact that some outcomes are more likely than others. Informally,

336-421: A → −∞ and b → ∞ does not exist: if the limits are taken so that a = − b , then the limit is zero, while if the constraint 2 a = − b is taken, then the limit is ln(2) . To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely , with E[ X ] left undefined otherwise. However, measure-theoretic notions as given below can be used to give

420-512: A finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration . In the axiomatic foundation for probability provided by measure theory , the expectation is given by Lebesgue integration . The expected value of a random variable X is often denoted by E( X ) , E[ X ] , or E X , with E also often stylized as E {\displaystyle \mathbb {E} } or E . The idea of

504-540: A given channel, which depends on its bandwidth and SNR. This relationship is described by the Shannon–Hartley theorem , which is a fundamental law of information theory. SNR can be calculated using different formulas depending on how the signal and noise are measured and defined. The most common way to express SNR is in decibels, which is a logarithmic scale that makes it easier to compare large or small values. Other definitions of SNR may use different factors or bases for

588-451: A million times stronger. When the signal is constant or periodic and the noise is random, it is possible to enhance the SNR by averaging the measurements. In this case the noise goes down as the square root of the number of averaged samples. When a measurement is digitized, the number of bits used to represent the measurement determines the maximum possible signal-to-noise ratio. This is because

672-488: A multidimensional random variable, i.e. a random vector X . It is defined component by component, as E[ X ] i = E[ X i ] . Similarly, one may define the expected value of a random matrix X with components X ij by E[ X ] ij = E[ X ij ] . Consider a random variable X with a finite list x 1 , ..., x k of possible outcomes, each of which (respectively) has probability p 1 , ..., p k of occurring. The expectation of X

756-445: A perfect input signal. If the input signal is already noisy (as is usually the case), the signal's noise may be larger than the quantization noise. Real analog-to-digital converters also have other sources of noise that further decrease the SNR compared to the theoretical maximum from the idealized quantization noise, including the intentional addition of dither . Although noise levels in a digital system can be expressed using SNR, it

840-462: A quantity proportional to power, as shown below: The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the strongest un- distorted signal on a channel and the minimum discernible signal, which for most purposes is the noise level. SNR measures the ratio between an arbitrary signal level (not necessarily the most powerful signal possible) and noise. Measuring signal-to-noise ratios requires

924-650: A random variable ( S ) to random noise N is: S N R = E [ S 2 ] E [ N 2 ] , {\displaystyle \mathrm {SNR} ={\frac {\mathrm {E} [S^{2}]}{\mathrm {E} [N^{2}]}}\,,} where E refers to the expected value , which in this case is the mean square of N . If the signal is simply a constant value of s , this equation simplifies to: S N R = s 2 E [ N 2 ] . {\displaystyle \mathrm {SNR} ={\frac {s^{2}}{\mathrm {E} [N^{2}]}}\,.} If

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1008-570: A real number μ {\displaystyle \mu } if and only if the two surfaces in the x {\displaystyle x} - y {\displaystyle y} -plane, described by x ≤ μ , 0 ≤ y ≤ F ( x ) or x ≥ μ , F ( x ) ≤ y ≤ 1 {\displaystyle x\leq \mu ,\;\,0\leq y\leq F(x)\quad {\text{or}}\quad x\geq \mu ,\;\,F(x)\leq y\leq 1} respectively, have

1092-455: A small circle of mutual scientific friends in Paris about it. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657) ) " De ratiociniis in ludo aleæ " on probability theory just after visiting Paris. The book extended

1176-611: A systematic definition of E[ X ] for more general random variables X . All definitions of the expected value may be expressed in the language of measure theory . In general, if X is a real-valued random variable defined on a probability space (Ω, Σ, P) , then the expected value of X , denoted by E[ X ] , is defined as the Lebesgue integral E ⁡ [ X ] = ∫ Ω X d P . {\displaystyle \operatorname {E} [X]=\int _{\Omega }X\,d\operatorname {P} .} Despite

1260-507: A value in any given open interval is given by the integral of f over that interval. The expectation of X is then given by the integral E ⁡ [ X ] = ∫ − ∞ ∞ x f ( x ) d x . {\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }xf(x)\,dx.} A general and mathematically precise formulation of this definition uses measure theory and Lebesgue integration , and

1344-469: A variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or E {\displaystyle \mathbb {E} } (in blackboard bold ), while a variety of bracket notations (such as E( X ) , E[ X ] , and E X ) are all used. Another popular notation is μ X . ⟨ X ⟩ , ⟨ X ⟩ av , and X ¯ {\displaystyle {\overline {X}}} are commonly used in physics. M( X )

1428-399: A very wide dynamic range , signals are often expressed using the logarithmic decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as and In a similar manner, SNR may be expressed in decibels as Using the definition of SNR Using the quotient rule for logarithms Substituting the definitions of SNR, signal, and noise in decibels into

1512-640: Is a Borel function ), we can use this inversion formula to obtain E ⁡ [ g ( X ) ] = 1 2 π ∫ R g ( x ) [ ∫ R e − i t x φ X ( t ) d t ] d x . {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\left[\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt\right]dx.} If E ⁡ [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]}

1596-415: Is a generalization of the weighted average . Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality. The expected value of a random variable with

1680-403: Is a uniformly distributed random signal with a peak-to-peak amplitude of one quantization level, making the amplitude ratio 2 /1. The formula is then: This relationship is the origin of statements like " 16-bit audio has a dynamic range of 96 dB". Each extra quantization bit increases the dynamic range by roughly 6 dB. Assuming a full-scale sine wave signal (that is, the quantizer

1764-541: Is any random variable with finite expectation, then Markov's inequality may be applied to the random variable | X −E[ X ]| to obtain Chebyshev's inequality P ⁡ ( | X − E [ X ] | ≥ a ) ≤ Var ⁡ [ X ] a 2 , {\displaystyle \operatorname {P} (|X-{\text{E}}[X]|\geq a)\leq {\frac {\operatorname {Var} [X]}{a^{2}}},} where Var

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1848-462: Is as in the previous example. A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. The probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function φ X {\displaystyle \varphi _{X}} by

1932-635: Is called the probability density function of X (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician , it follows that E ⁡ [ X ] ≡ ∫ Ω X d P = ∫ R x f ( x ) d x {\displaystyle \operatorname {E} [X]\equiv \int _{\Omega }X\,d\operatorname {P} =\int _{\mathbb {R} }xf(x)\,dx} for any absolutely continuous random variable X . The above discussion of continuous random variables

2016-431: Is clear and easy to detect or interpret, while a low SNR means that the signal is corrupted or obscured by noise and may be difficult to distinguish or recover. SNR can be improved by various methods, such as increasing the signal strength, reducing the noise level, filtering out unwanted noise, or using error correction techniques. SNR also determines the maximum possible amount of data that can be transmitted reliably over

2100-408: Is defined as E ⁡ [ X ] = x 1 p 1 + x 2 p 2 + ⋯ + x k p k . {\displaystyle \operatorname {E} [X]=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.} Since the probabilities must satisfy p 1 + ⋅⋅⋅ + p k = 1 , it is natural to interpret E[ X ] as

2184-476: Is designed such that it has the same minimum and maximum values as the input signal), the quantization noise approximates a sawtooth wave with peak-to-peak amplitude of one quantization level and uniform distribution. In this case, the SNR is approximately Expected value In probability theory , the expected value (also called expectation , expectancy , expectation operator , mathematical expectation , mean , expectation value , or first moment )

2268-399: Is different from Wikidata All article disambiguation pages All disambiguation pages Signal-to-noise ratio SNR is an important parameter that affects the performance and quality of systems that process or transmit signals, such as communication systems , audio systems , radar systems , imaging systems , and data acquisition systems. A high SNR means that the signal

2352-407: Is easily obtained by setting Y 0 = X 1 {\displaystyle Y_{0}=X_{1}} and Y n = X n + 1 − X n {\displaystyle Y_{n}=X_{n+1}-X_{n}} for n ≥ 1 , {\displaystyle n\geq 1,} where X n {\displaystyle X_{n}}

2436-433: Is employed to characterize sensitivity of imaging systems; see Signal-to-noise ratio (imaging) . Related measures are the " contrast ratio " and the " contrast-to-noise ratio ". Channel signal-to-noise ratio is given by where W is the bandwidth and k a {\displaystyle k_{a}} is modulation index Output signal-to-noise ratio (of AM receiver) is given by Channel signal-to-noise ratio

2520-564: Is equivalent to the representation E ⁡ [ X ] = ∫ 0 ∞ ( 1 − F ( x ) ) d x − ∫ − ∞ 0 F ( x ) d x , {\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,} also with convergent integrals. Expected values as defined above are automatically finite numbers. However, in many cases it

2604-436: Is finite if and only if E[ X ] and E[ X ] are both finite. Due to the formula | X | = X + X , this is the case if and only if E| X | is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations. The following table gives

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2688-626: Is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem , E ⁡ [ g ( X ) ] = 1 2 π ∫ R G ( t ) φ X ( t ) d t , {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,} where G ( t ) = ∫ R g ( x ) e − i t x d x {\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}

2772-1052: Is fundamental to be able to consider expected values of ±∞ . This is intuitive, for example, in the case of the St. Petersburg paradox , in which one considers a random variable with possible outcomes x i = 2 , with associated probabilities p i = 2 , for i ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has E ⁡ [ X ] = ∑ i = 1 ∞ x i p i = 2 ⋅ 1 2 + 4 ⋅ 1 4 + 8 ⋅ 1 8 + 16 ⋅ 1 16 + ⋯ = 1 + 1 + 1 + 1 + ⋯ . {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i}=2\cdot {\frac {1}{2}}+4\cdot {\frac {1}{4}}+8\cdot {\frac {1}{8}}+16\cdot {\frac {1}{16}}+\cdots =1+1+1+1+\cdots .} It

2856-399: Is given by Output signal-to-noise ratio is given by All real measurements are disturbed by noise. This includes electronic noise , but can also include external events that affect the measured phenomenon — wind, vibrations, the gravitational attraction of the moon, variations of temperature, variations of humidity, etc., depending on what is measured and of the sensitivity of the device. It

2940-407: Is more common to use E b /N o , the energy per bit per noise power spectral density. The modulation error ratio (MER) is a measure of the SNR in a digitally modulated signal. For n -bit integers with equal distance between quantization levels ( uniform quantization ) the dynamic range (DR) is also determined. Assuming a uniform distribution of input signal values, the quantization noise

3024-467: Is natural to say that the expected value equals +∞ . There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any nonnegative random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then

3108-411: Is often possible to reduce the noise by controlling the environment. Internal electronic noise of measurement systems can be reduced through the use of low-noise amplifiers . When the characteristics of the noise are known and are different from the signal, it is possible to use a filter to reduce the noise. For example, a lock-in amplifier can extract a narrow bandwidth signal from broadband noise

3192-407: Is only an approximation since E ⁡ [ X 2 ] = σ 2 + μ 2 {\displaystyle \operatorname {E} \left[X^{2}\right]=\sigma ^{2}+\mu ^{2}} . It is commonly used in image processing , where the SNR of an image is usually calculated as the ratio of the mean pixel value to the standard deviation of

3276-637: Is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables. The following three inequalities are of fundamental importance in the field of mathematical analysis and its applications to probability theory. The Hölder and Minkowski inequalities can be extended to general measure spaces , and are often given in that context. By contrast,

3360-593: Is the Fourier transform of g ( x ) . {\displaystyle g(x).} The expression for E ⁡ [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]} also follows directly from the Plancherel theorem . The expectation of a random variable plays an important role in a variety of contexts. In statistics , where one seeks estimates for unknown parameters based on available data gained from samples ,

3444-478: Is the variance . These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than

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3528-1673: Is then natural to define: E ⁡ [ X ] = { E ⁡ [ X + ] − E ⁡ [ X − ] if  E ⁡ [ X + ] < ∞  and  E ⁡ [ X − ] < ∞ ; + ∞ if  E ⁡ [ X + ] = ∞  and  E ⁡ [ X − ] < ∞ ; − ∞ if  E ⁡ [ X + ] < ∞  and  E ⁡ [ X − ] = ∞ ; undefined if  E ⁡ [ X + ] = ∞  and  E ⁡ [ X − ] = ∞ . {\displaystyle \operatorname {E} [X]={\begin{cases}\operatorname {E} [X^{+}]-\operatorname {E} [X^{-}]&{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\+\infty &{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\-\infty &{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty ;\\{\text{undefined}}&{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty .\end{cases}}} According to this definition, E[ X ] exists and

3612-514: Is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable. The expected value of any real-valued random variable X {\displaystyle X} can also be defined on the graph of its cumulative distribution function F {\displaystyle F} by a nearby equality of areas. In fact, E ⁡ [ X ] = μ {\displaystyle \operatorname {E} [X]=\mu } with

3696-458: Is used in Russian-language literature. As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider

3780-412: Is usually not included while measuring power or energy of a signal. This may cause some confusion among readers, but the resistance factor is not significant for typical operations performed in signal processing, or for computing power ratios. For most cases, the power of a signal would be considered to be simply An alternative definition of SNR is as the reciprocal of the coefficient of variation , i.e.,

3864-403: Is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2. More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract " Théorie analytique des probabilités ", where the concept of expected value was defined explicitly: ... this advantage in

3948-408: The sample mean serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased ; that is, the expected value of the estimate is equal to the true value of the underlying parameter. For a different example, in decision theory , an agent making an optimal choice in

4032-527: The Jensen inequality is special to the case of probability spaces. In general, it is not the case that E ⁡ [ X n ] → E ⁡ [ X ] {\displaystyle \operatorname {E} [X_{n}]\to \operatorname {E} [X]} even if X n → X {\displaystyle X_{n}\to X} pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on

4116-398: The Lebesgue theory of expectation is identical to the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable X is said to be absolutely continuous if any of the following conditions are satisfied: These conditions are all equivalent, although this is nontrivial to establish. In this definition, f

4200-446: The above equation results in an important formula for calculating the signal to noise ratio in decibels, when the signal and noise are also in decibels: In the above formula, P is measured in units of power, such as watts (W) or milliwatts (mW), and the signal-to-noise ratio is a pure number. However, when the signal and noise are measured in volts (V) or amperes (A), which are measures of amplitude, they must first be squared to obtain

4284-507: The concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability . In the foreword to his treatise, Huygens wrote: It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me

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4368-462: The corresponding theory of absolutely continuous random variables is described in the next section. The density functions of many common distributions are piecewise continuous , and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration . Sometimes continuous random variables are defined as those corresponding to this special class of densities, although

4452-490: The distinct case of random variables dictated by (piecewise-)continuous probability density functions , as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of measure theory and Lebesgue integration , which provide these different contexts with an axiomatic foundation and common language. Any definition of expected value may be extended to define an expected value of

4536-527: The expectation of a random variable with a countably infinite set of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that E ⁡ [ X ] = ∑ i = 1 ∞ x i p i , {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i},} where x 1 , x 2 , ... are

4620-492: The expected value can be defined as +∞ . The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable X , one defines the positive and negative parts by X = max( X , 0) and X = −min( X , 0) . These are nonnegative random variables, and it can be directly checked that X = X − X . Since E[ X ] and E[ X ] are both then defined as either nonnegative numbers or +∞ , it

4704-494: The expected value operator is not σ {\displaystyle \sigma } -additive, i.e. E ⁡ [ ∑ n = 0 ∞ Y n ] ≠ ∑ n = 0 ∞ E ⁡ [ Y n ] . {\displaystyle \operatorname {E} \left[\sum _{n=0}^{\infty }Y_{n}\right]\neq \sum _{n=0}^{\infty }\operatorname {E} [Y_{n}].} An example

4788-573: The expected value originated in the middle of the 17th century from the study of the so-called problem of points , which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics

4872-488: The expected values of some commonly occurring probability distributions . The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references. The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral . Note that

4956-401: The honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from

5040-1193: The indicator function of the event A . {\displaystyle A.} Then, it follows that X n → 0 {\displaystyle X_{n}\to 0} pointwise. But, E ⁡ [ X n ] = n ⋅ Pr ( U ∈ [ 0 , 1 n ] ) = n ⋅ 1 n = 1 {\displaystyle \operatorname {E} [X_{n}]=n\cdot \Pr \left(U\in \left[0,{\tfrac {1}{n}}\right]\right)=n\cdot {\tfrac {1}{n}}=1} for each n . {\displaystyle n.} Hence, lim n → ∞ E ⁡ [ X n ] = 1 ≠ 0 = E ⁡ [ lim n → ∞ X n ] . {\displaystyle \lim _{n\to \infty }\operatorname {E} [X_{n}]=1\neq 0=\operatorname {E} \left[\lim _{n\to \infty }X_{n}\right].} Analogously, for general sequence of random variables { Y n : n ≥ 0 } , {\displaystyle \{Y_{n}:n\geq 0\},}

5124-399: The infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable does not have finite expectation. Now consider a random variable X which has a probability density function given by a function f on the real number line . This means that the probability of X taking on

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5208-552: The inversion formula: f X ( x ) = 1 2 π ∫ R e − i t x φ X ( t ) d t . {\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.} For the expected value of g ( X ) {\displaystyle g(X)} (where g : R → R {\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}

5292-447: The letters "a.s." stand for " almost surely "—a central property of the Lebesgue integral. Basically, one says that an inequality like X ≥ 0 {\displaystyle X\geq 0} is true almost surely, when the probability measure attributes zero-mass to the complementary event { X < 0 } . {\displaystyle \left\{X<0\right\}.} Concentration inequalities control

5376-447: The likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable X and any positive number a , it states that P ⁡ ( X ≥ a ) ≤ E ⁡ [ X ] a . {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} [X]}{a}}.} If X

5460-419: The logarithm, depending on the context and application. One definition of signal-to-noise ratio is the ratio of the power of a signal (meaningful input) to the power of background noise (meaningless or unwanted input): where P is average power. Both signal and noise power must be measured at the same or equivalent points in a system, and within the same system bandwidth . The signal-to-noise ratio of

5544-401: The minimum possible noise level is the error caused by the quantization of the signal, sometimes called quantization noise . This noise level is non-linear and signal-dependent; different calculations exist for different signal models. Quantization noise is modeled as an analog error signal summed with the signal before quantization ("additive noise"). This theoretical maximum SNR assumes

5628-432: The newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values,

5712-422: The noise has expected value of zero, as is common, the denominator is its variance , the square of its standard deviation σ N . The signal and the noise must be measured the same way, for example as voltages across the same impedance . Their root mean squares can alternatively be used according to: where A is root mean square (RMS) amplitude (for example, RMS voltage). Because many signals have

5796-504: The noise level to 1 (0 dB) and measuring how far the signal 'stands out'. In physics, the average power of an AC signal is defined as the average value of voltage times current; for resistive (non- reactive ) circuits, where voltage and current are in phase, this is equivalent to the product of the rms voltage and current: But in signal processing and communication, one usually assumes that R = 1 Ω {\displaystyle R=1\Omega } so that factor

5880-434: The noise standard deviation σ {\displaystyle \sigma } does not change between the two states. The Rose criterion (named after Albert Rose ) states that an SNR of at least 5 is needed to be able to distinguish image features with certainty. An SNR less than 5 means less than 100% certainty in identifying image details. Yet another alternative, very specific, and distinct definition of SNR

5964-402: The pixel values over a given neighborhood. Sometimes SNR is defined as the square of the alternative definition above, in which case it is equivalent to the more common definition : This definition is closely related to the sensitivity index or d ' , when assuming that the signal has two states separated by signal amplitude μ {\displaystyle \mu } , and

6048-497: The possible outcomes of the random variable X and p 1 , p 2 , ... are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context. However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the Riemann series theorem of mathematical analysis illustrates that

6132-672: The random variables. To see this, let U {\displaystyle U} be a random variable distributed uniformly on [ 0 , 1 ] . {\displaystyle [0,1].} For n ≥ 1 , {\displaystyle n\geq 1,} define a sequence of random variables X n = n ⋅ 1 { U ∈ ( 0 , 1 n ) } , {\displaystyle X_{n}=n\cdot \mathbf {1} \left\{U\in \left(0,{\tfrac {1}{n}}\right)\right\},} with 1 { A } {\displaystyle \mathbf {1} \{A\}} being

6216-455: The ratio of mean to standard deviation of a signal or measurement: where μ {\displaystyle \mu } is the signal mean or expected value and σ {\displaystyle \sigma } is the standard deviation of the noise, or an estimate thereof. Notice that such an alternative definition is only useful for variables that are always non-negative (such as photon counts and luminance ), and it

6300-438: The same finite area, i.e. if ∫ − ∞ μ F ( x ) d x = ∫ μ ∞ ( 1 − F ( x ) ) d x {\displaystyle \int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx} and both improper Riemann integrals converge. Finally, this

6384-449: The same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed

6468-418: The same principle. But finally I have found that my answers in many cases do not differ from theirs. In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables . Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: That any one Chance or Expectation to win any thing

6552-403: The same term [REDACTED] This disambiguation page lists articles associated with the title S/n . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=S/n&oldid=1013667907 " Category : Disambiguation pages Hidden categories: Short description

6636-433: The selection of a representative or reference signal. In audio engineering , the reference signal is usually a sine wave at a standardized nominal or alignment level , such as 1 kHz at +4 dBu (1.228 V RMS ). SNR is usually taken to indicate an average signal-to-noise ratio, as it is possible that instantaneous signal-to-noise ratios will be considerably different. The concept can be understood as normalizing

6720-505: The sum hoped for. We will call this advantage mathematical hope. The use of the letter E to denote "expected value" goes back to W. A. Whitworth in 1901. The symbol has since become popular for English writers. In German, E stands for Erwartungswert , in Spanish for esperanza matemática , and in French for espérance mathématique. When "E" is used to denote "expected value", authors use

6804-460: The term is used differently by various authors. Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of X is given by the Cauchy distribution Cauchy(0, π) , so that f ( x ) = ( x + π ) . It is straightforward to compute in this case that ∫

6888-411: The theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for

6972-411: The value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely. For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely , which implies that

7056-434: Was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. He began to discuss the problem in the famous series of letters to Pierre de Fermat . Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on

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