Computation of the incomplete gamma function ratios and their inverse

DiDonato, A. R.; Morris, Alfred H.

doi:10.1145/22721.23109

Cited by 73 publications

(50 citation statements)

References 7 publications

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“…In [4] several algorithms are discussed. In [5] an algorithm in Fortran is given for the incomplete gamma functions and their inverses. In [6] asymptotic methods are used; in our notation: (1.3) is considered for small values of q, with a fixed.…”

Section: Numerical Examplesmentioning

confidence: 99%

Asymptotic Inversion of Incomplete Gamma Functions

Temme¹

1992

Mathematics of Computation

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Abstract. The normalized incomplete gamma functions P(a, x) and Q(a, x) are inverted for large values of the parameter a . That is, x-solutions of the equations P(a,x)=p, Q(a,x) = q, p e [0, 1], q = 1 -p, are considered, especially for large values of a . The approximations are obtained by using uniform asymptotic expansions of the incomplete gamma functions in which an error function is the dominant term. The inversion problem is started by inverting this error function term. Numerical results indicate that for obtaining an accuracy of four correct digits, the method can already be used for a = 2, although a is a large parameter. It is indicated that the method can be applied to other cumulative distribution functions.

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Section: Numerical Examplesmentioning

confidence: 99%

Asymptotic Inversion of Incomplete Gamma Functions

Temme¹

1992

Mathematics of Computation

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“…where λ * represents the optimum energy threshold (on the false alarm probability) for a predetermined target τ , while P −1 f (·) denotes the inverse function of P f (·), which can be efficiently calculated by using well-known inverse algorithms, e.g., [13]. In the sequel, the online detection probability can be directly computed by calculating P d (λ * ) using (20).…”

Section: B False Alarm Probability and Threshold Designmentioning

confidence: 99%

Simultaneous Spectrum Sensing and Data Transmission for Multi-User MIMO Cognitive Radio Systems

Miridakis

Tsiftsis²,

Alexandropoulos

et al. 2016

2016 IEEE Global Communications Conference (GLOBECOM)

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Abstract-We present a multi-user multiple-input multipleoutput (MIMO) cognitive radio system consisting of a secondary receiver that deploys spatial multiplexing to decode signals from multiple secondary transmitters, under the presence of primary transmissions. The secondary receiver carries out minimum mean-squared error detection to decode the secondary data streams, while it performs spectrum sensing at the remaining signal to capture the potential presence of primary activity. Assuming Rayleigh fading as well as the realistic cases of channel fading time variation and channel estimation errors, we present novel closed-form expressions for important system measures, namely, the detection and false-alarm probabilities as well as the transmission power of the secondary nodes. The enclosed numerical results verify the accuracy of the presented analysis.

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“…In the case where , is (24) This integral can be expressed using the incomplete gamma function denoted by . The false alarm rate in this situation is (25) An exact determination of the test threshold that achieves a desired false alarm rate requires inversion of (25). In situations where the false alarm rate is always fixed to a single value, a lookup table for versus may be appropriate.…”

Section: ) Properties Of the Likelihood Ratio Formentioning

confidence: 99%

“…If the application scenario requires that vary over time, the detector must be able to calculate the threshold numerically. Techniques to perform this calculation are described in [25].…”

Section: ) Properties Of the Likelihood Ratio Formentioning

confidence: 99%