First and Second Derivatives of a Periodic Function Using Discrete Fourier Transforms

a

1

1

ω

1

1

a

2

1

ω

2

1

base-2 log of number of sample points

8

derivative order

0

1

2

Consider the family of periodic functions

f(t)=

a

1

sin(

ω

1

t)+

a

2

cos(

ω

2

t)

, where

0≤t≤4π

. This Demonstration applies the discrete Fourier transform to compute the first and second derivatives of

f(t)

. (The derivative order 0 gives the original function.) The derivatives obtained analytically are shown in dashed red, while the numerical solutions are shown in blue. For a large number of sample points, there is close agreement between the two methods.