The probability density function for the output of an analog cross-correlator with correlated bandpass inputs

Miller, Leonard E.; Lee, Jhong

doi:10.1109/tit.1974.1055257

Cited by 25 publications

(5 citation statements)

References 26 publications

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“…From (4), (13), and (14), it can be concluded that the conditional random variable is the difference of two non-central chi-square variables. In these circumstances, as the cross-correlation coefficient between the noise terms and is equal to , the non-central chi-squares are independently distributed [37], [38], and in this way the conditional MGF of , can be written as the product of the MGFs of the conditional random variables and . As a consequence, the unconditional MGF of is given by (24) where (25) (26) with (27) with denoting the modified Bessel function of the first kind of order zero and , whereas is given by (27) with replaced by and replaced by , and…”

Section: A Moment Generating Function Derivationmentioning

confidence: 99%

Estimating the Performance of Direct-Detection DPSK in Optical Networking Environments Using Eigenfunction Expansion Techniques

Pires

Cancela

2010

J. Lightwave Technol.

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Abstract-In-band crosstalk, due to multiple interferers, has been identified as one of the most severe impairments in optical transparent networks, especially in the ones with a large number of nodes and a high wavelength density. Due to its robustness to in-band crosstalk differential phase-shift keying (DPSK) emerges as an attractive modulation scheme to be used in such environments. This paper proposes a rigorous formulation to estimate the performance of direct-detection DPSK receivers using an eigenfunction expansion technique in the time domain. The method takes into account both the in-band crosstalk, due to an arbitrary number of interfering terms, and the amplified spontaneous emission noise and is able to deal with any combination of optical and electrical filter shapes. Using this method the accuracy of an approximation based on the wideband optical filtering assumption was evaluated and shown that the approximation, although not providing reliable results for the error probabilities, can be used with confidence to compute power penalties due to in-band crosstalk. Furthermore, the crosstalk tolerance of DPSK over on-off keying was quantified and shown that this tolerance is reduced when the number of interferers increases.Index Terms-Differential phase-shift keying, eigenvalues and eigenfunctions, in-band crosstalk, optical networking.

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Section: A Moment Generating Function Derivationmentioning

confidence: 99%

Estimating the Performance of Direct-Detection DPSK in Optical Networking Environments Using Eigenfunction Expansion Techniques

Pires

Cancela

2010

J. Lightwave Technol.

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“…The purpose of this paper is to extend the results given in [ 1 ] when there is a bandwidth restriction on the signal design. Under a bandwidth restriction, the block orthogonal inner code may not be an efficient code in many cases.…”

Section: Introductionmentioning

confidence: 90%

The Probability Distribution of the Sum of N Sampled Baseband Outputs of a Product Detector Whose Inputs are Equal Power Coherent Narrow-Band Signals in a Gaussian Channel

Wilson

1979

IEEE Trans. Commun.

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The Probability Distribution of the Sum of N Sampled may then be written Baseband Outputs of a Product Detector Whose Inputs are Equal Power Coherent Narrow-Band z ( t ) = a 2 + a { n l ( t ) + n 3 ( t ) } + n l ( t ) n 3 ( t ) ROLAND WILSONThis output is sampled at times t = jT, j = 0, 1, 2, such that the noise samples are uncorrelated, i.e.Abstruct-General expressions for the probability distribution and density of the sum of N low-pass filtered outputs of a multiplier whose E{ni(t)nj(t + n~) }= 6, n 0 inputs are coherent, equal power noisy narrowband signals are derived in a finite series form. Graphs of the density for several values of N and two signal-noise ratios are presented and discussed.

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“…The P FA and P D expressions in (7) and (8) lend themselves to the evaluation of receiver operation characteristic (ROC) curves for product detectors such as coprime array and nested array while the P FA and P D expressions in (9) and (10) facilitate the computation of the ROC curves for CBF ULA and CBF non-uniform linear array (NULA). Fig.…”

Section: A Receiver Operation Characteristicmentioning

confidence: 99%

“…Hence, even though there is an interferer in the scenario, it is negligible because of its low power. Both coprime and nested arrays' ROCs were evaluated using (7) and (8) with the global adaptive quadrature method of numerical integration [37]. The green dash-dot line represents the standard ULA ROC curve evaluated using the analytical expressions for P FA,c and P D,c in (9) and (10) while the purple dot line represents the ROC curve for the non-uniform linear array (NULA) that has the same sensor locations as the coprime array but uses CBF processing.…”

Section: A Receiver Operation Characteristicmentioning

confidence: 99%

“…Several researchers have pointed out that while product processing can offer improved resolution over CBF processing for a given number of sensors, the product beampattern suffers from higher side lobes and reduction in array gain [2]- [6], [11]. Miller and Lee considered a special case of product arrays where a linear array is split into two subarrays that are joined end to end and derived the probability density functions (PDFs) of the product output for plane wave signals in presence of additive Gaussian noise [7], [8]. Miller and Lee compared detection performance of CBF and product processing for the specific product arrays…”

Section: Introductionmentioning

confidence: 99%

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Gaussian Signal Detection With Product Arrays

Adhikari

Buck

2020

IEEE Access

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The paper specifies the probability density function (PDF) for the detection statistic for a product processor of colinear arrays. The product processor's detection PDF is a scaled product of the detection statistic with modified Bessel functions. Using the PDF, this research compares the product processor's detection performance against a conventional beamforming (CBF) linear array with an equal number of sensors. For the basic detection case of a single known signal in Gaussian noise with a single Gaussian plane wave interferer, receiver operation characteristics curves and mean discriminating information over a range of interferer power illustrate that the product processor's performance is inferior to the CBF detector with an equal number of sensors over all possible interferer locations. The detection performance of a product processor matches or outperforms the CBF detector only for high power interferers in specific locations.

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The probability density function for the output of an analog cross-correlator with correlated bandpass inputs

Cited by 25 publications

References 26 publications

Estimating the Performance of Direct-Detection DPSK in Optical Networking Environments Using Eigenfunction Expansion Techniques

Estimating the Performance of Direct-Detection DPSK in Optical Networking Environments Using Eigenfunction Expansion Techniques

The Probability Distribution of the Sum of N Sampled Baseband Outputs of a Product Detector Whose Inputs are Equal Power Coherent Narrow-Band Signals in a Gaussian Channel

Gaussian Signal Detection With Product Arrays

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