Sunday, December 6, 2020

Calculus of the Sinc Function

Calculus of the Sinc Function


Introduction and Setup


The unnormalized Sinc function is defined as:


unsinc(x) = sin x / x


And the normalized Sinc function is defined as:


sinc(x) = sin( π x ) / ( π x )


Two things to assume about the sinc function:


1.  The function is not defined at x = 0, and


2.  The function uses radian angle measure.  


(x ≠ 0, and assume radians measure)



Let α be a real constant and define f(x) as:


f(x) = sin( α x ) / (α x)


When α = 1, f(x) becomes unsinc(x).  Likewise, when α = π, f(x) becomes sinc(x).  I am going to analyze this function f(x).


Limit


Limit of sin( α x ) / ( α x )


lim x → 0 ( sin( α x ) / ( α x ) ) → sin 0 / 0 → 0 / 0


This form of improper limit allows us to use L'Hôspital's Rule, which allows us to take the derivative of both the numerator function and denominator function:


lim x → 0 ( α * cos ( α x ) / α ) → lim x → 0 ( cos ( α x ) ) →  cos ( 0 ) → 1


Hence  lim x → 0 ( sin( α x ) / ( α x ) ) → 1


Derivative


Taking the derivative will call for use to use the quotient rule:


d/dx [n(x) / d(x)] = ( n '(x) * d(x) - n(x) * d '(x)) / (d^2(x))


Then:


d/dx [ sin( α x ) / ( α x ) ]:


n(x) = sin ( α x )

n'(x) = α * cos( α x )

d(x) = α * x 

d^2(x) = (α * x)^2

d'(x) = α



d/dx [ sin( α x ) / ( α x ) ]

= [ α * cos( α x ) * α * x - sin( α x ) * α ] / [ α^2 * x^2 ]

= [ α^2 * cos( α x ) * x - sin( α x ) * α ] / [ α^2 * x^2 ]

= [ α * cos( α x ) * x - sin( α x ) ] / [ α * x^2 ]

= cos( α x ) / x^2 - sin( α x ) / (α * x^2)


Integral


The integral of 


∫ sin( α x ) / ( α x ) dx


does not look like it can easily integrated.


Let's use the Taylor Series approach:


sin x = x - x^3 / 3! + x^5 / 5! - x^7 / 7! + x^9 / 9! + . . . 


sin( α x ) =  ( α x ) - ( α x )^3 / 3! + ( α x )^5 / 5! - ( α x )^7 / 7! + ( α x )^9 / 9! + ...


With x≠0


sin( α x ) / ( α x ) 

=  1 - ( α x )^2 / 3! + ( α x )^4 / 5! - ( α x )^6 / 7! + ( α x )^8 / 9! + ...

=  1 -  α^2 * x^2 / 3! + α^4 * x^4 / 5! - α^6 * x^6 / 7! + α^8 * x^8 / 9! + ...



Now integrate the series:


∫ sin( α x ) / ( α x ) dx

=  x - ( α^2 * x^3 ) / (3 * 3!) + ( α^4 * x^5 ) / (5 * 5!) - ( α^6 * x^7 ) / (7 * 7!) +  ( α^8 * x^9 ) / (9 * 9!) + ... + C

 =  x - ( α^2 * x^3 ) / 18 + ( α^4 * x^5 ) / 600 - ( α^6 * x^7 ) / 35280 +  ( α^8 * x^9 ) / 3265920 + ... + C


Eddie


All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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