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


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