FUZZY SETS OF QUASI-PERIODIC SIGNALS

Gonzalo Bailador1 Gracian Trivino2 Albert van der Heide2

1 Universidad Politecnica de Madrid, gonzalo.bailador@upm.es2 Europen Centre for Soft Computing, {gracian.trivino, albert.vdheide}@softcomputing.es

Abstract

A quasi-periodic signal is a signal that, hav-ing a repetitive recognizable periodic pat-tern, has variations in period and amplitude.Examples of this type of signals are, e.g.,the electrocardiogram, the accelerations pro-duced during the human gait, the vibrationof a musical instrument, etc.Currently these signals are analyzed using,among others, statistical methods, e.g. Hid-den Markov Models, or Soft Computingmethods, e.g. Neural Networks.This paper contributes to this field by in-troducing a method for creating a LinguisticModel of a set of these signals based on aFuzzy Finite States Machine. The obtainedmodel is more intuitive than the existing ap-proaches and its computational implementa-tion is easy to interpret. A practical exampleof application is provided.

Key words: Signal analysis, Quasi-periodicsignal, Fuzzy finite state machines

1 INTRODUCTION

Some signals obtained sampling physical phenomenaprovide a similar repetitive temporal pattern. Thesesignals are called quasi-periodic due to irregularitiesin period and amplitude. Popular approaches to dealwith these quasi-periodic signals vary from Waveletstransform [3], classical curve fitting methodologiesto Hidden Markov Models [7] and Neural Networks[2, 10]. A survey of different techniques that could beapplied in these cases can be found in [1].In this paper we present the fundamentals of a newtechnique for the analysis of quasi-periodical signals

based on Fuzzy Finite States Machines (FFSM). Weextend fuzzy logic modelling techniques [13] by defin-ing a linguistic model of the signal which is associatedwith a new kind of membership function. By creat-ing this type of linguistic model we want to contributeto the field ‘Computing with words and perceptions’,as defined by Zadeh [15]. The idea consists of aggre-gating a set of simple perceptions obtained during thetemporal sampling of a signal to create a more com-plex perception of the process in evolution, i.e. the se-quence of values of a linguistic variable that describea signal’s evolution is converted into a more generalconcept that we have called a Linguistic Model of thesignal. Fuzzy Finite State Machines can overcome thediscontinuous character of FSM [4]. FFSM allows asubset of possible states to be active to some degree atany moment in time [8] resulting in a system with con-tinuous flow from one state into another. Because theactivation of states is based upon fuzzy variables thishas the additional advantage that the system can ex-plain in human understandable terms what the causewas for current states in the model of the process. Theusefulness of fuzzy automata has been demonstratedin many areas, e.g. in fault diagnosis [9], ECG signalrecognition [12] and process monitoring [5]. The spe-cific contribution of this paper is the definition of theconcept of a fuzzy set of quasi-periodic signals, as wellas the idea of a Linguistic Model of the common char-acteristics of a group of similar quasi-periodic signals.The so obtained signal description and the associatedmonitoring system are easier to interpret than the onesobtained using classical methods.

2 INITIAL DEFINITIONS

Fuzzy set. Let X be a space of points (objects) witha generic element of X devoted by x. A fuzzy set(class) A in X is characterized by a membership func-tion fA(x) which associates with each point in X a realnumber in the interval [0, 1], with the value of fA(x)representing the ”grade of membership” of x in A [14].

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Quasi-periodic signal. A quasi-periodic signal is asignal that, having a repetitive recognizable periodicpattern, has variations in period and amplitude.Fuzzy set of quasi-periodic signals. Followingthe definition of fuzzy set we define a fuzzy set ofquasi-periodic signals U being LMQ(u) a membershipfunction that defines the degree with which a quasi-periodic signal u(t) belongs to U .We call (LMQ) the Linguistic Model of a set of quasi-periodic signals and it could be considered as an exten-sion of the concept of Linguistic Label, first introducedby Zadeh.Fuzzy finite state machine (FFSM). The con-cept of Finite State Machine can be extended us-ing fuzzy logic [6]. In general, a FFSM is a tuple:FFSM = {Q,X, Y, f, g, s} where:

• Q = {q1, q2, . . . , qn} is the set of states where qi

can take values in the interval [0,1]

• X is the set of fuzzy input variables

• Y is the set of fuzzy output variables

• f is the state transition functionQ(t) = f(X,Q(t − 1))

• g is the output functionY (t) = g(X,Q(t − 1))

• s is the initial state. (s = q1 in all the examplespresented in this paper)

In the rest of this paper we introduce a method, basedon the concept of FFSM, to create LMQ. In this in-troductory work we do not develop all the possibilitiesof the model, for example we do not use the set ofoutput variables Y and therefore we will not need toimplement the function g.

3 MODEL OF A SET OF PERIODIC

RECTANGULAR SIGNALS

In order to describe the method to create a LMQ, wewill start with a simple example, creating the Linguis-tic Model of a set of rectangular signals. Providedu1(t) = sin(ωt) the rectangular signal u2(t) is definedas:

u2(t) =

−1 if u1(t) ≤ −0.50 if −0.5 < u1(t) ≤ 0.51 if u1(t) > 0.5

See figure 1 where the signal is represented with ω = 1.

3.1 Define the states of the signal during aperiod

The first step of the method consists of choosing a setof states which represent the relevant aspects of the

0 2 4 6 8 10 12−2

−1

0

1

2

Time

Am

plitu

de

P

Z

N

Q1 Q2 Q3 Q4

Figure 1: Rectangular signal with related fuzzy labels.

repetitive pattern. How to choose and define states isa subjective matter and depends upon the application.In the rectangular signal we have identified four statesQ = {q1, q2, q3, q4}:

• q1 = High

• q2 = Medium

• q3 = Low

• q4 = Medium2

As can be seen in figure 1 the states correspond toclearly defined parts of the signal.

3.2 Identify the characteristics of differentstates

In the second step we must obtain from the signal u(t)the set of fuzzy variables xi that are needed to identifyevery state. This set of variables X = {x1, x2, . . . , xn}must be defined using their associated linguistic labelsAij . As we will see, depending on user requirementsand on the signal complexity, the number of input vari-ables can vary and the procedure to obtaining themfrom the original signal can differ.In the example of the rectangular signal we will use afuzzy input variable x1 that correspond directly withthe signal amplitude and we use a set of three linguis-tic labels {A11, A12, A13} to define the possible valuesof this variable. For simplicity we have renamed theselabels to {P,Z,N} meaning Positive, Zero, and Nega-tive. These labels are shown on the right vertical axisof the diagram in figure 1.

3.3 Define the transitions

In this step we define the temporal order with whichthe signal follows the states. Figure 2 shows the dia-gram of a FFSM that represents the temporal evolu-tion of the rectangular signal.

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Figure 2: States of the FFSM for the rectangular sig-nal.

3.4 Define the constraints

The next step of the method consists of using a setof fuzzy rules to define the state transition functionQ(t) = f(Q(t− 1), X(t)). We must define how the ac-tivation of states is influenced by the input variablesand by the degree of activation of the states at the pre-vious instant of time. We distinguish between rules tostay in a state i and rules to change:Rii : IF qi(t − 1) AND (x1 is Aii) THEN qi(t)Rij : IF qi(t − 1) AND (x1 is Aij) THEN qj(t)In the example, the current state of the signal is rep-resented by a vector of four states, Q = {q1, q2, q3, q4},and the set of input variables is X = {x1}.The following is the set of rules that describes the tran-sitions in the example of the rectangular signal:

R11 : q1(t − 1) ∧ (x = P ) → q1(t)R22 : q2(t − 1) ∧ (x = Z) → q2(t)R33 : q3(t − 1) ∧ (x = N) → q3(t)R44 : q4(t − 1) ∧ (x = Z) → q4(t)R12 : q1(t − 1) ∧ (x = Z) → q2(t)R23 : q2(t − 1) ∧ (x = N) → q3(t)R34 : q3(t − 1) ∧ (x = Z) → q4(t)R41 : q4(t − 1) ∧ (x = P ) → q1(t)

In general, it will be needed to incorporate temporalconstrains. We do this using two additional linguisticlabels (See figure 3):Time to stayi is the maximum time the signal is al-lowed to stay in a state i.Time to changeij is the minimum time that the sig-nal must be in state i before changing to state j.The following set of rules includes the temporal re-strictions in the model of the rectangular signal:

R11 : q1(t − 1) ∧ (x = P ) ∧ (d1 = Tstay1) → q1(t)R22 : q2(t − 1) ∧ (x = Z) ∧ (d2 = Tstay2) → q2(t)

R33 : q3(t − 1) ∧ (x = N) ∧ (d3 = Tstay3) → q3(t)R44 : q4(t − 1) ∧ (x = Z) ∧ (d4 = Tstay4) → q4(t)R12 : q1(t − 1) ∧ (x = Z) ∧ (d1 = Tchange12) → q2(t)R23 : q2(t − 1) ∧ (x = N) ∧ (d2 = Tchange23) → q3(t)R34 : q3(t − 1) ∧ (x = Z) ∧ (d3 = Tchange34) → q4(t)R41 : q4(t − 1) ∧ (x = P ) ∧ (d4 = Tchange41) → q1(t)

where di is the Timespan of the state qi. A detailedexplanation of the calculation of the Timespan is givenlater.

0 200 400 600 800 1000

0

0.2

0.4

0.6

0.8

1

TimeD

egre

e of

Mem

bers

hip

Time to stayTime to change

Figure 3: (a) Time to change, (b) Time to stay.

4 COMPUTATIONAL

IMPLEMENTATION

The objective is to implement the membership func-tion LMQ(u) that provides the degree of matching ofa signal with the Linguistic Model and therefore thedegree with which a signal u(t) belongs to the fuzzyset U . Figure 4 shows a diagram including the mainelements of the computational implementation of thismembership function.

Extraction of characteristics block consists of sen-sors, hardware processing, and software mechanismsthat obtain from the analyzed physical phenomena thesequence of numerical data, u(t), that will be used asinput to the Fuzzification block.

The Fuzzification block uses the linguistic labelsfor describing the relevant characteristics of the signalto obtain the set of fuzzy input variables X.

The f(Q,X) block consists of the implementation ofthe set of rules. They are Takagi-Sugeno-Kang infer-ence rules of zero-order with a constant output of valueequal to 1 [11].They can be expressed as:Rii : (qi(t − 1) = 1) ∧ (x = Aii) → qi(t) = 1Rij : (qi(t − 1) = 1) ∧ (x = Aij) → qj(t) = 1And are implemented as:

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Rii : qi(t) = MIN(qi(t − 1), Aii(x))Rij : qj(t) = MIN(qi(t − 1), Aij(x))where;qi(t) is the activation of state i,Aii(x) is the degree with which the signal matches theconditions to stay in the state i,Aij(x) is the degree with which the signal matches theconditions to change from state i to j,qi(t) is the normalization of qi(t) that is calculated inthe Feedback block.The total degree of activation of a state qi is calculatedas:

qi(t) = maxN

r=1(qir) (1)

where;qir is the contribution to the degree of activation ofthe state i delivered by the rule r. Note that defuzzi-fication is not needed.

Figure 4: Computational implementation of LMQ.

The Feedback block performs three functions,namely, Normalization, Timespan calculation and Lowpass filtering.We need as input to the FFSM the current distributionof activation between all the possible states. Using thefollowing formula we perform such normalization (re-distribution) by processing the values obtained at theFFSM output.

qi(t) =qi(t)

n∑

j=1

qj(t)

(2)

where n is the number of states.In addition, the Feedback block calculates a vectorD = (d1, d2, . . . , dn) which represents the Timespanin each state; di is calculated as the time that qi > θ.

Where θ is a parameter depending upon the specificimplementation.

Finally, in order to minimize the influence of noise onthe states evolution the Feedback Block implementsa low-pass filter that introduces certain inertia to thestate. The value of qi is modified by the formula:

qi(t) = αqi(t − 1) + (1 − α)qi(t) (3)

where α is a parameter called inertia factor. Thehigher the value the more resistant to change.

Membership calculation block calculates to whatdegree the current signal u(t) is a member of theset of signals U . At every instant the vectorq1(t), q2(t), ..., qn(t) represents the degree with whichthe signal is matched by the model.

We define the Instantaneous Degree of Matchingwith the model of a signal u(t) as:

IDM(u, t) = ORn

1(qi(t)) = maxn

1(qi(t)) (4)

where n is the number of states.Based on this measure we define the AccumulatedDegree of Matching as:

ADM(u, t) =1

t

t∑

s=1

IDM(u, s) (5)

And finally we define the Degree of Membership ofthe signal u(t) to the fuzzy set U as the AccumulatedDegree of Matching after a relatively long period oftime, e.g. several periods:

LMQ(u) =1

T

T∑

s=1

IDM(u, s) (6)

5 FIRST RESULTS

As an illustration of how the model works let us toconsider u3(t), which is obtained by adding a smallsinusoidal signal to u2(t).

u3(t) = u2(t) + sin(100ωt) (7)

Figure 5 shows the curves corresponding withIDM(u3), and ADM(u3). See how IDM(u3) changeswhen the signal does not match the model.

The calculation of the membership of function u2(t)and u3(t) to the set of functions provides:LMQ(u2) = 1LMQ(u3) = 0.87Signal u2(t) is clearly a prototypical signal of ourmodel and therefore has a higher membership valueto the set of rectangular signals than u3.

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0 5 10 15 20 25 30

−1

0

1

Time

Am

plitu

de

0 5 10 15 20 25 300

0.5

1

Time

IDMADM

Figure 5: IDM(u3), and ADM(u3).

In addition, as an illustration of how the model workswith the temporal constrains, figure 6 shows a signalu4(t) with increasing period. It shows how IDM(u4),and ADM(u4) decreases when timespan in a state islonger than the defined by Time to Stay.

0 5 10 15 20 25 30

−1

0

1

Time

Am

plitu

de

0 5 10 15 20 25 300

0.5

1

Time

IDMADM

Figure 6: IDM(u4), and ADM(u4)

6 A PRACTICAL EXAMPLE

In music, the note ‘A4’ corresponds with a sine with afrequency 440Hz (figure 7). However when we produce‘A4’ with different musical instruments the resultantacoustic signals will be different from the perfect signaland will have a different degree of membership to thefuzzy set defined around the ‘pure’ signal. We havefollowed the steps described above to create a mem-bership function that calculates the degree with whicha signal belongs to the set ‘Note-A4’ of quasi-periodicsignals.

0 0.5 1 1.5 2x 10

−3

−1

0

1

Time

Am

plit

ud

e

0 0.5 1 1.5 2x 10

−3

0

0.5

1

Time

IDMADM

Q1 Q2 Q3 Q4

Figure 7: Sinusoidal signal

6.1 Define the states

As before the states correspond to parts in the signalperiod. A depiction of the states is given in figurefigure 7 where the states are shown at the top. Thedifferent states are defined as:

• q1 = Positive increasing; this state matches thefirst part of the signal, when the amplitude is pos-itive and increasing.

• q2 = Positive decreasing; this state matches thesecond part of the signal, when the signal haspassed its positive peak and is decreasing towardsan amplitude of zero.

• q3 = Negative decreasing; the amplitude is neg-ative and getting more negative.

• q4 = Negative increasing; the amplitude is neg-ative, and getting smaller (going towards zero).

6.2 Identify characteristics

In order to distinguish the states in this case, we use asan additional input variable x2 based on the derivativeof u(t). The derivative makes it possible to distinguishbetween the increasing and decreasing parts of the sig-nal. We will use two linguistic labels to assign fuzzyvalues to the signal amplitude x1(t), namely, Positive(P) and Negative (N) and we will use two linguisticlabels to assign fuzzy values to x2, namely, Decreasing(D), and Increasing (I).

6.3 Define the transitions

A basic sequential ordering of states is sufficient. Infact the state transitions will be identical to rectangu-lar signal (figure 1).

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6.4 Define the constraints

R11 : q1(t − 1) ∧ (x1 = P ) ∧ (x2 = I) ∧ (d1 = Tstay) → q1(t)

R22 : q2(t − 1) ∧ (x1 = P ) ∧ (x2 = D) ∧ (d2 = Tstay) → q2(t)

R33 : q3(t − 1) ∧ (x1 = N) ∧ (x2 = D) ∧ (d3 = Tstay) → q3(t)

R44 : q4(t − 1) ∧ (x1 = N) ∧ (x2 = I) ∧ (d4 = Tstay) → q4(t)

R12 : q1(t − 1) ∧ (x1 = P ) ∧ (x2 = D) ∧ (d1 = Tchange) → q2(t)

R23 : q2(t − 1) ∧ (x1 = N) ∧ (x2 = D) ∧ (d2 = Tchange) → q3(t)

R34 : q3(t − 1) ∧ (x1 = N) ∧ (x2 = I) ∧ (d3 = Tchange) → q4(t)

R41 : q4(t − 1) ∧ (x1 = P ) ∧ (x2 = I) ∧ (d4 = Tchange) → q1(t)

The parameters used in this experiment are: for inertiaα = 0.5 and for the timespan θ = 0.01. Note that dueto the symmetry of sinus function Tchange and Tstay

are equal for all rules.

6.5 Experimental results

We have calculated the membership of various A4-tonesignals: a perfect A4-tone, an A4-tone produced by apiano, an A4-tone produced by a violin, and an A4-tone produced by a tuning fork. Additionally we cal-culated the membership of a A5-tone signal producedby a piano to the set of A4-tone signals.

LMQ(ReferenceA4) = 0.9LMQ(PianoA4) = 0.74LMQ(V iolinA4) = 0.56LMQ(ForkA4) = 0.87LMQ(PianoA5) = 0

As can be seen in the obtained results the similar sig-nals produce high degrees of membership while thelast signal obtains a low degree. In fact the last signalfails to match the first state and therefore does notsynchronize with the model.

7 CONCLUSIONS

We have introduced the fundamentals for a new tech-nique of analysis of quasi-periodical signals. The use oflinguistic labels to define the set boundaries makes iteasier to design and understand the model. The resultsso far have been promising and there are many possi-bilities for further improvements and research. Futureinvestigation will focus on how to define with more pre-cision the Linguistic Models corresponding with com-plex sets of signals.

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