j000spectrum.fodt

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   <text:h text:style-name="Heading_20_1" text:outline-level="1">Power Spectrum</text:h>
   <text:p text:style-name="Text_20_body">Based on Numerical Recipes in C, Second Edition, pp 496.</text:p>
   <text:p text:style-name="Text_20_body">Fourier Transform:</text:p>
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   <text:p text:style-name="Text_20_body">Inverse Fourier Transform:</text:p>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Derivatives</text:h>
   <text:p text:style-name="Text_20_body">Fourier Transform:</text:p>
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          </mrow>
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        </mrow>
        <annotation encoding="StarMath 5.0">H^&apos;(f)
=
int from -infinity to infinity dot h(t)
e^{i2%pi ft}
dt
=
i(2{%pi}f) H(f)
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   <text:p text:style-name="Text_20_body">Inverse Fourier Transform:</text:p>
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         <mrow>
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          <mrow>
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          <mrow>
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          </mrow>
         </msup>
         <mrow>
          <mi mathvariant="italic">df</mi>
          <mo stretchy="false">=</mo>
          <mrow>
           <munderover>
            <mo stretchy="false">∫</mo>
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            </mrow>
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            <mfrac>
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        </mrow>
        <annotation encoding="StarMath 5.0">h(t)
=
int from -infinity to infinity H(f)
e^{-i2%pi ft}
df
=
int from -infinity to infinity 
sqrt {{P_{h}(f)} over 2}
e^{i%theta(f)}
e^{-i2%pi ft}
df</annotation>
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=
int from -infinity to infinity 
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over
{i(2%pi f)}}
e^{-i2%pi ft}
df
=
int from -infinity to infinity 
{sqrt {{{P_{dot h}(f)} over 2}}
e^{i%theta(f)}
over
{i(2%pi f)}}
e^{-i2%pi ft}
df</annotation>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Fast Fourier Transform</text:h>
   <text:p text:style-name="Text_20_body">Define</text:p>
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=
h(t_k),
~~~~~~~~~
t_k=k%DELTA,
~~~~~~~~~
k=0,1,2,...,N-1</annotation>
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   <text:p text:style-name="Text_20_body">Assume periodicity</text:p>
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        <annotation encoding="StarMath 5.0">h_{N + k}
=
h_k,
~~~~~~~~~
k=0,1,2,...,N-1</annotation>
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   <text:p text:style-name="Text_20_body">Define</text:p>
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          <mi>f</mi>
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          <mi>N</mi>
          <mn>2</mn>
         </mfrac>
        </mrow>
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=
N over 2</annotation>
       </semantics>
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    </draw:frame></text:p>
   <text:p text:style-name="Text_20_body">Maximum period is</text:p>
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       <semantics>
        <mrow>
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          <mo stretchy="false">=</mo>
          <msub>
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           <mi>N</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <mi>N</mi>
         </mrow>
         <mrow>
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          <mfrac>
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          </mfrac>
          <mo stretchy="false">=</mo>
          <mfrac>
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            <mi>f</mi>
           </msub>
           <msub>
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            <mi mathvariant="italic">max</mi>
           </msub>
          </mfrac>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">T
=
t_N
=
N%DELTA
=
1 over df
=
N_f
over
f_max</annotation>
       </semantics>
      </math>
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   <text:p text:style-name="Text_20_body">The FFT produces amplitudes at discrete frequencies:</text:p>
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         <mrow>
          <mrow>
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         <mi>,</mi>
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         <mi>,</mi>
         <mn>0</mn>
         <mi>,</mi>
         <mn>...</mn>
         <mi>,</mi>
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          <mi>f</mi>
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        <annotation encoding="StarMath 5.0">H_n
=
H(f_n),
~~~~
f_n
=
n(df)
=
{n}
over
{N%DELTA}
=
n over T,
~~~~
n
=
-
N_f,
...,
0,
...,
N_f
</annotation>
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   <text:p text:style-name="Text_20_body">Maximum frequency is</text:p>
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=
f_{N_f}
=
N_f df
=
{1}
over
{2%DELTA}
=
N_f over T
=
{nitalic {Nyquist frequency}}</annotation>
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   <text:p text:style-name="Text_20_body">Useful relationships:</text:p>
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=
{1}
over
{N%DELTA}
=
f_max
over
{N_f}
=
1 over T</annotation>
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   <text:p text:style-name="Text_20_body">The Fourier transform</text:p>
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=
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e^{i2%pi f_n t}
dt</annotation>
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              <mrow>
               <mi>N</mi>
               <mi mathvariant="normal">Δ</mi>
              </mrow>
             </mfrac>
             <mi>k</mi>
             <mi mathvariant="normal">Δ</mi>
            </mrow>
           </msubsup>
          </mrow>
          <mo stretchy="false">=</mo>
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         </mrow>
         <mrow>
          <munderover>
           <mo stretchy="false">∑</mo>
           <mrow>
            <mi>k</mi>
            <mo stretchy="false">=</mo>
            <mn>0</mn>
           </mrow>
           <mrow>
            <mi>N</mi>
            <mo stretchy="false">−</mo>
            <mn>1</mn>
           </mrow>
          </munderover>
          <msubsup>
           <mi>h</mi>
           <mrow>
            <mi>k</mi>
            <mi>e</mi>
           </mrow>
           <mrow>
            <mi>i</mi>
            <mn>2</mn>
            <mi mathvariant="normal">π</mi>
            <mfrac>
             <mi mathvariant="italic">kn</mi>
             <mi>N</mi>
            </mfrac>
           </mrow>
          </msubsup>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">H(f_n)
approx
sum from k=0 to N-1 h_k
e^{i2%pi f_n t_k}
%DELTA
=
%DELTA
sum from k=0 to N-1 h_k
e^{i2%pi {{n}
over
{N%DELTA}} k%DELTA}
=
%DELTA
sum from k=0 to N-1 h_k
e^{i2%pi kn over N}
</annotation>
       </semantics>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Text_20_body">Define Discrete Fourier Transform</text:p>
   <text:p text:style-name="Text_20_body"><draw:frame draw:style-name="fr1" draw:name="Object16" text:anchor-type="as-char" svg:width="1.202in" svg:height="0.4626in" draw:z-index="13"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>H</mi>
          <mi>n</mi>
         </msub>
         <mo stretchy="false">=</mo>
         <mrow>
          <munderover>
           <mo stretchy="false">∑</mo>
           <mrow>
            <mi>k</mi>
            <mo stretchy="false">=</mo>
            <mn>0</mn>
           </mrow>
           <mrow>
            <mi>N</mi>
            <mo stretchy="false">−</mo>
            <mn>1</mn>
           </mrow>
          </munderover>
          <msubsup>
           <mi>h</mi>
           <mrow>
            <mi>k</mi>
            <mi>e</mi>
           </mrow>
           <mrow>
            <mi>i</mi>
            <mn>2</mn>
            <mi mathvariant="normal">π</mi>
            <mfrac>
             <mi mathvariant="italic">kn</mi>
             <mi>N</mi>
            </mfrac>
           </mrow>
          </msubsup>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">H_n
=
sum from k=0 to N-1 h_k
e^{i2%pi kn over N}
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   <text:p text:style-name="Text_20_body">Therefore</text:p>
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approx
%DELTA
H_n
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   <text:p text:style-name="Text_20_body">Inverse Fourier Transform</text:p>
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        </mrow>
        <annotation encoding="StarMath 5.0">h(t_k)
=
int from -infinity to infinity H(f)
e^{-i2%pi f t_k}
df</annotation>
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         <mrow>
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             <msub>
              <mi>N</mi>
              <mi>f</mi>
             </msub>
            </mrow>
           </mrow>
           <msub>
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           </msub>
          </munderover>
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        <annotation encoding="StarMath 5.0">h(t_k)
approx
sum from n=-{N_f} to {N_f}
H(f_n)
e^{-i2%pi f_n t_k}
df
=
df
sum from n=-{N_f} to {N_f}
H(f_n)
e^{-i2%pi kn over N}
</annotation>
       </semantics>
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   <text:p text:style-name="Text_20_body">Inverse FFT</text:p>
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         <mrow>
          <mrow>
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           <mrow>
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             <mi>k</mi>
            </msub>
           </mrow>
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          <mo stretchy="false">=</mo>
          <mrow>
           <munderover>
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            <mrow>
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             <mo stretchy="false">=</mo>
             <mrow>
              <mo stretchy="false">−</mo>
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              </msub>
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            </mrow>
            <msub>
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             <mi>f</mi>
            </msub>
           </munderover>
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         <msubsup>
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          <mrow>
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           <mi>e</mi>
          </mrow>
          <mrow>
           <mrow>
            <mo stretchy="false">−</mo>
            <mi>i</mi>
           </mrow>
           <mn>2</mn>
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           <msub>
            <mi>f</mi>
            <mi>n</mi>
           </msub>
           <msub>
            <mi>t</mi>
            <mi>k</mi>
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            <mi mathvariant="normal">Δ</mi>
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          </mfrac>
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           <msub>
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            <mi>f</mi>
           </msub>
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         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">h(t_k)
=
sum from n=-{N_f} to {N_f}
%DELTA H_n
e^{-i2%pi f_n t_k}
1 over {N %DELTA}
=
1 over N
sum from n=-{N_f} to {N_f}
H_n
e^{-i2%pi kn over N}
</annotation>
       </semantics>
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