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# Qfunction - The Complementary Unit Gaussian Distribution...

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The Complementary Unit Gaussian Distribution Function Q ( x ) Let φ ( u ), where φ ( u ) = 1 2 π exp - u 2 2 , denote the probability density function (pdf) of a standard (or unit) Gaussian random variable. The cumulative probability distribution function (CDF) of this random variable is denoted Φ( x ) where Φ( x ) = Z x -∞ φ ( u ) du = Z x -∞ 1 2 π exp - u 2 2 du. Φ( x ) is also known as the unit Gaussian distribution function and Q ( x ) = 1 - Φ( x ) = Z x φ ( u ) du = Z x 1 2 π exp - u 2 2 du is called the complementary unit Gaussian distribution function. Note that since Φ( x ) is a monotone increasing function rising from 0 at -∞ to 1 at , Q ( x ) is a monotone decreasing function falling from 1 at -∞ to 0 at . In many applications in communications and signal processing, Q ( x ) is slightly more convenient to use than Φ( x ). For example, the bit error rate in some communication systems can be expressed as Q ( SNR) where SNR is the signal-to-noise ratio . Since Q ( x ) is a decreasing function of its argument, maximizing the SNR is an important objective in communication system design.

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Qfunction - The Complementary Unit Gaussian Distribution...

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