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Cell[CellGroupData[{
Cell["Introduction to Discrete Fourier Transforms", "Title"],

Cell["\<\
APAM 1601y
Columbia University\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
 "Last week, we introduced the Joseph Fourier's transform as a technique to \
solve partial differential equations (PDEs). The Fourier transform changes \
the way functions are represented. For example, instead of representing a \
function, ",
 StyleBox["f",
  FontSlant->"Italic"],
 "[",
 StyleBox["x",
  FontSlant->"Italic"],
 "], as a curve along ",
 StyleBox["x",
  FontSlant->"Italic"],
 "; we can represent ",
 StyleBox["f",
  FontSlant->"Italic"],
 " with a number of coefficients, ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["a", "i"], TraditionalForm]]],
 "and ",
 Cell[BoxData[
  FormBox[
   SubscriptBox["b", "i"], TraditionalForm]]],
 ", that multiply a known set of orthogonal \"modes\". "
}], "Text"],

Cell["\<\
The Fourier transform expands functions in terms of sine and cosine functions \
of different wavelengths or frequencies. The Fourier transform is like \
\"spectrum analyzer.\" It takes a function and returns a spectrum: a number \
proportional to the fraction of the function that is represented by each \
wavelength or frequency. \
\>", "Text"],

Cell["\<\
The \"discrete Fourier transform\" became in use in the 1960's as the world \
entered the  \"computer age,\" Today, information is primarily represented \
(and transmitted) in digital form, and the field of \"digital signal \
processing\" is at the core of all forms of digitally-recorded music, video, \
and information.\
\>", "Text"],

Cell[TextData[{
 "This notebook introduces the discrete Fourier transform (DFT) and \
illustrates its use. We also describe the recursive \"fast Fourier \
transform\" (FFT) that is built into ",
 StyleBox["Mathematica. ",
  FontSlant->"Italic"]
}], "Text"],

Cell[CellGroupData[{

Cell["John W. Tukey (1915-2000)", "Subsection"],

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John Wilder Tukey was born in New Bedford, Massachusetts, on June 16, 1915.  \
He earned bachelor's and master's degrees in chemistry at Brown University in \
1936 and 1937, respectively. He then went to Princeton, where, in 1938 he was \
a Jacobus Fellow, and upon receiving his doctorate in mathematics from \
Princeton in 1939, he was appointed Henry B. Fine Instructor in Mathematics.  \
A decade later, at age 35, he was became a full professorship.  He directed \
Princeton's Statistical Techniques Research Group from its founding in 1956.  \
 At the same time, he was a Member of Technical Staff at AT&T Bell \
Laboratories in 1945, advancing to Assistant Director of Research, \
Communications Principles in 1958 and, in 1961, to Associate Executive \
Director-Research Information Sciences, which position he held until his \
retirement in 1985.\
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John Tukey has attracted international attention for his studies in \
mathematical and theoretical statistics and their applications to a wide \
variety of scientific and engineering disciplines.  He has led the way in the \
now-burgeoning fields of Exploratory Data Analysis and Robust Estimation, and \
his contributions to the Spectrum Analysis of Time Series and other aspects \
of Digital Signal Processing have been widely used in engineering and \
science.  He has been credited with coining the word \"bit\", a contraction \
of \"binary digit\", which refers to a unit of information, often as \
processed by a computer.\
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improving the access of the scientist to the scientific literature, \
particularly through the development of citation and permutation indices to \
the literature of statistics and probability",
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In 1965, Tukey was named the first recipient of the Samuel S. Wilks Award of \
the American Statistical Association.  He received the National Medal of \
Science in 1973 \"for his studies in mathematical and theoretical statistics\
\[Ellipsis]and for his outstanding contributions to the applications of \
statistics to the physical, social, and engineering sciences\".  Tukey \
received the Shewhart Medal of the American Society for Quality Control in \
1977; the Institute of Electronic and Electrical \
Engineers\[CloseCurlyQuote]1982 Medal of Honor , for the Cooley-Tukey Fast \
Fourier Transform (FFT) Algorithm; and the American Society for Quality \
Control\[CloseCurlyQuote]s Deming Medal in 1983.  In 1989 he was elected a \
Foreign Member of The Royal Society (London) .  Princeton University honored \
him in 1984 with the James Madison Medal , given annually to an alumnus of \
the Graduate School who has had a distinguished career, who has advanced the \
cause of graduate education, or who has achieved a record of outstanding \
public service. In 1989 he received the Monie A. Ferst Award of Sigma Xi, and \
in 1990 the Educational Testing Service Award for Distinguished Service to \
Measurement.\
\>", "Text"],

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Tukey has taught on both the undergraduate and graduate levels and is widely \
sought as a seminar leader and lecturer.  He holds honorary degrees from Case \
Institute of Technology, the University of Chicago and Brown, Temple and Yale \
Universities.\
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President of the Institute for Mathematical Statistics in 1960,he has also \
served as Vice President of the American Statistical Association and as \
Chairman of both its Biometrics Section and its Section on Physical and \
Engineering Sciences.He has been a Member of the Council of the Biometric \
Society,and served on the NationalResearch Council for more than a decade.He \
is a member of the National Academy of Sciences,and is a member of the \
American Philosophical Society (vice president,1975-77) and the American \
Academy of Arts and Sciences,Honorary Fellow of the Royal Statistical Society \
(London),and a Foreign Member of The Royal Society (London).Among his other \
affiliations are the International Statistical Institute,the American \
Association for the Advancement of Science (vice president,Section A,1972 and \
Section U,1974) and Sigma Xi.\
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