Example_Real_application_Joint_pdf

# Example_Real_application_Joint_pdf - is the n th real root...

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EEE 554 Random Signal Theory Real Application Example: Random Variable Transformation Question: A difuse sound Feld is a a Feld in which re±ected waves, arriving equally ²rom all directions, combine to produce uni²orm sound pressure at all points within the Feld. The arrival angle Θ is thus uni²ormly distributed, Θ U ( - π, π ). ³or sound traveling as a plane-wave and arriving ²rom a given angle between two sensors, the time delay T o² the signal between the two sensors is given by T = ( d/c ) cos(Θ), where d is the distance and c is the speed o² sound. Note that θ = 0 is end-Fre arrival and θ = π/ 2 is broadside arrival. ³ind the probability density ²unction f T ( t ), o² the delay o² the difuse sound Feld. Solution: ³or the trans²ormation T = g (Θ) = ( d/c ) cos( θ ), the probability density ²unction o² T in terms o² the probability density ²unction o² Θ is f T ( t ) = s n f Θ ( θ n ) / | g ( θ n ) | , (1) where θ n
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Unformatted text preview: is the n th real root o² t = g ( θ ) and g ′ ( θ ) = dg ( θ ) dθ . We consider two possible regions o² t values: • When | t | > d/c , the ²unction t = ( d/c ) cos( θ ) has no solutions; hence f T ( t ) = 0. • When | t | < d/c , and-π < θ < π , the ²unction t = ( d/c ) cos( θ ) has two solutions, θ 1 = arccos( c t/d ) , θ 2 = 2 π-arccos( c t/d ) . (2) The derivative o² g ( θ ) is given by g ′ ( θ ) = (-d/c ) sin( θ ). So, using (2), the denominator o² (1) becomes | g ′ ( θ n ) | = r ( d/c ) 2-T 2 , n = 1 , 2. Also, f Θ ( θ ) = 1 / 2 π within-π to π . Thus, (1) becomes f T ( t ) = 2 2 π r ( d/c ) 2-t 2 , | t | < d/c. The resulting probability density ²unction o² T is given by: f T ( t ) = 2 2 π r ( d/c ) 2-t 2 , | t | < d/c , | t | > d/c 1...
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## This note was uploaded on 01/26/2011 for the course EEE 554 taught by Professor Duman during the Spring '10 term at ASU.

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