PROCEEDINGS OF THE IEEE, VOL.
66,
NO. 1, JANUARY 1978
51
On
the Use
of
Windows
for
Harmonic
Analysis
with
the Discrete
Fourier
Transform
FREDRIC J.
HARRIS,
MEMBER, IEEE
Ahmw-This Pw!r mak=
available
a
concise review of data
win-
compromise consists of applying windows to the sampled
daws
pad the^
affect
the
Of
in
data set, or equivalently, smoothing
the spectral samples.
'7
of
aoise9
m
the
ptesence
sdroag
bar-
The two
operations
to which we subject the
data
are
momc mterference.
We
dm
call
attention
to
a
number
of common
-=
be
rp~crh
windows
den
used
with the
fd
F~-
sampling
and
windowing.
These
operations can
be performed
transform.
This
paper includes a comprehensive
catdog
of data win-
in either order. Sampling is well understood, windowing is less
related to sampled
windows for DFT's.
I. INTRODUCTION
HERE
IS
MUCH signal
processing devoted to detection
and estimation. Detection is the task
of determiningif
a specific signal set is present in an observation, while
estimation is the task of obtaining
the values
of the parameters
describing the signal.
Often the signal is complicated
or
is
corrupted by interfering
signals
or noise.
To facilitate the
detection
and
estimation of
signal
sets, the
observation
decomposed by a basis set which spans the signal
space
[
11.
For many problems of
engineering interest, the class
of
signals
being sought are periodic which
leads
quite naturally to a
decomposition
by a basis consisting of
simple periodic func-
tions, the sines
and
cosines.
The classic Fourier
transform is
the mechanism
by
which we are able to perform this decom-
position.
By necessity, every
observed
signal
we process must be
finite extent. The extent may
be
adjustable and selectable,
but it must
be finite. Processing a finite-duration observation
imposes interesting and interacting considerations on the har-
monic
analysis.
These
considerations
include
detectability
of tones in the presence
of nearby
strong tones, resolvability
of similarstrength
nearby
tones, resolvability
of shifting tones,
and
biases in
estimating
the
parameters
of any of the afore-
mentioned signals.
For practicality, the
data we process
are
N
uniformly spaced
samples
of the observed
signal. For convenience,
N
is highly
composite, and we
will
assume
N
is even. The
harmonic
estimates we obtain
through
the discrete Fourier
transform
(DFT) are
N
uniformly spaced
samples
of the associated
periodic spectra.
This approach is elegant
and
attractive
when the processing
scheme is cast
as
a spectral decomposition
in an N-dimensional
orthogonal
vector space
[
21. Unfortu-
nately, in many practical situations, to obtain meaningful
results this elegance
must
be
compromised.
One such
Manuscript received September 10,
1976; revised April 11,
1977 and
September
1,
1977.
This work was supported by Naval Undersea
Center (now Naval Ocean Systems Center) Independent Exploratory
Development Funds.