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HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing
Spring 2007
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View Full Document HarvardMIT Division of Health Sciences and Technology
HST.582J: Biomedical Signal and Image Processing, Spring 2007
Course Director: Dr. Julie Greenberg
HST582J/6.555J/16.456J
Biomedical Signal and Image Processing
Spring 2007
Chapter 4  THE DISCRETE FOURIER TRANSFORM
±
c Bertrand Delgutte and Julie Greenberg, 1999
Introduction
The Fourier representation of signals that we studied in Chapter 3 is important for understand
ing how ﬁlters work and what a spectrum is, but it is not a practical tool because the DTFT
is a continuous function of frequency and therefore its computation would in general require an
inﬁnite number of operations. The purpose of this chapter is to introduce another representation
of discretetime signals, the
discrete Fourier transform (DFT),
which is closely related to the
discretetime Fourier transform, and can be implemented either in digital hardware or in soft
ware. The DFT is of great importance as an eﬃcient method for computing the discretetime
convolution of two signals, as a tool for ﬁlter design, and for measuring spectra of discretetime
signals. While computing the DFT of a signal is generally easy (requiring no more than the
execution of a simple program) the
interpretation
of these computations can be diﬃcult because
the DFT only provides a complete representation of
ﬁniteduration
signals.
4.1
Deﬁnition of the discrete Fourier transform
4.1.1
Sampling the Fourier transform
It is not in general possible to compute the discretetime Fourier transform of a signal because
this would require an inﬁnite number of operations. However, it is always possible to compute a
ﬁnite number of
frequency samples
of the DTFT in the hope that, if the spacing between samples
is suﬃciently small, this will provide a good representation of the spectrum. Simple results are
obtained by sampling in frequency at regular intervals. We therefore deﬁne the
N
point discrete
Fourier transform
X
[
k
] of a signal
x
[
n
] as samples of its transform
X
(
f
) taken at intervals of
1
/N
:
∞
X
[
k
] =
±
X
(
k/N
)=
±
x
[
n
]
e
−
j
2
πkn/N
for 0
k
N
1(
4
.
1)
n
=
≤
≤
−
−∞
Because
X
(
f
) is periodic with period 1,
X
[
k
] is periodic with period
N
, which justiﬁes only
considering the values of
X
[
k
] over the interval [0
,N
−
1].
4.1.2
Condition for signal reconstruction from the DFT
An important question is whether the DFT provides a complete representation of the signal,
that is, if the signal can be reconstituted from its DFT. From what we know about sampling,
Cite as: Julie Greenberg, and Bertrand Delgutte. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image
Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
we expect that this will only be possible under certain conditions. Speciﬁcally, we have seen
in Chapter 1 that, if we take
N
samples
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This note was uploaded on 11/07/2011 for the course AERO 16.422 taught by Professor Juliegreenberg during the Spring '07 term at MIT.
 Spring '07
 JulieGreenberg

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