# An Adaptive Resolution Computationally Efficient Short-Time Fourier Transform

Authors: Saeed Mian Qaisar; Laurent Fesquet; Marc Renaudin
Journal: Research Letters in Signal Processing
 (2008)

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## Abstract

The short-time Fourier transform (STFT) is a classical tool, used for characterizing the time varying signals. The limitation of the STFT is its fixed time-frequency resolution. Thus, an enhanced version of the STFT, which is based on the cross-level sampling, is devised. It can adapt the sampling frequency and the window function length by following the input signal local characteristics. Therefore, it provides an adaptive resolution time-frequency representation of the input signal. The computational complexity of the proposed STFT is deduced and compared to the classical one. The results show a significant gain of the computational efficiency and hence of the processing power.


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## Body

## 1. Introduction

Most of the real-life signals like
speech, Doppler, seismic, and  biomedical signals are time
varying in nature. The spectral contents of these signals vary with time, which
is a direct consequence of the signal generation process [1]. The STFT is a
classical tool for characterizing such signals [2]. The limitation with the
STFT is that it provides a fixed resolution time-frequency representation of
the input signal. This fixed resolution is the reason for the creation of the multiresolution
analysis (MRA) techniques [3–5], which provide a good frequency but a poor
time resolution  for the low-frequency events and a good time but apoor
frequency resolution for the  high-frequency
events. This type of analysis is
well suited for most of the real-life signals  [3].In this article, the fixed
resolution dilemma is resolved to a certain extent by revising the STFT. The
motivation behind the proposed STFT is to achieve a smart time-frequency
representation of the time varying signals. The idea is to adapt the
time-frequency resolution along with the computational load by following the
input signal local characteristics. An efficient solution is proposed by
smartly combining the features of both uniform and nonuniform signal processing
tools.


## 2. Proposed Adaptive Resolution STFT

The block diagram of the proposed
STFT is shown in Figure1. The description of different blocks is given below.Figure 1
Block diagram of the
proposed STFT. “—” represents the signal flow, “……”
represents the control flow, and “- - - - -” represents the parameters flow, at
system different stages.


### 2.1. Asynchronous Analog to Digital Converter (AADC)

According to [6], the sampling instants of a nonuniformly sampled
signal obtained with the level crossing sampling scheme (LCSS) are defined by (1).
Where tn is the current sampling instant, tn−1 is the previous one, and dtn is the time delay between
the current and the previous sampling instants (cf. (2)).The LCSS drastically reduces the
activity of the post processing chain, because it only captures the relevant
information [7–9]. In this context, analog to digital converters based
on the LCSS have been developed [10–12]. The AADC [10] is employed for
digitizing x(t). An M-bit resolution AADC has 2M−1 quantization levels which are uniformly disposed according to x(t) amplitude
dynamics. The AADC has a finite
bandwidth. Thus, to assure a proper signal capturing a band pass filter with pass band
[fmin;fmax] is employedat the AADC input. Let ΔVin and Δx(t) be the AADC and x(t) amplitude dynamics, respectively.
In order to avail the complete AADC resolution in the studied case, Δx(t) is always adapted to match ΔVin. For an AADC, the maximum and the minimum sampling frequencies [7] are defined by (3) and (4),
respectively. Where, Fsmax and Fsmin are the maximum and the minimum sampling frequencies of the AADC. fmax is the bandwidth and fmin is the fundamental frequency of x(t):(1)tn=tn−1+dtn,(2)dtn=tn−tn−1,(3)Fsmax=2⋅fmax⋅(2M−1),(4)Fsmin=2⋅fmin⋅(2M−1).


### 2.1.1. Enhanced Activity Selection Algorithm (EASA) and Window Selector

The
relevant parts of the nonuniformly sampled signal obtained
with the AADC are selected—corresponds to the
variable length rectangular window—by the EASA. The EASA is defined as shown in Algorithm1.
T0=1/fmin is the fundamental period of x(t). T0 and dtn detect parts of the nonuniformly sampled signal with activity. The condition on dtn is chosen in order to satisfy the Nyquist criterion for fmin,
when sampling x(t) nonuniformly with the AADC [13]. Ni represents the number of nonuniform samples lie in the ith selected window, which lie on the jth active part of the nonuniformly
sampled signal. Where, i and j both belong to the set of natural
numbers N*. Nref represents the
upper bound on Ni. The choice of Nref depends on x(t) characteristics and on system parameters. The above
described loop repeats for each selected window, which occurs during the
observation length of x(t). Every time before repeating the loop, i is incremented and Ni is initialized to zero.The
EASA displays interesting features with the LCSS, which
are not available in the classical case. It selects only the active parts of
the nonuniformly sampled signal, obtained with the AADC. Moreover, it
correlates the length of the selected window with the signal local
characteristics.The
window selector implements the condition given by expression (5). Jointly, the EASA and the window selector provide an efficient
spectral leakage reduction in the case of transient signals [13]. Indeed,
spectral leakage occurs due to the signal truncation problem. Usually an appropriate smoothening (cosine)
window function is employed to reduce the signal truncation. For the proposed
case, as long as the condition 5 is true, the leakage problem is resolved by
avoiding the signal truncation. As no signal truncation occurs so no cosine
window is required. In this case the window decision Di=1,
which makes the switch state 1 (cf. (Figure 1)). Otherwise, an appropriate
cosine window is employed to reduce the signal truncation problem. In this case Di=0, which makes the switch state 0. In
expression 5, t1i represents the 1st sampling instant of the ith selected window and tendi−1 represents the last sampling instant of the (i−1)th selected window.For proper spectral representation,
the condition given by expression (6) should be satisfied [13]. Where, Li is the length in seconds of the ith selected window. In order to satisfy
this condition for the worst case, which occurs for Fsmax, Nref is calculated for an appropriate reference window length Lref.
Where, Lref satisfies the condition Lref≥T0. The process
is given by (7) as follows:
(5)if(Ni≤Nrefand(Tdi=t1i−tendi−1)>T02),(6)Li≥T0,(7)Nref=Lref⋅Fsmax.
The lower and the upper bounds on Lref are posed, respectively, by T0 and the system resources (the
maximum sample frame which the system can  process at once). For Nref (cf. (7)), the
condition 6 holds for all selected windows except for the case when the actual
length of the jth activity is less than T0.


### 2.1.2. Adaptive Sampling Rate

The AADC sampling frequency is correlated tox(t) local
variations [7, 13]. Let Fsi represent the AADC sampling
frequency for the ith selected window. Fsi can be calculated by using (8). Where, tmaxi and tmini are the final and the initial times of the ith
selected window. The upper and the lower bounds on Fsi are
posed by Fsmax and Fsmin, respectively:(8)Li=tmaxi−tmini.Fsi=NiLi.The selected data obtained with the
EASA can be used directly for further nonuniform digital processing [8, 14].
However in the studied case, the selected data is resampled uniformly. It
enables to take advantage of both nonuniform and uniform signal processing
tools [7, 13]. Due to this resampling there will be an additional error.
Nevertheless, prior to this transformation, one can take advantage of the
inherent oversampling
of the relevant signal parts in the system [7]. Hence, it adds to the accuracy
of the post resampling process [11]. The nearest neighbour resampling
interpolation (NNRI) is employed for data resampling. The reasons of
inclination towards NNRI are discussed in [13, 15].A reference sampling frequencyFref is chosen, such as itremains greater than and closest to the FNyq=2⋅fmax. Depending upon the values of Fref and Fsi, the resampling frequency Frsi (cf. (Figure 1)) can be adapted for the ith selected window. For the
case, Fsi>Fref, Frsi is chosen as: Frsi=Fref. It is done
in order to resample the selected data, lie in the ith selected
window closer to the Nyquist frequency. It avoids the unnecessary
interpolations during the data resampling process and so reduces the
computational load of the proposed technique.For the case,Fsi≤Fref,Frsi  is chosen as: Frsi=Fsi. In this case, it appears that the data lie in the ith selected window may be resampled at a frequency which is less than FNyq and so it can cause aliasing. Since, the sampling rate of the AADC varies
according to the slope of x(t) [10]. A high-frequency signal part has a
high slope and the AADC samples it at a higher rate and vice versa. Hence, a
signal part with only low-frequency components can be sampled by the AADC at a subNyquist
frequency of x(t). But still this signal part is locally oversampled in
time with respect to its local bandwidth [7]. Hence, there is no danger of
aliasing. This statement is further illustrated with the results summarized in
Table 1.Table 1
Summary of parameters of
the selected windows.
Selected windowLiFsiNiFrefFrsiNri(Sec)(kHz)(Smp)(kHz)(kHz)(Smp)1st4.990.735001.250.735002nd0.492.814001.251.256253rd0.587.040961.251.257314th0.587.040961.251.257315th0.437.030051.251.25536


### 2.1.3. Adaptive Resolution Analysis

The STFT of a sampled signalxn is determined by computing
the discrete Fourier transform (DFT) of an N samples segment centred on τ,  which
describes the spectral contents of xn around the instant τ. Where N is defined as: N=L⋅Fs. Here, L is
the effective length in seconds of the window function wn and Fs is
the sampling frequency. The STFT can be expressed mathematically by (9). In Equation (9), f is the frequency index, which is  normalized with respect to Fs.L controls the STFT time and frequency resolution [2]. In the classical case, the
input signal is sampled at a fixed sampling frequency Fs, regardless of
its local variations. Thus, a fixed L results into a fixed N. In
this case, the time resolution Δt and the
frequency resolution Δf of the STFT
can be defined by (10) and (11), respectively. Equation (11) shows that for a
fixed Fs,Δf can be increased by increasing N. But increasing N requires to increase L which will reduce Δt (cf. (10)). Thus, a larger L provides a better Δf but a poor Δt, and vice
versa. This conflict between Δf and Δt is the
reason for the creation of the MRA techniques [3–5].The proposed STFT is a smart
alternative of the MRA techniques. It performs adaptive time-frequency
resolution analysis, which is not attainable with the classical STFT. It is
achieved by adapting theFrsi,Li, and Nri according to the local variations of x(t). Nri is the number of resampled data points that lie in the ith selected window. Thus, the time
resolution Δti and the frequency resolution Δfi of
the proposed STFT can be specific for the ith selected window and are defined by (12) and (13), respectively. Because of this
adaptive resolution, the proposed STFT will be named as the adaptive resolution
STFT, (ARSTFT) throughout the following parts of this article. The adaptation
of Frsi,Li,
and Nri also adds to the
computational gain of the ARSTFT, compared to the classical one. It is achieved
firstly by avoiding the unnecessary samples to process and secondly by avoiding
the use of the cosine window function as far as the condition 5 is true. The
ARSTFT is defined by (14). In (14), τi and fi are the central
time and the frequency index of the ith selected window, respectively. fi is normalized with respect to Frsi. n is the index of the resampled data
points lie in the ith selected
window. The notation wni represents that the window function length Li and shape (rectangle or cosine) can be adapted for the ith selected window:(9)X[τ,f]=∑n=τ−L/2τ+L/2[xn⋅wn−τ]⋅e−j⋅2π⋅f⋅n.(10)Δt=L.(11)Δf=FsN.(12)Δti=Li.(13)Δfi=FrsiNri.(14)X[τi,fi]=∑n=τi−Nri/2τi+Nri/2[{Resample(xn,tn)}⋅wn−τii]⋅e−j⋅2π⋅fi⋅n.


## 2.1. Asynchronous Analog to Digital Converter (AADC)

According to [6], the sampling instants of a nonuniformly sampled
signal obtained with the level crossing sampling scheme (LCSS) are defined by (1).
Where tn is the current sampling instant, tn−1 is the previous one, and dtn is the time delay between
the current and the previous sampling instants (cf. (2)).The LCSS drastically reduces the
activity of the post processing chain, because it only captures the relevant
information [7–9]. In this context, analog to digital converters based
on the LCSS have been developed [10–12]. The AADC [10] is employed for
digitizing x(t). An M-bit resolution AADC has 2M−1 quantization levels which are uniformly disposed according to x(t) amplitude
dynamics. The AADC has a finite
bandwidth. Thus, to assure a proper signal capturing a band pass filter with pass band
[fmin;fmax] is employedat the AADC input. Let ΔVin and Δx(t) be the AADC and x(t) amplitude dynamics, respectively.
In order to avail the complete AADC resolution in the studied case, Δx(t) is always adapted to match ΔVin. For an AADC, the maximum and the minimum sampling frequencies [7] are defined by (3) and (4),
respectively. Where, Fsmax and Fsmin are the maximum and the minimum sampling frequencies of the AADC. fmax is the bandwidth and fmin is the fundamental frequency of x(t):(1)tn=tn−1+dtn,(2)dtn=tn−tn−1,(3)Fsmax=2⋅fmax⋅(2M−1),(4)Fsmin=2⋅fmin⋅(2M−1).


## 2.1.1. Enhanced Activity Selection Algorithm (EASA) and Window Selector

The
relevant parts of the nonuniformly sampled signal obtained
with the AADC are selected—corresponds to the
variable length rectangular window—by the EASA. The EASA is defined as shown in Algorithm1.
T0=1/fmin is the fundamental period of x(t). T0 and dtn detect parts of the nonuniformly sampled signal with activity. The condition on dtn is chosen in order to satisfy the Nyquist criterion for fmin,
when sampling x(t) nonuniformly with the AADC [13]. Ni represents the number of nonuniform samples lie in the ith selected window, which lie on the jth active part of the nonuniformly
sampled signal. Where, i and j both belong to the set of natural
numbers N*. Nref represents the
upper bound on Ni. The choice of Nref depends on x(t) characteristics and on system parameters. The above
described loop repeats for each selected window, which occurs during the
observation length of x(t). Every time before repeating the loop, i is incremented and Ni is initialized to zero.The
EASA displays interesting features with the LCSS, which
are not available in the classical case. It selects only the active parts of
the nonuniformly sampled signal, obtained with the AADC. Moreover, it
correlates the length of the selected window with the signal local
characteristics.The
window selector implements the condition given by expression (5). Jointly, the EASA and the window selector provide an efficient
spectral leakage reduction in the case of transient signals [13]. Indeed,
spectral leakage occurs due to the signal truncation problem. Usually an appropriate smoothening (cosine)
window function is employed to reduce the signal truncation. For the proposed
case, as long as the condition 5 is true, the leakage problem is resolved by
avoiding the signal truncation. As no signal truncation occurs so no cosine
window is required. In this case the window decision Di=1,
which makes the switch state 1 (cf. (Figure 1)). Otherwise, an appropriate
cosine window is employed to reduce the signal truncation problem. In this case Di=0, which makes the switch state 0. In
expression 5, t1i represents the 1st sampling instant of the ith selected window and tendi−1 represents the last sampling instant of the (i−1)th selected window.For proper spectral representation,
the condition given by expression (6) should be satisfied [13]. Where, Li is the length in seconds of the ith selected window. In order to satisfy
this condition for the worst case, which occurs for Fsmax, Nref is calculated for an appropriate reference window length Lref.
Where, Lref satisfies the condition Lref≥T0. The process
is given by (7) as follows:
(5)if(Ni≤Nrefand(Tdi=t1i−tendi−1)>T02),(6)Li≥T0,(7)Nref=Lref⋅Fsmax.
The lower and the upper bounds on Lref are posed, respectively, by T0 and the system resources (the
maximum sample frame which the system can  process at once). For Nref (cf. (7)), the
condition 6 holds for all selected windows except for the case when the actual
length of the jth activity is less than T0.


## 2.1.2. Adaptive Sampling Rate

The AADC sampling frequency is correlated tox(t) local
variations [7, 13]. Let Fsi represent the AADC sampling
frequency for the ith selected window. Fsi can be calculated by using (8). Where, tmaxi and tmini are the final and the initial times of the ith
selected window. The upper and the lower bounds on Fsi are
posed by Fsmax and Fsmin, respectively:(8)Li=tmaxi−tmini.Fsi=NiLi.The selected data obtained with the
EASA can be used directly for further nonuniform digital processing [8, 14].
However in the studied case, the selected data is resampled uniformly. It
enables to take advantage of both nonuniform and uniform signal processing
tools [7, 13]. Due to this resampling there will be an additional error.
Nevertheless, prior to this transformation, one can take advantage of the
inherent oversampling
of the relevant signal parts in the system [7]. Hence, it adds to the accuracy
of the post resampling process [11]. The nearest neighbour resampling
interpolation (NNRI) is employed for data resampling. The reasons of
inclination towards NNRI are discussed in [13, 15].A reference sampling frequencyFref is chosen, such as itremains greater than and closest to the FNyq=2⋅fmax. Depending upon the values of Fref and Fsi, the resampling frequency Frsi (cf. (Figure 1)) can be adapted for the ith selected window. For the
case, Fsi>Fref, Frsi is chosen as: Frsi=Fref. It is done
in order to resample the selected data, lie in the ith selected
window closer to the Nyquist frequency. It avoids the unnecessary
interpolations during the data resampling process and so reduces the
computational load of the proposed technique.For the case,Fsi≤Fref,Frsi  is chosen as: Frsi=Fsi. In this case, it appears that the data lie in the ith selected window may be resampled at a frequency which is less than FNyq and so it can cause aliasing. Since, the sampling rate of the AADC varies
according to the slope of x(t) [10]. A high-frequency signal part has a
high slope and the AADC samples it at a higher rate and vice versa. Hence, a
signal part with only low-frequency components can be sampled by the AADC at a subNyquist
frequency of x(t). But still this signal part is locally oversampled in
time with respect to its local bandwidth [7]. Hence, there is no danger of
aliasing. This statement is further illustrated with the results summarized in
Table 1.Table 1
Summary of parameters of
the selected windows.
Selected windowLiFsiNiFrefFrsiNri(Sec)(kHz)(Smp)(kHz)(kHz)(Smp)1st4.990.735001.250.735002nd0.492.814001.251.256253rd0.587.040961.251.257314th0.587.040961.251.257315th0.437.030051.251.25536


## 2.1.3. Adaptive Resolution Analysis

The STFT of a sampled signalxn is determined by computing
the discrete Fourier transform (DFT) of an N samples segment centred on τ,  which
describes the spectral contents of xn around the instant τ. Where N is defined as: N=L⋅Fs. Here, L is
the effective length in seconds of the window function wn and Fs is
the sampling frequency. The STFT can be expressed mathematically by (9). In Equation (9), f is the frequency index, which is  normalized with respect to Fs.L controls the STFT time and frequency resolution [2]. In the classical case, the
input signal is sampled at a fixed sampling frequency Fs, regardless of
its local variations. Thus, a fixed L results into a fixed N. In
this case, the time resolution Δt and the
frequency resolution Δf of the STFT
can be defined by (10) and (11), respectively. Equation (11) shows that for a
fixed Fs,Δf can be increased by increasing N. But increasing N requires to increase L which will reduce Δt (cf. (10)). Thus, a larger L provides a better Δf but a poor Δt, and vice
versa. This conflict between Δf and Δt is the
reason for the creation of the MRA techniques [3–5].The proposed STFT is a smart
alternative of the MRA techniques. It performs adaptive time-frequency
resolution analysis, which is not attainable with the classical STFT. It is
achieved by adapting theFrsi,Li, and Nri according to the local variations of x(t). Nri is the number of resampled data points that lie in the ith selected window. Thus, the time
resolution Δti and the frequency resolution Δfi of
the proposed STFT can be specific for the ith selected window and are defined by (12) and (13), respectively. Because of this
adaptive resolution, the proposed STFT will be named as the adaptive resolution
STFT, (ARSTFT) throughout the following parts of this article. The adaptation
of Frsi,Li,
and Nri also adds to the
computational gain of the ARSTFT, compared to the classical one. It is achieved
firstly by avoiding the unnecessary samples to process and secondly by avoiding
the use of the cosine window function as far as the condition 5 is true. The
ARSTFT is defined by (14). In (14), τi and fi are the central
time and the frequency index of the ith selected window, respectively. fi is normalized with respect to Frsi. n is the index of the resampled data
points lie in the ith selected
window. The notation wni represents that the window function length Li and shape (rectangle or cosine) can be adapted for the ith selected window:(9)X[τ,f]=∑n=τ−L/2τ+L/2[xn⋅wn−τ]⋅e−j⋅2π⋅f⋅n.(10)Δt=L.(11)Δf=FsN.(12)Δti=Li.(13)Δfi=FrsiNri.(14)X[τi,fi]=∑n=τi−Nri/2τi+Nri/2[{Resample(xn,tn)}⋅wn−τii]⋅e−j⋅2π⋅fi⋅n.


## 3. Illustrative Example

In order to illustrate the ARSTFT an
input signalx(t), shown on the left part of Figure 2 is
employed. Its total duration is 30 seconds and it consists of three active
parts. Each activity is a sinusoid of 0.9 v amplitude and of 50, 200, and 500 Hz frequency, respectively. The time length of each activity is 5, 0.5, and 1.6
seconds, respectively. x(t) is band
limited between 50 to 500 Hz and it
is sampled by employing a 3-bit resolution AADC. Thus, Fsmax and Fsmin become 7 kHz and 0.7 kHz, respectively (3), (4). Fref = 1.25 kHz and ΔVin = 1.8 v are chosen. The selected data obtained with the EASA is shown on
the right part of Figure 2. By following the criteria given in Section 2, Nref=4096 is chosen, which
leads to 5 selected windows. First, two selected windows correspond to the
first two activities and the remaining corresponds to the third activity. The
last three selected windows are not distinguishable on the right part of Figure 2, because they lie consecutively on the third activity. The parameters of each
selected window are summarized in Table 1.Input signal (left) and
selected signal (right).Table1 exhibits the interesting
features of the ARSTFT. Fsi represents
the sampling frequency adaptation by following the local variations of x(t). It is achieved due to the smart
features of the AADC and the EASA. Ni shows that the relevant signal parts are locally oversampled in time with
respect to their local bandwidths [7]. Frsi shows the adaptation of the resampling frequency for each selected window. It
further adds to the computational gain of the ARSTFT, by avoiding the
unnecessary interpolations during the resampling process. Nri shows how the adaptation of Frsi avoids the processing of unnecessary samples during
the spectral computation. Li exhibits the EASA dynamic feature, which is to correlate the window
function length with the local variations of x(t). Adaptation of Li, Frsi and Nri leads to the adaptive time-frequency resolution, which is clear from the values
of Δti and Δfi in Table 2.Table 2
Time and frequency
resolution of the selected windows.
Window1st2nd3rd4th5thΔti(Sec)4.990.490.580.580.43Δfi(Hz)0.22.01.711.712.33Table2 demonstrates that ARSTFT
adapts its time-frequency resolution by following the local variations of x(t). It provides a good time but a poor
frequency resolution for the high frequency parts of x(t), and vice
versa. It is the type of analysis, well suited for most of the real-life
signals [3]. The spectrum of each selected window is computed and plotted with
respect to τi on Figure 3. Figure 3 shows the fundamental and the
periodic spectrum peaks of each selected window. In this case, the spectrum
periodic frequency for the ith selected window fpi  is equal to Frsi. It shows the
adaptation of Frsi, which can
be visualized from Figure 3.Figure 3
The ARSTFT of the selected
windows.The ARSTFT also adapts  the window
shape (rectangle or cosine) for theith selected window. The condition 5 remains true for the first two selected
windows, which sets   Di=1. As no signal truncation occurs so no
cosine window is required in this case. On the other hand, the number of
samples for the fourth activity is 11200. Therefore, Nref=4096 leads to the three selected windows for the
time span of the fourth activity. The condition 5 becomes false in this case,
which sets Di=0. As signal truncation occurs so
suitable length cosine (Hanning) windows are employed to reduce this effect.In the classical case, ifFs=Fref is chosen, in order to satisfy the Nyquist sampling criterion for x(t).
Then the whole signal will be sampled at 1.25 kHz, regardless of its
local variations. It will produce unnecessary samples than required. Moreover,
the windowing process is not able to select only the active parts of the
sampled signal. In addition, L remains static and is not able to adapt
with the signal local variations. Thus, it causes the system to process
needless samples and so causes an increased computational activity than the
proposed case. For classical case, fixed N = 4096 will produce nine fixed L = 3.3
second windows, for the total x(t) time span of 30 seconds. It will lead
to fix Δt=3.3 seconds
and Δf=0.31Hz for
all nine windows (cf. (10) and (11) .


## 4. Computational Complexity

This section compares the
computational complexity of the ARSTFT with the classical STFT. The complexity
evaluation is made by considering the number of operations executed to perform
the algorithm.In the classical case,Fs is fixed. In this case, a time
invariant, fixed L, cosine window
function is employed to window the sampled data. If N is the number of samples lie in the window then the windowing
operation will perform N multiplications between wn and xn (cf. (9)). The
spectrum of the windowed data is obtained by computing its DFT. A complex term
is involved in the DFT computation. The DFT complexity is calculated by taking
the real and the imaginary parts separately. The DFT performs 2⋅(N)2 additions and 2⋅(N)2 multiplications, thus
operations count becomes 4⋅(N)2 for N output frequencies. The combined computational
complexity C1 of the STFT
is given by (15). Where, A is the
total number of windows occurs for the observation length of x(t).For the
proposed ARSTFT,Fsi, Frsi,
and  wni are not fixed and are adapted according to the
local variations of x(t). The EASA performs 2⋅Ni comparisons and Ni increments for the ith selected
window (cf. (Section 2)). The choice of Frsi and window shape requires three comparisons. The selected signal is
resampled before computing its DFT. The
NNRI is employed for the resampling purpose. The NNRI only requires a
comparison operation for each resampled observation. Therefore, the resampler
performs Nri comparisons.
If Di=0, then a cosine
window function is applied on the resampled data, which performs Nri multiplications (cf. (Figure 1)). The DFT
performs 4⋅(Nri)2 operations for the ith selected window. The combine computational complexity C2 of the ARSTFT is given by (16). Where i=1,2,…,K represents the index of
the selected window. α is a
multiplying factor, its value is 1 for Di=0 and 0 for Di=1. The computational
gain of the ARSTFT over the classical one is calculated by employing the
results of the illustrative example. The results are summarized in Table 3.
(15)C1=A⋅{N+4⋅(N)2}.(16)C2=K⋅3+∑i=1K3⋅Ni+α⋅Nri+Nri+4⋅(Nri)2.Table 3
Summary of the
computational gain.
Time (seconds)1st activity2nd activity3rd activityGain =C1/C22.7743.4312.46Table3 shows the computational gain
of the ARSTFT over the STFT for each x(t) activity. It shows that the
ARSTFT leads to a significant reduction of the total number of operations as
compared to the classical one. This reduction in operations is achieved by
adapting Fsi,  Frsi, and wni according to the local variations of x(t).


## 5. Conclusions

A new tool for the adaptive
resolution time-frequency analysis is proposed. The ARSTFT is especially well
suited for the low activity sporadic signals like electrocardiogram, phonocardiogram,
seismic signals, and so forth. It is shown thatFsi and Li adapt by following the x(t) local
variations. Criteria to choose the appropriate Fref and Nref are developed. A complete methodology of adapting Frsi and wni for the ith selected window has been
demonstrated.The ARSTFT outperforms the STFT. The
advantages of the ARSTFT over the STFT are the adaptive time-frequency
resolution and the computational gain. These smart features of the ARSTFT are
achieved due to the joint benefits of the AADC, the EASA, and the resampling as
they enable to adaptFsi, Frsi, Ni, Nri, and wni by exploiting the local variations of x(t).
The employment of fast algorithms in place of the DFT for the spectrum
computation is in progress, it will further add up to the computational efficiency
of the ARSTFT. Moreover, the performance comparison of the ARSTFT with other
MRA techniques, in terms of computational complexity and quality, opens the way
to new research prospective.Algorithm 1:Enhanced Activity Selection Algorithm (EASA).
While(dtn≤T0/2andNi≤Nref)Ni=Ni+1;end



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*Source: 932068-2008-05-22.xml*