Mathematical Preliminaries

 [Fourier Series and Boundary Value Problems,R.V. Churchill, McGraw-Hill, 1941]

The following is a sketch of the mathematicians steps from the definition of a Fourier series defined on the range -π to π to the starting definition of Fourier Integrals in Press's Numerical Recipes.

 

Section 26 definition of a Fourier Series

 Eqn 1

is a Fourier series provided

        Eqn 2

Section 30.  Other Forms of Fourier Series

For F(z) defined in the interval (-π,π) the Fourier series is

Eqn 3

Substituting  and writing f(x) for F(πx/L)

  Eqn 4

Now moving to section 40 and noting that

 

this can be written as

  Eqn 5

The convergence of the series is to  

     Assuming that the integral of f(x) converges for large L, the division by L makes the first term go to zero, as L becomes very large.  Putting , the remaining terms can be written

  Eqn 6

The last series has the form of a trap rule integral of α = nΔα

so that

   Eqn 7

Now write the cosine in exponential form

 Eqn 8

where in the last step α takes on negative values.

- Bob's derivations begin here

Sine and Cosine Transforms

     The large L limit could have been taken without converting to a cos(x-x') form.  Start from Equation 4 above

and again drop the first term as L becomes very large

   Eqn 9

Now define

       Eqn 10

Then

   Eqn 11

Phase Corrected form

     Convert the equations to h(t) and H(f).

     Eqn 12

and define

              Eqn 13

so that

       Eqn 14

then also define (f_n) such that

      Eqn 15

so that

     Eqn 16

which is also equal to

           Eqn 17

The only problem with this is that  is subjected to sudden changes when parts of |B| switch signs.  Noting that

 and using it to keep  continuous, the phase corrected form of the transform ready to be switched to an integral is

            Eqn 18

The switch to an integral occurs by taking dn=Ldf/π so that

            Eqn 19

Exponential Transform

            1. Discrete Transforms (in frequency space only)

Begin with equation 5  Eqn 20

Note that at this point an imaginary part has been introduced into both of the above integrals.  These are explicitly equal and opposite in sign and thus cancel.

Separate the exponentials

      Eqn 21

Now define  and

    Eqn 22

so that

   Eqn 23

Note that there is an imaginary part to F(α_n) that needs to be combined with the imaginary parts in exp(-jαx) to cancel for a real f(x). To be specific including the imaginary parts of F, there are twice as many constants on the right side of equation 15 as on the right side of equation 11.

            2. Continuous Limit

Begin with equation 8 and separate the parts of the exponential to yield

       Eqn 24

Define

               Eqn 25

So that

            Eqn 26

Converting to Press Notation

Start with equation 16 which is equation 8 with the exponential separated into a part in each integral.  Then let  .  Also change the f which began as Churchill's notation to h so that f can be used for the frequency with less confusion. Doing this the transform relation becomes

Eqn 27

Change α' to f and x' to t and define

      Eqn 28

so that

     Eqn 29

It is always useful to note that

        Eqn 30

Which establishes the relationship

             Eqn 31

Finally in equation 19, change to f' = f/2π so that df = 2πdf', and then change f' back to f so that

      Eqn 32

 

     Eqn 33

 

  Eqn 34

 

   Eqn 35

Determining B(f) from H(f)

 

     The normal definition of the Fourier transform is that given in equation 28.  If the exponent is expanded it becomes

      

For a real h(t), this tells us that the cosine transform is the real part of H while the sin transform is the imaginary part of H.  Thus

         Eqn 36

and the angles are

     Eqn 37

These are the definitions that the code nfindphi.for works from.  It returns phi and a signed B such that phi is continuous.  The original signal is given by equation 35 as

     Eqn 38

This expression contains only positive frequencies.  The total power out of the system is given by

       Eqn 39

For a real h(t)

     Eqn 40

Thus

         Eqn 41

This means that the integral can be doubled and used over positive frequencies only to yield.

     eqn 42

The  is what is plotted as the power spectrum, though frequently the square root of this is also plotted.  The advantage of B(f) over |H(f)| occurs in places where B(f) switches signs.