Applied Mathematical Modelling 34 (2010) 968–977

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Random telegraph signals with time dependent capture and emissionprobabilities: Analytical and numerical results

Roberto da Silva *, Gilson I. WirthInstituto de Informática – UFRGS, Av. Bento Gonçalves, 9500 Bloco IV, CEP 91501-970 Porto Alegre, BrazilDepartamento de Engenharia Eletrica – UFRGS, Av. Osvaldo Aranha, 103 CEP 90035-190 Porto Alegre, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 August 2008Received in revised form 6 July 2009Accepted 23 July 2009Available online 30 July 2009

Keywords:Statistical modelingCyclo-stationary noiseIntegrated power spectral density

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.07.012

* Corresponding author.E-mail address: rdasilva@inf.ufrgs.br (R. da Silva

In this work, we study the noise spectra of random telegraph signals (RTS) with timedependent capture and emission probabilities. The capture and emission probabilitiesare made time dependent by cyclo-stationary excitation. Rigorous analytical calculationsare performed to derive the model equations, and Monte Carlo simulations are performedto explore the noise behavior and validate the analytical model. Two study cases areexplored: input signals modeled by periodic square wave and sinusoidal waves. We pres-ent clear situations where the integrated noise power is smaller than stationary case for asuitable choice of parameters in both cyclo-stationary inputs studied.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Low-frequency (LF) noise in semiconductor devices [1] is a performance limiting factor in analog circuits and therefore itsunderstanding and control is a fundamental point to the design of high performance integrated circuits. Recent experimentalworks have shown that the LF-noise performance of modern small area MOS devices is dominated by random telegraph sig-nals (RTS).

RTS noise arises from the capture and subsequent emission of charge carriers at discrete trap levels near the Si–SiO2 inter-face. In many practical applications the MOS device is not biased at steady state, but periodically switched, and LF-noise un-der cyclo-stationary excitation is of great interest. The periodic switching of the device bias changes the trap occupancyprobability, leading to capture and emission probabilities that are a periodic function of time. In previous works, we haveproposed a probabilistic modeling approach for stationary excitation [2,3].

Although RTS noise under cyclo-stationary excitation [4] has yet received considerable attention [5], general models, validfor any excitation frequency and signal, are not available yet. Moreover, there is a lack in the literature for a simple analyticaldeduction and formulation for the RTS power spectral density (PSD) under cyclo-stationary excitation. In this communica-tion an alternative derivation of power spectral density is proposed. It is valid at noise frequencies below the excitation fre-quency, for any kind of cyclo-stationary excitation. Our derivation is an extension of the Machlup derivation [1], since theMachlup derivation was performed for the case where emission and capture probability do not change along time.

Our main goals in this work are:

(1) A detailed analysis of two important cases of cyclo-stationary excitation of practical interest: square wave and sinu-soidal excitation. The power spectral density (PSD) of RTS is obtained in a regime of small T (T � hki�1 and T � hli�1,where hki and hli are the average under period of capture and emission probabilities, respectively), and Monte Carlo

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).

R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977 969

simulations are performed for suitable comparisons. For large T (regime where trap occupancy can change many timesduring a cycle, i.e., Thki or Thli are measurable quantities) we discuss the analytical results in detail and perform com-parison to MC simulations, resulting in an interesting analysis of the divergence between high and low T regimes.

(2) The integrated PSD is analyzed as function of parameters of the cyclo-stationary input considered. Our results are ableto model the conditions which lead to the largest integrated PSD for each kind of cyclo-stationary excitation studied.

Our paper is organized as follows: in Section 2 we derive an extension of Machlup results for a general cyclo-stationaryexcitation. In Section 3 we perform a study of the two cases of excitation waveforms of practical interest: square wave andsinusoidal. The numerical results (Monte Carlo simulations and numerical Fourier transforms) and the their comparisonswith analytical results are presented in Sections 4 and 5. Finally, our main conclusions and summaries are presented inthe Section 6.

2. The model and our extension

Let us consider X 2 Sq ¼ fx1; . . . ; xqg, a discrete q-state random variable at instant p. Let us denote pðtÞxi ;xjðsÞ ¼

PrðXs ¼ xjjXt ¼ xiÞ. For the particular case of a two-state random variable X 2 f0;1g, like a random telegraph signal, pðtÞ1;1ðsÞdenotes the probability to have a even number of transitions starting from Xt ¼ 1. We can write, supposing the natural nor-malization pðtÞ1;0ðkÞ ¼ 1� pðtÞ1;1ðkÞ, that the simple equation to describe the continuous time evolution of pðtÞ1;1ðsÞ,

dpðtÞ1;1

ds¼ �kðsÞpðtÞ1;1ðsÞ þ 1� pðtÞ1;1ðsÞ

h ilðsÞ ¼ � kðsÞ þ lðsÞ½ � pðtÞ1;1ðsÞ þ lðsÞ

where pð0! 1Þds ¼ kðsÞds and pð1! 0Þds ¼ lðsÞds are respectively the transition probabilities. This equation is a naturalextension of Machlup equation [1] in the sense of the transition probabilities are time dependent.

The general solution of this equation is given by:

pðtÞ1;1ðsÞ ¼ CðtÞe�R s

kðyÞþlðyÞ dy þ e�R s

kðyÞþlðyÞ dyZ s

eR x

kðyÞþlðyÞ dylðxÞdx

and from initial conditions is obtained

CðtÞ ¼ 1� e�R t

kðyÞþlðyÞdyR teR x

kðyÞþlðyÞ dylðxÞdx

e�R t

kðyÞþlðyÞdy¼ eR t

kðyÞþlðyÞdy �Z t

eR x

kðyÞþ lðyÞ dylðxÞdx;

due to pðtÞ1;1ðtÞ ¼ 1.The complete solution is now written as:

pðtÞ1;1ðsÞ ¼ eR t

kðyÞþlðyÞdy �Z t

eR x

kðyÞþlðyÞ dylðxÞdx� �

e�R s

kðyÞþlðyÞdy þ e�R s

kðyÞþlðyÞdyZ s

eR x

kðyÞþlðyÞdylðxÞdx

¼ e�R s

tkðyÞþlðyÞdy þ

Z s

teR x

kðyÞþlðyÞdylðxÞdx� �

e�R s

kðyÞþlðyÞ dy:

The knowledge of this probability allows computing the autocorrelation of the generated signal, which is calculated through:

Aðt; sÞ ¼ XtXs ¼X

xi ;xj2f0;1g

xixjPrðXs ¼ xjjXt ¼ xiÞPrðXt ¼ xiÞ ¼ PrðXs ¼ 1jXt ¼ 1ÞPrðXt ¼ 1Þ ¼ pðtÞ1;1ðsÞPrðXt ¼ 1Þ

Let us consider p1ðtÞdt the probability that state 1 will not make a transition for time t, then will make one between times tand t þ dt. Thus we have

p1ðtÞdt ¼ uðtÞkðtÞdt

and uðtÞ is the probability that after time t state 1 will not have made a transition and kðtÞ the probability (per unit of time)of making a transition to state 0 at time t. However we have

uðt þ dtÞ ¼ uðtÞð1� kðtÞdtÞ

that du=dt ¼ �uðtÞkðtÞ; and under initial conditions we have uðtÞ ¼ exp �R t

0 kðsÞds� �

, which leads to

p1ðtÞ ¼ kðtÞ exp �Z t

0kðsÞds

� �� �

what is normalized sinceR1

0 p1ðtÞdt ¼ �R1

0ddt exp �

R t0 kðsÞds dt ¼ 1� exp �

R10 kðsÞds

� ¼ 1.

Similarly, if ptð0! 1Þ ¼ lðtÞ, we have p0ðtÞ ¼ lðtÞ exp �R t

0 lðsÞds� �

. To determine the probability PrðXt ¼ 1Þ in the time

dependent case is very complicate. However, as a first approximation we can write:

970 R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977

PrðXt ¼ 1Þ ¼ th i1th i0 þ th i1

¼R1

0 tp1ðtÞdtR10 t½p1ðtÞ þ p0ðtÞ�dt

; ð1Þ

where hti1 is the average residence time in state 1 and hti0 the same for state 0.In Machlup’s model, kðtÞ and lðtÞ are constants, and are made kðtÞ ¼ 1=sc and lðtÞ ¼ 1=se, where sc and se are time con-

stants (emission and capture, respectively) since in this case hti1 ¼ ð1=scÞR1

0 te�t=sc dt ¼ sc and hti2 ¼ ð1=seÞR1

0 te�t=se dt ¼ se

which implies in PrðXt ¼ 1Þ ¼ sc=ðse þ scÞ.On the other hand, if kðtÞ and lðtÞ are periodic functions of time, a more detailed analysis is required. Considering a period

T, we have:

th i1 ¼Z 1

0tkðtÞ exp �

Z t

0kðsÞds

� �dt ¼ �

Z 1

0t

ddt

exp �Z t

0kðsÞds

� �dt

¼ t exp �Z t

0kðsÞds

� �1

0þZ 1

0exp �

Z t

0kðsÞds

� �dt ¼

Z 1

0exp �

Z t

0kðsÞds

� �dt

¼X1k¼0

Z ðkþ1ÞT

kTexp �

Z t

0kðsÞds

� �dt ¼

X1k¼0

Z T

0exp �kT kh i �

Z nt

0kðsÞds

� �dnt ¼

R T0 exp �

R nt

0 kðsÞds� �

dnt

1� expð�T kh iÞ

and similarly

th i0 ¼

R T0 exp �

R nt

0 lðsÞds� �

dnt

1� expð�T lh iÞ

with h�i ¼ ð1=TÞR T

0 � dx denotes the time average on the period T.Therefore, we have a complete expression for PrðXt ¼ 1Þ:

PrðXt ¼ 1Þ ¼1�e�T lh i

1�e�T kh i

� ��R T

0 exp �R nt

0 kðsÞds� �

dnt

1� e�T lh ið ÞR T

0 e�R nt

0kðsÞds

� �dnt þ 1� e�T kh ið Þ

R T0 e

�R nt

0lðsÞds

� �dnt

ð2Þ

For t ¼ 0 the autocorrelation is then given by

AðsÞ ¼ PrðX0 ¼ 1Þ e�R s

0kðyÞþlðyÞdy þ

Z s

0eR x

kðyÞþlðyÞdylðxÞdx� �

e�R s

kðyÞþlðyÞ dy� �

¼ PrðX0 ¼ 1Þ e�R s

tkðyÞþlðyÞdy þ

Z s

teR x

0kðyÞþlðyÞdylðxÞdx

� �e�R s

0kðyÞþlðyÞ dy

� �;

and a complete formula is

AðsÞ ¼1�e�T lh i

1�e�T kh i

� ��R T

0 exp �R nt

0 kðtÞdt� �

dnt 1þR s

0 eR x

0kðyÞþlðyÞdylðxÞdx

� �

1� e�T lh ið ÞR T

0 e�R nt

0kðtÞdt

� �dnt þ 1� e�T kh ið Þ

R T0 e

�R nt

0lðtÞdt

� �dnt

e�R s

0kðyÞþlðyÞ dy

: ð3Þ

2.1. Approximated results: a general formula for T small if compared to transition probabilities

If many transitions can occur during a cycle ðTÞ, to solve the autocorrelation of a cyclo-stationary noise is a complex task,because this corresponds to a ‘‘turbulent” regime and in this case MC simulations are more appropriate than the existinganalytical results. However, for the regime where hliT and hkiT are small values we can obtain an analytical and interestingapproximation. In this context, the meaning of ‘‘small” is also investigated by MC simulations. The situations in which theresults from MC simulations become different from the analytical formulation are elucidated, and for this situations we canpropose a suitable formulation, showing that modulation phenomena can appear.

A first order approximation is to consider that for hkiT small: exp �R nt

0 kðsÞds� �

� expð�nthkiÞ resulting in:

th i1 ¼R T

0 exp �nt kh ið Þdnt

1� expð�T kh iÞ ¼1kh i

and similarly hti0 ¼ 1hli, which leads to

PrðXt ¼ 1Þ � lh ilh i þ kh i ;

from (1).

R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977 971

In this section we will obtain a general formulation, independent from the waveform of the cyclo-stationary signal, validin the regime of small T. So, it is convenient to consider the parameterization s ¼ nT , where n ¼ 0;1;2 . . .. The reader mustobserve that this approximation is interesting only in case of small values of T. For this we can evaluate the simple interme-diate expression:

Z s

0kðyÞ þ lðyÞdy ¼

Xn�1

k¼0

Z ðkþ1ÞT

kTkðyÞdyþ

Xn�1

k¼0

Z ðkþ1ÞT

kTlðyÞdy ¼ nT � kh i þ lh ið Þ

On the other hand, if x ¼ kT þ nx, with 0 < nx < T , we have that:

Z s

0eR x

0kðyÞþlðyÞdylðxÞdx ¼

Xn�1

k¼0

Z ðkþ1ÞT

kTeR x

0kðyÞþlðyÞdylðxÞdx ¼

Xn�1

k¼0

Z T

0eR kTþnx

0kðyÞþlðyÞdylðkT þ nxÞdnx

¼Xn�1

k¼0

Z T

0eR kTþnx

0kðyÞþlðyÞdylðnxÞdnx ð4Þ

if lðtÞ is a periodic function of t with period T.For small values of T, a first approximation is

R kTþnx

0 kðyÞ þ lðyÞdy � kTðhki þ hliÞ. The autocorrelation can be written as

AðsÞ ¼ lh ilh i þ kh i 1þ T lh i

Xn�1

k¼0

ekT kh iþ lh ið Þ

" #e�s kh iþ lh ið Þ ¼ lh i

lh i þ kh i 1þ T lh i enT kh iþ lh ið Þ � 1eT kh iþ lh ið Þ � 1

� �e�s kh iþ lh ið Þ

¼ lh ilh i þ kh i 1� T lh i

eT kh iþ lh ið Þ � 1

� �e�s kh iþ lh ið Þ þ lh i

lh i þ kh iT lh i

eT kh iþ lh ið Þ � 1

where in the first passage it must be observed that we have a geometrical sum of n terms, and similarly a Taylor expansionleads to: eTðhkiþhliÞ � 1 � Tðhki þ hliÞ and we have

AðsÞ ¼ lh ilh i þ kh i 1� lh i

kh i þ lh ið Þ

� �e�s kh iþ lh ið Þ þ lh i2

lh i þ kh i½ �2ð5Þ

The Fourier transform of AðsÞ leads to the power spectral density, where AðsÞ ¼ Að�sÞ. In these conditions we have:

SðxÞ ¼ 12p

Z 1

�1AðsÞeixsds ¼ 1

plh i

lh i þ kh i 1� lh ikh i þ lh ið Þ

� �kh i þ lh i

kh i þ lh ið Þ2 þx2þ dðxÞ

and we have then:

SðxÞ ¼ 1p

lh i kh ikh i þ lh ið Þ �

1

kh i þ lh ið Þ2 þx2þ lh i2

lh i þ kh i½ �2dðxÞ ð6Þ

where dðxÞ is the delta de Dirac function.The Machlup results [1] are recovered if we consider the emission and absorption probabilities to be constant values

along time, particularly making l ¼ 1=s and k ¼ 1=r we have the autocorrelation:

AMðsÞ ¼r

ðrþ sÞ2s exp � 1

rþ 1

s

� �s

� �þ r

� �ð7Þ

and the power spectral density:

SMðxÞ ¼1p

rsrþ sð Þ2

1rþ 1

s

� rþsrs

� 2 þx2h iþ r2

ðrþ sÞ2dðxÞ: ð8Þ

In the next sections we will consider two kinds of cyclo-stationary excitation for a detailed analysis of the power spectraldensity: square wave and sinusoidal wave.

3. Study cases – analytical results

3.1. Square wave excitation (approximated results)

In this case we have the bias voltage alternating abruptly and periodically between two states, i.e.

lðyÞ ¼1=son if nT 6 y < ðnþ aÞT1=soff if ðnþ aÞT 6 y 6 ðnþ 1ÞT

�

972 R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977

and

kðyÞ ¼1=ron if nT 6 y < ðnþ aÞT1=roff if ðnþ aÞT 6 y P ðnþ 1ÞT

�

where son, soff , ron and roff are constants and 0 < a < 1 is the duty cycle, i.e., the fraction of the period in which the device isin the on state. In this section analytical results for this input signal are explored.

Since hli ¼ asonþ ð1�aÞ

soffand hki ¼ a

ronþ ð1�aÞ

roffand we write the autocorrelation from Eq. (5)

AðsÞ ¼a

sonþ ð1�aÞ

soff

� �a

ronþ ð1�aÞ

roff

� �a ronþson

ronson

� �þ ð1� aÞ roffþsoff

roff soff

� �h i2 e� a ronþson

ronsonð Þþð1�aÞ roffþsoffroff soff

� �h isþ

asonþ ð1�aÞ

soff

� �2

a ronþsonronson

� �þ ð1� aÞ roffþsoff

roff soff

� �h i2

So we can obtain the noise power calculating the Fourier transform SðxÞ ¼ 1p

R10 AðsÞ cosðxsÞds since AðsÞ ¼ Að�sÞ by def-

inition. The power spectral density is then:

SðxÞ ¼ 1p

asonþ ð1�aÞ

soff

� �a

ronþ ð1�aÞ

roff

� �axon þ ð1� aÞxoff½ � �

1

axon þ ð1� aÞxoffð Þ2 þx2þ dðxÞ ð9Þ

where xon ¼ ronþsonronson

and xoff ¼ roffþsoffroff soff

.This formulae for the PSD of the square wave excitation must satisfy two important properties:

(1) SðwÞ is symmetric under changes of ron $ roff and son $ soff if a ¼ 1=2.(2) For ron ¼ roff ¼ r and son ¼ soff ¼ s the Machlup formulation, independent of a, must be obtained.

Although this formula was obtained supposing a regime of small T, the real validity of this formula seems to be numer-ically more relaxed and not yet understood as a whole. The behavior for large values of T is discussed in details. We will ver-ify the validity of this formula comparing the exact formulae with MC simulations in Section 4.

3.2. Sinusoidal excitation

In this section we developed an approach to understand the PSD for sinusoidal input signals. For that, let us considerkðtÞ ¼ c1½2� sinðx0t þu1Þ� and lðtÞ ¼ c2½2� sinðx0t þu2Þ� where x0 ¼ 2p=T. By physical arguments a natural choice isu1 ¼ 0 and u2 ¼ p is considered, because experimental results claim that kðtÞ must increase when lðtÞ decreases and con-versely. So, we have then kðtÞ ¼ c1½2� sinðx0tÞ� and lðtÞ ¼ c2½2þ sinðx0tÞ� and the first integral needed for the evaluation ofthe autocorrelation can be easily calculated:

Z s

0kðyÞ þ lðyÞdy ¼ c1

Z s

02� sinðx0yÞdyþ c2

Z s

02þ sinðx0yÞdy

¼ 2ðc1 þ c2Þs� c1

Z s

0sinðx0yÞdyþ c2

Z s

0sinðx0yÞdy ¼ 2ðc1 þ c2Þsþ

ðc1 � c2Þx0

cosðx0sÞ � 1ð Þ

Here, it is important to mention that we need to parameterize s as nT, simply because in this case the Fermi-level changesaccording to a continuous function, and therefore we can use Eq. (3) instead of (5). However, this does no imply that thisresult cannot be slightly distinct from MC simulations if extreme parameters are used, given that this will change onlythe constant in the power spectral density equation, as we will see at the end of this subsection.

For the sake of simplicity, we can consider the same amplitude for both sinusoidal inputs, i.e., c1 ¼ c2 ¼ c. In this case wehave

R s0 kðyÞ þ lðyÞdy ¼ 4cs and the other integral needed according to our approach is given by

Z s

0eR x

0kðyÞþlðyÞdylðxÞdx ¼ c

Z s

02þ sinðx0xÞ½ �e4cxdx

¼ cx0

16c2 þx20

� 12

� �þ 16c2 þx2

0 � 2cx0 cosðx0sÞ þ 8c2 sinðx0sÞ2ð16c2 þx2

0Þ

� �e4cs:

From Eq. (2) we have:

PrðXt ¼ 1Þ ¼

R T0 exp �2cnt � c

x0cosðx0ntÞ þ c

x0

� �dntR T

0 exp �2cnt � cx0

cosðx0ntÞ þ cx0

� �dnt þ

R T0 exp �2cnt þ c

x0cosðx0ntÞ � c

x0

� �dnt

h i

since hkðtÞi ¼ hlðtÞi ¼ 2c, the autocorrelation is represented by

R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977 973

AðsÞ ¼ PrðXt ¼ 1Þ cx0

16c2 þx20

þ 12

� �þ 16c2 þx2

0 � 2cx0 cosðx0sÞ þ 8c2 sinðx0sÞ2ð16c2 þx2

0Þ

� �e4cs

� �e�4cs

¼ PrðXt ¼ 1Þ cx0

16c2 þx20

þ 12

� �e�4cs þ PrðXt ¼ 1Þ 16c2 þx2

0 � 2cx0 cosðx0sÞ þ 8c2 sinðx0sÞ2ð16c2 þx2

0Þ

� �

If PrðXt ¼ 1Þ � h li=ðh ki þ h liÞwhen h kiT � 1 and h liT � 1 (which in this case corresponds to cT � 1), taking the Fouriertransform and observing that the second term results in a singular term (not shown here), we have

SðxÞ ¼ 1=28pc

cx0

16c2 þx20

þ 12

� �:

1

1þ ðx=4cÞ2�cT�1 1=2

8pc1

1þ ðx=4cÞ2ð10Þ

with corner frequency estimated as x� ¼ 4c. We must observe that the last line of this equation corresponds to Eq. (6) asexpected.

An interesting amount is the integrated power spectral density given by

I ¼Z 1

0SðxÞdx ¼ 1

4cx0

16c2 þx20

þ 12

� �ð11Þ

4. Results I: square wave excitation

First of all we perform simulations to obtain the PSD for the case of square wave excitation. We use the ‘‘unadulterated”parameters r / expðEt � EonÞ=kBT and s / expðEon � EtÞ=kBT such that ron ¼ r and roff ¼ r �m and son ¼ s and soff ¼ s=m,where m ¼ expðEon � Eoff Þ=kBT and kB is the Boltzmann constant, T is the temperature, Et is the energy of trap and Eon; Eoff

are the Fermi-level in state on and off, respectively (see, for instance, [6]). Experimental results show that as the gate voltageis decreased roff increases and simultaneously soff decreases, i.e., the probability of a trap becoming filled decreases and theprobability of a trap becoming empty increases (in n-channel MOSFETS) justifying the introduction of a division and multi-plication by m, respectively, as used in [7]. In our simulations we choose m as an integer number. However, it is just neededthat m P 1.

We first analyze the effect of symmetry on the pure parameters. The left plot in the Fig. 1, shows the PSD for differentvalues of m when symmetry is assumed, and in this case r ¼ s ¼ 800. The symbols correspond to MC simulation resultsand the continuous line corresponds to the analytical results obtained from Eq. (9), since a low T regime was assumed(T ¼ 100 time units). As can be seen in this plot, making a symmetric RTS ðr=s ¼ 1Þ cyclo-stationary always leads to a reduc-tion in the noise power. For non-symmetric RTS the situation is different. This case is shown in the right plot of Fig. 1. In thiscase, we have used r ¼ 700 and s ¼ 900 time units. As can be seen, the noise power can increase or decrease, which is inagreement to the results from [7].

For the Monte Carlo simulations N ¼ 219 time units were used, and the power spectral density was average over a sampleof Nrun ¼ 300 different runs of our algorithm. For each run a time series of RTS noise is calculated and an efficient algorithmcomputes its Fourier transform (fast Fourier transform – FFT), (see for example [8, p. 501]). For curve smoothing a simpleaveraging method was implemented, using a frequency-lag of Dn ¼ 10. It is important to remark the need for a large numberof points, in order to get a fine frequency resolution. The Fig. 2 shows the numerical approximation into the exact value (ana-lytical) for the PSD, for different values of N. We can observe the need for using a large number of points (here we user ¼ s ¼ 800, m ¼ 10 and T ¼ 100).

Fig. 1. Right plot: spectral density function in the symmetric case. Left plot: spectral density function in the non-symmetric case.

Fig. 2. Power spectral density generated for Monte Carlo simulation using different number of points in the fourier transform.

974 R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977

Starting from (9) we can determine the integrated power spectral density for the case of small T as being

Fig. 3.a ¼ 0:8

IðaÞ ¼Z 1

0S0ðxÞdx ¼

12

asonþ ð1�aÞ

soff

� �a

ronþ ð1�aÞ

roff

� �axon þ ð1� aÞxoff½ � :

Fig. 3 shows the behavior of the integrated PSD as function of a. The plot shows two situations, symmetric caseron ¼ son ¼ 600 and roff ¼ 600 � 5 and roff ¼ 600=5 ðm ¼ 5Þ and a non-symmetric case ron ¼ 400 and son ¼ 800 androff ¼ 400 � 5 and soff ¼ 800=5.

An interesting fact to be observed is that the integrated noise power steadily decreases for a decreasing from 1 to 0 in thesymmetric case. This does not occur for the non-symmetric case, since for this case at a � 0:82 a maximum is found.

Finally, we analyze the effects of large periods for the input signals. In this case is interesting to introduce thenomenclature:

x0on ¼ron þ son

ronsonand x0off ¼

roff þ soff

roffsoff

and the cyclo-stationary noise spectral frequency:

x0cyc ¼ ax0on þ ð1� aÞx0off :

When the period T of the excitation signal is not much shorter than capture and emission times, low-frequency noise behav-ior is no longer described by Eq. (9). In this case where the period of the excitation signal is not much shorter than captureand emission times, the RTS PSD becomes a superposition of the spectral density in the on and off states. Fig. 4 depicts thelow-frequency noise for high and low excitation frequencies. As can be seen in this figure, for low excitation frequency thePSD is no longer a pure Lorentzian, but simply the superposition (sum) of two Lorentzians, each one corresponding to thespectral density of the RTS signals in the on and off states. The condition that delimits both situations is the relation betweenx0 ¼ 2p=T and x0on and x0off . If x0 x0on and x0 x0off the resulting spectral density is a pure Lorentzian given by Eq. (9)On the other hand if x0 < x0on and x0 < x0off , the resulting spectral density is simply a superposition of the two Lorentz-ians, as predicted by simple modulation theory. In this figure, T large corresponds to an excitation frequency x0 ¼ 10 rad=s.

A comparisom between the integrated PSD in a symmetric case and a non-symmetric case. A inflection in the non-symetric case is observed nearly.

Fig. 4. A PSD study for small and large T. For small value of T ðxs ¼ 106 rad=sÞwe can observe a fine agreement between MC simulations and the theoreticalprediction. For large T ðxs ¼ 10 rad=sÞ the MC simulations are in a fine agreement with the superposition of two Lorentzians with corner frequenciesrespectively given by: xon ¼ 2200 rad=s and xoff ¼ 25000 rad=s.

R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977 975

In this case two inflection points can be observed, one corresponding to a corner frequency x0on � 2200 rad=s and the otherone corresponding to the corner frequency of the second Lorentzian x0off � 25000 rad=s. The symbols corresponding to theMC simulations, while the continuous curve for this regime ðx0 x0on and x0 x0offÞ is a superposition of two Lorentz-ians exactly with these corner frequencies:

Fig. 5.of T for

Sðx0x0on ;x0off Þ �

ðxÞ ¼ Aaðsonronx3

on�1

1þ ðx=xonÞ2þ ð1� aÞðs0ff roffx3

off Þ�1

1þ ðx=xoffÞ2

" #ð12Þ

where A is a constant numerically adjusted. We can see an excellent agreement between MC simulations and the superpo-sition of Lorentzians. Heuristically, we can think that in this regime the power spectra strongly dependent of time and can beheuristically written instantaneously as StðxÞ ¼ lðtÞkðtÞ

ðlðtÞþkðtÞÞ31

1þðx=½lðtÞþkðtÞ�Þ2, where hSðx0>>x0on ;x0off ÞiðxÞ ¼ 1

T

R T0 StðxÞdt except by

constant A. It must be observed that if the noise is slightly cyclo-stationary (Thki and Thli are small), by averaging S we dem-onstrated that Eq. (6) corresponds to replacing the functions lðtÞ and kðtÞ by their time averages. This is may be seen as akind of adiabatic approximation. However in the non adiabatic case, the average is performed integrating over the PSD: inthis case, heuristically, we have the modulation given by Eq. (12).

On the other hand, for small T ðx0 ¼ 106 rad=sÞ the agreement between the theoretical description (Eq. (9)) depicted bycontinuous curve and MC simulations is perfect, and as shown above, no modulation is observed. To make comparison easy,we show in a same figure the simulation for small T and the non trivial large T regime.

5. Results II: sinusoidal wave excitation

For sinusoidal excitation, we also performed an analysis similar to the described above. First of all, we analyze the agree-ment between the MC simulations and our theoretical formulae, which depends on c and T. Our analysis is for T ¼ 100 time

In the right plot a study of different values of c are analyzed for PSD if the inputs are sinusoidal signals for T ¼ 100. The left plot shows independencec ¼ 10�3.

Fig. 6. Theoretical prediction: large value of T (left plot), small value of T (right plot). For c ¼ 10�3, MC simulations and theoretical prediction agree well.

Fig. 7. Integrated PSD of RTS for sinusoidal excitation as function of c. The plot is considered up to c ¼ 0:01 and shows that the integrated PSD decreases asthe period T increases.

976 R. da Silva, G.I. Wirth / Applied Mathematical Modelling 34 (2010) 968–977

units, while amplitude is varied. Simulations for c ¼ 10�3, 10�2 and 10�1 (left plot Fig. 5) shows that agreement between the-oretical prediction (10) and Monte Carlo simulations is good, observing a small disagreement just for a very high value ofc(10�1, that corresponds hkiT ¼ hliT ¼ 20, an extremely high value). We have tested the effect of T for a specific small valueof c ¼ 10�3 from MC simulations. No differences are observed for all values of T simulated (T ¼ 3, 10, 100, and 1000) as can beobserved in the left plot of the Fig. 5.

However, it is interesting to test the agreement between the MC simulations and theoretical prediction also for c ¼ 10�3

in two distinct situations: a large T (104 time units) and a small T (3 time units), shown in the left and right plots of the Fig. 6,respectively.

The agreement is swiftly better for T ¼ 3 but good in both cases, showing the robustness of results for absorption/emis-sion probabilities of Oð10�3Þ, although up to Oð10�2Þ the MC simulations are well reproduced by theoretical prediction, asseen in Fig. 5.

Finally, we analyze the integrated PSD obtained by Eq. (11) for the interval 0 < c < 10�2. We can observe a sensible dif-ference between the integrated PSD between the small value ðT ¼ 3Þ and large value ðT ¼ 100Þ, which is amplified if c is in-creased (see Fig. 7).

6. Conclusions and summaries

A rigorous analytical analysis of the noise spectral density of RTS under cyclo-stationary excitation is provided. Numericalsimulations to validate the model are performed. The work restores the generality of the original derivation by Machlup forthe noise spectral density of stationary RTS. The derived formulation is valid for arbitrary periodic excitation signals. Thebehavior of the RTS power spectral density at different excitation frequencies is numerically explored by MC simulations.We have studied two different kind of excitation signals: square wave and sinusoidal wave. For square waves our analyticalresults show a good agreement with numerical simulations for small periods of cyclo-stationary input. However for largeperiods the MC simulations are reproduced as a superposition of two Lorentzians with characteristic corner frequencies.

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For symmetric RTS ðr ¼ sÞ, a decrease of integrated power density is always observed as the duty cycle a decreases. In thenon-symmetric case a maximum of integrated power density may be found. For the sinusoidal excitation similar analysis isperformed, good agreement between MC simulations and analytical results is observed, in a regime of small amplitude0 < c < 10�2. For c ¼ 10�3 there is no dependency on the period, and the integrated PSD increases as function of c for differ-ent periods of the input signal.

Acknowledgement

R. da Silva and Gilson Wirth would like to thank the Council Researsh CNPq/Brazil by partial financial support under pro-jects: 480258/2008-2, 577473/2008-5 and CESUP (Centro de Super Computação da UFRGS) for major part of computationalresources used to perform this work.

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