Int’l J. of Communications, Network and System Sciences, 2011, 4, 364-371
doi:10.4236/ijcns.2011.46042 Published Online June 2011 (http://www.SciRP.org/journal/ijcns)
Multi-Resolution Fourier Analysis
Part I: Fundamentals
Nourédine Yahya Bey
Faculté des Sciences et Techniques, Département de physique, Université Franço is Rabelais, Tours, France
Received September 2, 2010; revised September 25, 2010; accepted November 5, 2010
Abstract
In the first paper of this series, we propose a multi-resolution theory of Fourier spectral estimates of finite
duration signals. It is shown that multi-resolution capability, achieved without further observation, is ob-
tained by constructing multi-resolution signals from the only observed finite duration signal. Achieved reso-
lutions meet bounds of the uncertainty principle (Heisenberg inequality). In the forthcoming parts of this se-
ries, multi-resolution Fourier performances are observed, applied to short signals and extended to time-fre-
quency analysis.
Keywords: Multi-Resolution, Spectral Analysis, Fourier Analysis, Resolution, Frequency Resolution
1. Introduction
Analyzing single realization of noisy short-time signals
(short data records), multi-resolution and space-frequen-
cy or time-frequency approaches, estimating frequencies
of multiple signals in noise, reduction of noise variance,
recovery of missing parts of signals and so on, are im-
portant topics in many areas of sciences and industries
(radar and sonar data processing, communications, geo-
physical and seismic exploration, biomedical engineer-
ing, non destructive testing, and so on). In pertinent lit-
erature, multi-resolution analysis is now considered as a
standard tool by researchers in image and signal proc-
essing. One finds, for example, that resolving sinusoidal
signals in noise with nearby frequencies is of special
interest [1-5]. An other example of this importance is
the well known development of various parametric spec-
tral estimation methods [6,7] and wavelet theories
[8-10].
In real-world applications, one acquires only finite
duration signals. These signals can be viewed as being
obtained by windowing infinite signals with boxcar
functions. Obtained signals are therefore assumed to
vanish outside the observation interval. Many problems
of the Fourier spectral estimation are traced to this as-
sumption made about the data outside the observation
interval. The overall transform includes a convolution of
the desired transform with that representing the window
function. The main lobe width between 3-dB levels of
the window transform, approximately the inverse of the
time interval T, determines the frequency resolution. Al-
though important works proposed solutions that limit the
impact of time windowing effects [11,12] and others that
provide minimum-error band-limited approximation of
non band-limited signals [13], performance limitations
and their consequences as poor frequency and amplitude
estimations remain non-recoverable. Moreover, the im-
portant frequency extent (multiple of the reciprocal of
the observation interval) of spectral leakage perturbs
amplitude estimation and masks weak components.
Skillful selection of windows reduces only its amplitudes
with no effect on its spectral extent.
Parameter identification approach (autoregressive
(AR), moving average (MA), ARMA) was used to avoid
deficiencies of Fourier spectral estimation since unrealis-
tic assumption about the nature of the signal (zero or
cyclic) outside the observation interval is eliminated.
Important improvements over Fourier spectral estimation
is reported by pertinent literature especially for short
finite duration signals (higher resolution and lack of
side-lobes). However, well known drawbacks of this
approach are excessive sensitivity to observation noise
(resolution varies as a function of the signal-to-noise
ratio), important computation times with respect to FFT
analysis, computational complexity [14-16], multiplicity
of algorithms estimating model parameters and the ne-
cessity of subjective judgement in the selection of the
order [17]. Contamination of parametric spectra by spu-

N. YAHYA BEY
365
rious peaks is an inherent problem to parametric model-
ing. In [18], we proposed AR modeling of signal defined
for low signal-to-noise ratios with an adapted model or-
der selection. We have shown that sensitivity to observa-
tion noise of AR modeling is drastically reduced whereas
computation times remain important.
Pisarenko harmonic decomposition, extended Prony’s
method and Prony spectral line decomposition [19-21]
depict analysis modelings similar to those of the pa-
rameter identification approach (ARMA or AR proc-
esses). Performances are dependent on the order, usually
unknown, and remain sensitive to high level of observa-
tion noise. Deficiencies, mentioned above, are therefore
encountered by these methods. On the other hand, the
algorithm “MUSIC” for “Multiple Signal Classification”
[22] detects frequencies in a signal by performing an
eigen decomposition on the covariance matrix of a data
vector of samples obtained from the samples of a con-
sidered signal. MUSIC assumes known the number of
samples and the number of frequencies. This algorithm is
attractive provided the available signal-to-noise (SNR) is
high to resolve two distinct peaks in the estimated spec-
trum. One finds, however, that devised various methods
[23] to overcome drawbacks such as weak robustness to
both modeling errors and the presence of a strong back-
ground noise add to computational complexity and re-
quires high enough SNRs.
An alternative non parametric approach for the resolu-
tion of mentioned problems is developed by wavelets. It
is well known that wavelet transforms have remarkable
resolution properties but trail some drawbacks: 1) wave-
lets capture only few oscillations and therefore act as
local magnifiers independently of the nature of the signal
under analysis (stationary, quasi-stationary or not); 2) the
necessity of skillful selection of appropriate wavelets to
the signal under analysis; 3) the problem of interpretation
of wavelets spectra is made quite difficult since the Fou-
rier spectrum of a wavelet is in itself a complex one. Let
us note that Fourier coefficients are not only concepts but
have physical evidence; 4) information on frequency is
only approximative since a wavelet does not have a pure
frequency as a sine wave. Characterization of a precise
frequency content is therefore not suitable by means of
wavelets; 5) when treating extraction of signals from
noise [24] we have shown in [25] that wavelet denoising,
fail or yield notably perturbed results when spectral den-
sities of colored noise (Gaussian or not) and the signal
overlap.
In this work, we propose multi-resolution theory of
Fourier spectral estimates. The key idea is based on the
fact that observed signals carry information on their un-
observed or missing parts and the difficulty of the task is
to let these signals reveal this hidden information by us-
ing the simplest possible theory. Our main effort, de-
scribed here, is to construct signals from the only ob-
served one able to reveal in the frequency domain re-
sulting transforms whose main lobe-widths between
3-dB levels, and therefore resolutions, decrease as
lengths of constructed signals increase. The number of
resolution levels is defined as a function of the length of
the corresponding multi-resolution signal in order to de-
pict detailed or global views. Multi-resolution signals
can be viewed as wavelets composed of versions of the
signal itself analyzed by means of FFT spectral estima-
tion. Advantages of the proposed approach are:
1) Resolution at any desired level is applied simulta-
neously to all corresponding points of the frequency axis.
The whole frequency axis is magnified.
2) Contraction of spectral leakage and improvement of
frequency estimation proportionally to levels of multi-
resolution signals (second part of this series).
3) Easier and efficient implementation since the popu-
lar FFT algorithm remains used for all computations.
Important reduction of computation times are expected
when compared to those required by parametric or
wavelet approaches.
4) Reduction of the spectral variance of resolved noisy
spectral estimates as a function of the applied resolution
level. This helps simple and efficient denoising of a sin-
gle noisy realization of a short signal (third part of this
series).
5) Unlike models based on estimation of correlation
lags, here, phase information is not destroyed. Inverse
transformation recovers missing parts of observed sig-
nals (second part of this series).
6) Performances of the denoising tools [24-27] based
on FFT algorithm can be used for extraction of buried
resolved spectral estimates (third part of this work).
7) Extension of obtained results to a novel time-fre-
quency analysis is proposed in the fourth part of this work.
8) Extraction of buried time-varying spectra in noise is
treated in the fifth part of this work.
9) One can also apply the theory to a novel image
processing.
It is crucial to notice that our main focus of attention
in the first part of this work is to derive expressions of
multi-resolution signals. In section III, we precise basics
of multi-resolution Fourier analysis by constructing dou-
ble resolution signals and generalizing the theory to
higher frequency resolution levels. Resolution properties,
contraction of spectral leakage, improvement of fre-
quency estimation and recovering of missing parts of
short signals are discussed and observed in the forth-
coming parts of this series.

366 N. YAHYA BEY
2. Signal Representation
Consider a continuous-time real signal

x
t for <


and let
<t

X

be its bandpass spectrum de-
fined by,

min max
=0,X   . (1)
where min

and max

are the bounds of the spectral
support of

X

.
A finite duration signal


T
x
t “cut out” from


x
t
in the time interval is given by,
[0, ]T
 

,[0,
==
0, otherwise.
TT
]
x
tt T
xt xtt 
 (2)
where is the rectangular time window whose

Tt

length T is denoted by its lower script.
The notion of resolution used here is related to the
ability of the developed theory to discriminate between
two or more pure sinusoids. It is well known that within
the framework of the definition given by (2), two sinu-
soids of respective angular frequencies 1

and 0

are
barely resolvable if,
10
=2T  . (3)
Sinusoids are discriminated if they are more than
10
=

  apart in the frequency domain and simi-
larly, this discrimination is achieved in the time domain
if sinusoids are more than apart. In other words, the
signal and its Fourier transform cannot be both highly
concentrated. The uncertainty principle (3) is called also
“Rayleigh criterion” [1].
T
In this work, given the time interval , we are inter-
ested in signals for which two angular frequency com-
ponents
T
1

and 0

are unresolvable, i.e.,
2T

. (4)
However, according to the indeterminacy principle (or
Heisenberg inequality widely known for its applications
in quantum mechanics [28], signal processing [29] and
various other theories [30]) the resolution in angular fre-
quency and observation time cannot be arbitrarily small,
i.e.,
12T

 . (5)
3. Multi-Resolution Fourier Analysis
3.1. Double Resolution Fourier Analysis
Here, we use the only observed signal

T
x
t
T to con- struct a double resolution signal for which =

 is
satisfied. This double resolution ability requires an an-
gular frequency axis whose locations are separated by
the mutual distance T

. How to create these locations
by using the only available time interval, ? We pro-
pose hereafter a two-step procedure using the “one-
point” interpolation followed by zeros insertion in the
frequency domain. This helps to derive expression of the
double resolution window, describe its properties and
represent double resolution signals.
T
3.1.1. One-Point Interpolat ion
The “one-point” interpolation used here in the frequency
domain means one-point insertion between two existing
frequency locations separated by the mutual distance
T

. Hence, given the finite duration observed signal


T
x
t, we can form the overall signal defined in [0, 2T]
by writing,



  
22
ˆTTTT
T
x
ttxtz
t


T, (6)
where


0
T
zt

,


0,tT and 2T is the rec-
tangular window of length 2T. The upper script T in



t


2
ˆTT
x
t denotes the length of the most narrow time win-
dow whereas the lower script 2T is the length of the time
extension obtained here by addition of zeros in the time
interval of length T.
Now, it is crucial to notice that the overall signal


2
ˆTT
x
t can be written as a function of the true signal


2T
x
t. An equivalent form of (6) using


2T
x
t is
therefore given by,



 
22
2
ˆTTT
T

T
x
ttxt t. (7)
One can see easily that (7) can be put under the form,




 
222
22
2
ˆ
2,
TT
TTT
T
TT
T
xt xtt
xtt t








 (8)
where




2
T
TTT
ttt 

T


 .
The window


2t

T
T extends over the interval [0, 2T]
(lower script) and the length of its most narrow time
sub-window is T (upper script). The window


2
TTt
 is
“tailed-window” whose tail (or zero-padding), repre-
sented by


T
T
zt

, as defined above, has the same
length as the observed signal.
3.1.2. Zeros Insertion in the Frequency Domain
The transform of the tailed-window constructed
above defines the angular frequency locations

2
TTt

{0, T

,


,2,TT}

  The second step in the construction
aims at eliminating angular frequency representation of
the signal at these locations. This means that sought
transform of the constructed window depicts zeros at
these locations.
We propose to double the length of the interval in
which


2
ˆTT
x
t is defined by substituting 4T for 2T in the
subscripts of (8). As the signal

2
ˆTT
x
t is defined in the

N. YAHYA BEY
367
interval [0, 2T], doubling the length of its time interval is
obtained by introducing a local period of length 2T so
that the length of the time interval reaches 4T. This
gives,
 
1
42
0
ˆˆ 2
TT
TT
p
x
txtp


T. (9)
By using (8), we can write,
 
1
44 2
0
ˆ2
TT
TT T
p
x
ttxtp 


T, (10)
where,



44
42
T
T
T
ttt 



T
T


tpT
, (11)
and,
 
3
4
0
1p
T
TT
p
t 

 
. (12)
One can see easily that angular frequency axis of the
transform of the signal


4
ˆTT
x
t depicts angular fre-
quency locations separated by the mutual distance π/T as
follows: each angular frequency interval [nL, (n + 1)L],
where n is an integer and L = 2π/T, is divided into four
sub-intervals defined by the locations,
 
,, 14, 12,34, 1nnLnLnLnLn L  
with imposed zeros at {nL, (n + 1/2)L, (n+1)L}. The
transform of the signal in [nL, (n + 1)L] is represented at
the two locations (n + 1/4)L and (n + 3/4)L separated by
L/2. We have therefore an angular frequency axis with
locations separated by the mutual distance π/T.
Let us consider hereafter properties of the derived
window represented by (11).

4
TTt

3.1.3. Properties in the Frequency Domain
The window is defined in the interval [0, 4T]
and has a local period given by 2T. It is easy to see that
, resulting from the addition of

4
TTt


4
TTt



4Tt
 and
, has the following transform,

4
T
Tt

 
(4,1) 4 (4,1)
=WHH




 , (13)
where

4
H

 and

(4,1)
H


are respectively Fourier
transforms of and , i.e.,

4Tt
 
4
T
Tt





 
44
=4 2
H
TST



 
(4,1) =42sin 2cos

H
iTS TTT


   , (14)
where




=sinSxx x. The complex exponential


4



represents phase induced by the absolute position of the
time interval on the time scale.
One can see from (14) that the width, defined here by
the interval for which


2=ST  0 for


=2 4kT  
where k ± 1, is given by T

. The length T

is the
width of the most narrow frequency response of the
window


4
TT
wt.
Let us note that,





4(4,1) (4,1)
=2 =2 =0
==
TT
HH H



  0
(15)
The amplitude spectrum

(4,1)
H


reaches an extre-
mum in the interval


0, 2T



 for which,

(4,1)
=0
d
[0,2],= 0
d
H
T



  , (16)
where


dd
H
xx represents the derivative of


H
x
with respect to
x
.
Here some algebra yields

0. This means
that this extremum is achieved at the midway bounds
defining the interval
4T  

0, 2T



. Now, let us focus on
the bandwidth


of (4,1)

W

. The bandwidth is
defined here as the main lobe width between 3-dB or
12levels of the resulting transform of the window.
According to (14) and (16),
 
(4,1)4 =
=0
0
HH






 
. (17)
It follows that the bandwidth of )(
(4,1)

W is so that,
 
(4,1) 4
BB=wW wHT 


 


2, (18)
where Bw


f




 represents the bandwidth of


f

.
Now, let us use, in the following, the result (18) in
order to find the bandwidth of

4
ˆTT
x
t as defined by
(10).
3.1.4. Doubl e Res ol ution Proper ties
Here, we derive the bandwidth of the constructed overall
signal as defined by (10). Hence, by considering (10) in
the frequency domain, we can write,
 
 
1
2 (4,1)
0
2 (4,1)
2
,2 2
,
T
p
XTFTxtpTW
XHW




 



 






*
*
(19)
where the second argument

2T

 of ˆ
X
represents
the bandwidth we are searching for and []
F
Tx is the
Fourier transform of
x
. Here

X

and


2
H


are respectively Fourier transforms of the signal


x
t
and the rectangular window 2

H
t
 of length 2T in
which it is observed. Here also, the bandwidth of 2
H

is roughly T

. The term

2

 gathers phases

368 N. YAHYA BEY
resulting from time translation.
The convolution between the windows


H2

 and

(4,1)
W

yields a window with the broadest of the two
bandwidths. According to (18), this gives,
 
2 (4,1)
2
,=HTH W

 

*, (20)
where the bandwidth of


,
H
T   is given by its
second argument T.
By using (20), (19) becomes,
 


ˆ,,

X
XH  *T, (21)
only if,
=2

.
This defines the frequency resolution,
=T

.
It is crucial to notice that the true spectrum

,

X
T   can be extracted from

ˆ,
X
T   as
shown in the second part of this series.
3.1.5. Expression of Double Resolution Signals
Expression of a double resolution signal as a function of
the only observed finite duration signal


T
x
t is
obtained by considering (10). One can see easily that,
 
 
1
44 2
0
1
40
ˆ2
2.
TT
TT T
p
T
Tp
x
ttxtp
txtpT






 
T
(22)
Components of the overall signal (22) are


T
x
t and
its translated version

2
T

x
tT. Notice that for resolv-
ing potential ambiguities [6] or smoothing the appearance
of the spectral estimates, one can apply zero-padding to
the double resolution signal as defined by (22) and not to
the signal depicted by (7) (see the second part of this
series).
3.2. Fourfold Frequency Resolution
3.2.1. Quadruple Resolution Window
It can be seen immediately that the quadruple resolution
window can be obtained from the double resolution one
(13) by substituting 8T for 4T in the lower scripts and
2T for T in upper scripts. The expression corresponding
overall signal of length is given by, T8
 
1
22
88 4
0
ˆ4
TT
TT T
p
x
ttxtp 


T, (23)
where,




2
88
82
T
T
T
ttt  
2T
T
. (24)
Conclusions in the frequency domain for this
quadruple resolution window are therefore straight-
forward and one finds easily that the frequency reso-
lution is given by, =2T


. The main concern
hereafter is to find the expression in the time domain of
the quadruple resolution signal as a function of the only
observed one


T
x
t.
3.2.2. The Overall Signal
The half-period restriction (in the interval [0 ) of
(24) yields,
, 4]T






2
44
4
=2
T
T
T
wtt t  
2T
T
. (25)
As the window


2
4
T
T
t  

]T

t
4T imposes zeros in
the interval [2, then, the overall signal , 4T


2
4
ˆT
T
x
t
(in the interval ) is given by, [0, 4]T



 
2
42
ˆ2
T
TTT T

x
txtqtTztT, (26)
where


2=0 T
zt in the interval [0 and , 2]T


T
qt is
an unknown signal.
3.2.3. Sign al I denti fi cation
Now, the difficulty of the task depicted by (26) is the
identification of the unknown signal


T
qt by using the
only known signal


T
x
t. The half-period restriction of
(23), yields,






22
4444
2
ˆTT
TTTT
T

x
txtwtxt t
. (27)
By using (26), (27) becomes,






 
22
2
=
=.
TTT T
TTT
xtqtTx tt
x
tt tT

 


 (28)
Since




2= T

T
T
x
ttxt

, then the unknown
signal


T
qt is specified by,


=,
TT
qt xt


, (29)
where
 represents an unknown phase that characterizes
the signal




2TT
x
ttT


“cut-out” from


2T
x
t



2Tt

with respect to

T

x
t.
In the following, we identify the expression of the
signal


T
qt by using its amplitude and phase spectra
together with the sign of its angular frequency.
3.2.4. Ampl i tude Spectrum
The spectrum of


T
qt , as defined by (29), is given by,


()
=e
i
TT
QX
 
. (30)
According to (30),


=
TT
QX

.
3.2.5. Ph ase Spe ctru m
Let us consider the amplitude spectrum of the sum of


T
x
tand


T
qt. By assuming that



=
TT
XX
()
ei


and using (30), we have,

N. YAHYA BEY
369



[( )( )]
=1e=2
i
TT TT
XQ XX
  
 .
(31)
Here (31) is satisfied only if,



=T   . (32)
By using (32), the spectrum of


T
qt
, as depicted by
(30), yields,

=e
iT
TT
QX

. (33)
3.2.6. The Angular Frequency
By setting =

 and =



T
, we find respectively
by applying inverse Fourier transformation to (33) the
following four signals
x
tT, , )( TtxT


T
x
Tt

and

T
x
Tt . Since,

[0,],== 0
TT
tTxTtxtT , (34)
then the unknown signal is the time reversed signal of
the observed one, i.e.,

[0, ],=0
TT
tTqtxTt . (35)
3.2.7. Expression of Fourfold Frequency Resolution
Signals
The expression of fourfold frequency resolution signals
is now easily obtained by combining (26) and (35), i.e.,
 
1
2
880
ˆ442
T
TTT
Tp
x
ttxtpTxpT




 t
 .
(36)
Zero-padding is applied to the fourfold resolution
signal as depicted by (36). Analyzed signals are depicted
with angular frequency separations given by 2T.
3.3. Threefold and Quintuple Frequency
Resolution Signals
Threefold and quintuple frequency resolution signals can
be immediately deduced from above developments on
respectively double and fourfold resolution signals as
shown below.
3.3.1. Resolu tion Windows
By generalizing overall signals as given by (8) and (23)
to threefold and quintuple overall signals, we have,

 


2
ˆ2
Is TIs T
sT sTsT
sT
xtxt tt 



2
, (37)
where s = 3 and s = 5 are for respectively threefold and
quintuple resolution signals. Here


2
I
s represents the
integer part of 2
s
.
Let us introduce the local period
s
T and rewrite (37)
in the interval [0, 2]
s
T as follows,



 
1
22
22
0
ˆIs TIs T
sTsT sT
p
x
twtxtspT


. (38)
The obtained resolution window is therefore given by,

 


 
 
1
2
222
0
21
=
2.
Is T
sT sTI sT
p
Is TT
wt ttspT
tspIs T








 

(39)
In the frequency domain,


2
2
Is T
sT
wt yields,
 







 

 
2
2, 22,[/2
2
2,[ 2
=
=2
ˆcos2,
s
sIs sIs
s
sIs
WHH
sTS s T
Hs



 




T

(40)
where




=sinSxx x and

2s


 is a phase
induced by absolute position on the time scale. Here



2,[ /2
ˆsIs
H


is so that,


 
2
2, 2
ˆs
sIs
HH




. (41)
Let us note that,



2, 2=
=0
sIs sT
W


 , (42)
and the bandwidth of )(
/2])[,(2

sIs
W is given by,
 

2, 22
Bw Bw
sIs s
WH 






sT (43)
One can see immediately that Fourier transformation
of (38) yields,



 
 

  
2,2
2
ˆ,2
,2 ,
ssIs
s
XTXHW
XH sT








**
*

(44)
where


s

 are phases induced by the time position
on the time scale and represents the convolution. *
Equality (44) is satisfied only if,
,= s

and the bandwidth of the spectrum, as depicted by (44)
exhibits therefore the resolution,
=2π.Ts


3.3.2. Expression of Threefold and Quintuple
Resolution Signals
By using (38) and (39), expressions of threefold and
quintuple resolution signals in the interval [0, 2sT], are

370 N. YAHYA BEY
respectively specified for s = 3 and s = 5 by,

 
 


1
2
220
ˆ
21 2
Is T
sT T
sT p
T
xt txtspT
.
I
sxspTt




 

  (45)
Here also zero-padding can be applied to resolution
signals as defined by (45).
4. Optimum Resolution Signal
Here we propose to find the finest level of resolution and
the corresponding expression of the resolution signal by
using the lower bound of the uncertainty principle.
4.1. Variation of the Resolution
It is well known that when a function is scaled




g
tgat
>1a where , then it is contracted if and expanded when . Above expressions of
multi-resolution signals show that we have a contrac-
tion-expansion effect evaluated by increasing or de-
creasing the length of the local period of the
multi-resolution window by the constant quantity
>0aa<1
T
representing the original observation interval of the
signal


x
t. We have therefore a discrete variation of
the resolution,
s
T, where s = 2, 3, 4, 5. Now, the
question of interest is the achievable finest limit of
frequency resolution. This is discussed hereafter.
4.2. Levels of Multi-Resolution Signals
Let us consider the two following important facts :
1) Frequency spacing between adjacent frequency
components of the double resolution signal is


2T.
Clearly, this frequency spacing is greater than the
spacing


12T defined by the lower bound of the
uncertainty principle. Similarly, for the quadruple
frequency signal, we have the spacing


4T that
remains greater than


12T. By generalizing these
results, one finds that any frequency spacing of any
multi-resolution signal is conditioned by,
 
,12
s
sT T . (46)
This immediately yields,
2[]
s
I

, (47)
where


I
x is the integer part of
x
.
As the lower limit of the indeterminacy principle
cannot be arbitrarily small, this means that we cannot
build resolution signals for which (47) is not fulfilled.
Resolution levels,
s
, are therefore limited by 6s

.
2) Here we discuss the significance of the upper bound
provided by the inequality (47) by considering the
indeterminacy principle as an uncertainty principle. To
see this, let us compare the variation between two
adjacent frequency resolution levels with the lower
bound of the uncertainty principle (given by 0.5). The
variation between two adjacent resolutions and
is given by,
6=s
5=s
6=s






=5 =6
=0
ss
TT T    .2. (48)
It can be seen that depicting frequency separations by
using quintuple () resolution signal instead of the
sextuple resolution signal () yields errors smaller
than the lower bound of the uncertainty principle. This
means that the optimal resolution signal has the
quintuple form given by (45) for which,
5=s
6=s
0.4T


. (49)
Here, (49) represents the achieved optimal Fourier
frequency resolution.
4.3. Expression of Multi-Resolution Signals
By generalizing expression depicted by (45) to double
and fourfold resolution signals and generalization of the
equivalent form (38) initially written for and
to and , we can write,
3=s
5=s2=s4=s




 
 





2
1
2
22
0
()
=
21 2
=.
T
Is T
sT sTsT
p
TT
sT
xt
wtxtspT t
2
1
=0
ˆIs
sT
p
x
tspTIsx spIsTt
xt












(50)
Notice that for the sake of simplicity, the resolution
signal constructed from


x
t is relabeled


()sT
x
t




where


()sT
x
t



 is the resolution operator of level
s
(lower script) applied to


T
x
t.
5. Conclusions
We proposed multi-resolution theory of Fourier spectral
estimates. We have shown that multi-resolution capabil-
ity, achieved without further observation, is obtained by
constructing multi-resolution signals from the only ob-
served finite duration signal. Obtained frequency resolu-
tions are not limited by the length of the observation in-
terval and meet bounds of the indeterminacy principle or
Heisenberg inequality. Observation results and applica-
tion of the Fourier multi-resolution theory to short sig-
nals and time-frequency analysis are reported in the
forthcoming parts of this series.

N. YAHYA BEY
371
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