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Correlated Noise Description Using HDLs
8th Vienna International Conference on Mathematical Modelling 8th Vienna International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna Vienna18International Conference on Mathematical Mathematical Modelling 8th Conference on Modelling February 18International - 20, 2015. Vienna University of Technology, Available online at Vienna, www.sciencedirect.com Austria February of Vienna, February 18 18 -- 20, 20, 2015. 2015. Vienna Vienna University University of Technology, Technology, Vienna, Austria Austria Austria
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IFAC-PapersOnLine 48-1 (2015) 556–561
Correlated Correlated Correlated
Noise Noise Noise
Description Description Description
Using Using Using
HDLs HDLs HDLs
Joachim Haase and Andr´ e Lange Joachim Haase and Andr´ e Lange Joachim e Joachim Haase Haase and and Andr´ Andr´ e Lange Lange Fraunhofer IIS/EAS, Dresden, Germany Fraunhofer Dresden, Fraunhofer IIS/EAS, IIS/EAS, Dresden, Germany Germany e-mail: (joachim.haase | andre.lange)@eas.iis.fraunhofer.de Fraunhofer IIS/EAS, Dresden, Germany e-mail: (joachim.haase | andre.lange)@eas.iis.fraunhofer.de e-mail: (joachim.haase || andre.lange)@eas.iis.fraunhofer.de andre.lange)@eas.iis.fraunhofer.de e-mail: (joachim.haase
Abstract: There exist several problems where only a frequency-domain characteristic is given, Abstract: There exist several problems where only a frequency-domain characteristic is given, Abstract: exist problems where only characteristic is for instanceThere the description the terminal by frequency-dependent parameters, Abstract: There exist several severalof problems wherebehaviour only aa frequency-domain frequency-domain characteristic is given, given, for instance the description of the terminal behaviour by frequency-dependent parameters, for instance of the behaviour by frequency-dependent parameters, and where wethe aredescription only interested interminal the results of a small-signal frequency-domain analysis. for instance the description of the terminal behaviour by frequency-dependent parameters, and where we are only interested in the results of aa small-signal frequency-domain analysis. and we are interested the results frequency-domain Therefore, how to make usein an extended modelling approach to analysis. describe and where where we weshow are only only interested inof the results of ofsmall-signal a small-signal small-signal frequency-domain analysis. Therefore, we show how to make use of an extended small-signal modelling approach to Therefore, we show show how to to make make use of of an an extended extended small-signal modelling approach description to describe describe any small-signal behaviour in particular correlatedsmall-signal noise sources using hardware Therefore, we how use modelling approach to describe any small-signal behaviour in particular correlated noise sources using hardware description any small-signal in particular correlated noise sources using hardware languages (HDLs)behaviour such as the Verilog-AMS and VHDL-AMS behavioural modelling description languages. any small-signal behaviour in particular correlated noise sources using hardware description languages (HDLs) such as the Verilog-AMS and VHDL-AMS behavioural modelling languages. languages languages (HDLs) (HDLs) such such as as the the Verilog-AMS Verilog-AMS and and VHDL-AMS VHDL-AMS behavioural behavioural modelling modelling languages. languages. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Simulation languages, Circuit models, Noise analysis, Spectral density, Correlation Keywords: Simulation Simulation languages, Circuit Circuit models, Noise Noise analysis, Spectral Spectral density, Correlation Correlation Keywords: Keywords: Simulation languages, languages, Circuit models, models, Noise analysis, analysis, Spectral density, density, Correlation 2. EXTENDED SMALL-SIGNAL FREQUENCY 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 2. SMALL-SIGNAL FREQUENCY 1. 2. EXTENDED EXTENDED SMALL-SIGNAL FREQUENCY ANALYSIS USING HDLS 1. INTRODUCTION 2. EXTENDED SMALL-SIGNAL FREQUENCY ANALYSIS USING HDLS ANALYSIS USING HDLS ANALYSIS USING HDLS When a description in the small-signal frequency-domain When aa description description in in the the small-signal frequency-domain frequency-domain 2.1 Determination of the Current Frequency When is available, the standard approach that is well-known 2.1 Determination of the Current Frequency When a description in the small-signal small-signal frequency-domain is available, the standard approach that is well-known well-known 2.1 is available, the standard approach that is Determination of of the the Current Current Frequency Frequency and widely used in Spice-like simulation engines for the 2.1 Determination is available, the standard approach that is well-known and widely used in Spice-like simulation engines for the and widely used in Spice-like simulation engines for the We summarize how access to the frequency in analysis of electrical networks is to translate the frequencyand widely used in networks Spice-likeissimulation for the We summarize how access to the current current frequency in analysis of electrical electrical to translate translateengines the frequencyfrequencysummarize how access to the current frequency in analysis of networks to the small-signal analysis is supported in different hardware domain descriptions into linearis lumped element circuits or We We summarize how access to the current frequency in analysis of electrical networks is to translate the frequencysmall-signal analysis is supported in different hardware domain descriptions into linear lumped element circuits or small-signal analysis is supported in different hardware domain descriptions into linear lumped element circuits or description languages. state space equations. This procedure may not circuits be simple small-signal analysis is supported in different hardware domain descriptions into linear lumped element or state space space equations. equations. This This procedure procedure may may not not be be simple simple description languages. state and isspace oftenequations. inaccurate, particularmay when description languages. languages. state Thisin procedure notfrequencybe simple description and is often inaccurate, in particular when frequencyand is often inaccurate, in particular when frequencyVerilog-AMS Small-signal analyses use a complex sysdomain characteristics based on measurements are used. and is often inaccurate, in particular when frequencyVerilog-AMS Small-signal analyses use sysdomain characteristics based on measurements are used. Verilog-AMS Small-signal analyses use aaa complex complex sysdomain characteristics on are tem of equations obtained by linearisation around systhe In particular, problems based occur when the frequency-domain Small-signal analyses use complex domain characteristics based on measurements measurements are used. used. Verilog-AMS tem of equations obtained by linearisation around the In particular, problems occur when the frequency-domain of equations by linearisation around In particular, problems occur when the frequency-domain operating point. A obtained small-signal sinusoidal stimulus is the debehaviour cannot be described by broken rational transfer tem tem of equations obtained by linearisation around the In particular, problems occur when the frequency-domain operating point. A small-signal sinusoidal stimulus is debehaviour cannot be described by broken rational transfer operating A small-signal sinusoidal stimulus is debehaviour cannot described by broken rational transfer scribed bypoint. magnitude and phase. Uncorrelated smallfunctions. It is notbe reasonable to apply such a transformaoperating point. A small-signal sinusoidal stimulus is debehaviour cannot be described by broken rational transfer scribed by magnitude and phase. Uncorrelated smallfunctions. It is not reasonable to apply such a transformascribed by magnitude and phase. Uncorrelated smallfunctions. It is not reasonable to apply such a transformasignal noise sources can be described by their spectral tion if only a frequency-domain analysis is required. Therescribed by magnitude and phase. Uncorrelated smallfunctions. Ita is not reasonable toanalysis apply such a transformasignal noise sources can be described by their spectral tion if only frequency-domain is required. Therenoise can by spectral tion ifitonly aa frequency-domain analysis is Theredensities. Wesources can branch in described a Verilog-AMS model to a fore, makes sense to provide features in behavioural signal noise sources can be be described by their their spectral tion frequency-domain analysis is required. required. There- signal densities. We can branch in aa Verilog-AMS model to a fore, ifit itonly makes sense to to provide provide features in behavioural behavioural densities. We can branch in Verilog-AMS model fore, makes sense features in specific description of the small-signal frequency behaviour modelling languages that allow to features provide models that are densities. We can branch in a Verilog-AMS model to to a a fore, it makes sense to provide in behavioural specific description of the small-signal frequency behaviour modelling languages that allow to provide models that are specific description of the small-signal frequency behaviour modelling languages that allow to provide models that are by evaluating the Verilog-AMS analysis type "ac". The only used for frequency analysis and avoid the transformaspecific description of the small-signal frequency behaviour modelling languages that allow to provide models that are the analysis type "ac". The only used used for for frequency frequency analysis analysis and and avoid the the transformatransforma- by by evaluating evaluating the Verilog-AMS Verilog-AMSreturns analysis type "ac". The only function call analysis("ac") onetype (1) "ac". if the curtion from frequency to time-domain models. instance, by evaluating the Verilog-AMS analysis The only used for frequency analysis and avoid avoid theFor transformafunction call analysis("ac") returns one (1) if the curtion from frequency to time-domain models. For instance, function call analysis("ac") returns one (1) if the curtion from frequency to time-domain models. For instance, rent analysis matches small-signal .AC analysis (Accelera, these problems in particular occur when electromagnetic function call analysis("ac") returns one (1) if the curtion from frequency to time-domain models. For instance, analysis matches small-signal .AC analysis (Accelera, these problems problems in in particular particular occur occur when when electromagnetic electromagnetic rent small-signal these 2014,analysis Section matches 4.6). compatibility problems in when complex systems are rent rent analysis matches small-signal .AC .AC analysis analysis (Accelera, (Accelera, these problems(EMC) in particular occur electromagnetic 2014, Section 4.6). compatibility (EMC) problems in complex systems are 2014, Section 4.6). compatibility (EMC) problems in complex systems are under investigation. 2014, Section 4.6). compatibility (EMC) problems in complex systems are under investigation. investigation. The routine vpi get analog freq() is a Verilog Proceunder The routine vpi get analog analog freq() freq() is is aa Verilog Verilog ProceProceunder The routine vpi get Interface routine freq() that is part the VerilogIn thisinvestigation. paper, we show how the problem can be handled dural The routine vpi(VPI) get analog is a of Verilog Procedural Interface (VPI) routine that is part of the VerilogIn this paper, we show how the problem can be handled Interface (VPI) routine is part of the In this paper, we show how the problem can be handled AMS standard (Accelera, 2014,that Section 12.8) thatVerilogcan be with special emphasis on small signal-noise analysis. We dural dural Interface (VPI) routine that is part of the VerilogIn this paper, we show how the problem can be handled AMS standard (Accelera, 2014, Section 12.8) that can be with special special emphasis emphasis on on small small signal-noise signal-noise analysis. analysis. We We AMS standard (Accelera, 2014, Section 12.8) that can be with used to determine the current frequency used in can smalldescribe how emphasis an arbitrary standard (Accelera, 2014, Section 12.8) that be with special on frequency-domain small signal-noisebehaviour analysis. can We AMS used to determine the current frequency used in smalldescribe how an arbitrary frequency-domain behaviour can used determine current frequency in smalldescribe how an arbitrary frequency-domain behaviour can signalto analysis. Thethe function returns zero used during DC or be described using a system task that makes the current used to determine the current frequency used in smalldescribe how an arbitrary frequency-domain behaviour can signal analysis. analysis. The The function function returns returns zero zero during during DC DC or or be described described using using aa system system task task that that makes makes the the current current signal be analyses. Thus, access to the frequency is, or in simulation frequency available. This approach is in current partic- transient signal analysis. The function returns zero during DC be described using a system task that makes the transient analyses. Thus, access to the frequency is, in simulation frequency available. This approach is in partictransient analyses. Thus, access to the frequency is, in simulation frequency available. This approach is in particprinciple, possible. The user can define a (tool-dependent) ular interesting when the behaviour at higher frequencies transient possible. analyses. The Thus, access to the frequency is, in simulation frequency available. This approach isfrequencies in partic- principle, user can define a (tool-dependent) ular interesting when the behaviour at higher The can define ular when the at frequencies task that possible. allows access to the and that can be is of interesting interest. Such problems occur when investigating the principle, principle, The user user canfrequency define aa (tool-dependent) (tool-dependent) ular when the behaviour behaviour at higher higher frequencies task that that possible. allows access access to the the frequency and that that can can be be is of of interesting interest. Such Such problems occur when when investigating the task to frequency and is interest. problems occur investigating the used that in theallows description of the model behaviour. However, EMC behaviour. This approach also allows for handling task allows access to the frequency and that can be is of interest. Such problems occur when investigating the used in the description of the model behaviour. However, EMC behaviour. This approach also allows for handling in description of the behaviour. However, EMC behaviour. approach allows the VPI routine vpi get freq() is currently not small-signal noise This straightforwardly. we can used used in the the description of analog the model model behaviour. However, EMC behaviour. approach also alsoFurthermore, allows for for handling handling the VPI VPI routine vpi get get analog freq() is currently currently not small-signal noise This straightforwardly. Furthermore, we can can the routine vpi analog freq() is not small-signal noise straightforwardly. Furthermore, we supported in some widely used commercial simulation establish a general approach to handle correlated noise the VPI routine vpi widely get analog freq() is currently not small-signal noise straightforwardly. Furthermore, wenoise can supported in some used commercial simulation establish a general approach to handle correlated some used simulation establish aa general approach to noise tools. The in Verilog-AMS language reference manual does sources. This is important in modelling thecorrelated noise behaviour supported in some widely widely used commercial commercial simulation establish general approach to handle handle correlated noise supported tools. The Verilog-AMS language reference manual does sources. This is important in modelling the noise behaviour The reference manual does sources. This is in behaviour not define correspondinglanguage system task. Mierzwinski al. in systems (Domizioli et al., 2010) andthe thenoise noise of tran- tools. tools. The aaVerilog-AMS Verilog-AMS reference manual et does sources. This is important important in modelling modelling behaviour not define define correspondinglanguage system task. task. Mierzwinski et al. in systems systems (Domizioli et al., al., 2010) and andthe thenoise noise of trantran- not a corresponding system Mierzwinski et al. in (Domizioli et 2010) the noise of (2010) report on a non-standard Verilog-A(MS) system sistors at higher frequencies (McAndrew et al., 2005). We not define a corresponding system task. Mierzwinski et al. in systems (Domizioli et al., 2010) and the noise of tran(2010) report on a non-standard Verilog-A(MS) system sistors at higher frequencies (McAndrew et al., 2005). We report aa non-standard Verilog-A(MS) sistors higher et We task $realfreq that provides the current frequency system during describe basicfrequencies ideas in the(McAndrew main text. Technical details (2010) report on on non-standard Verilog-A(MS) system sistors at atthe higher frequencies (McAndrew et al., al., 2005). 2005). We (2010) task $realfreq that provides the current frequency during describe the basic ideas in the main text. Technical details $realfreq that current frequency during describe the basic in text. small-signal analysis. The toolthe Agilent EESof shall support and basics the ideas underlying theory areTechnical deferred details to the task task $realfreq that provides provides current frequency during describe theof in the the main main text. small-signal analysis. The tool toolthe Agilent EESof shall support support and basics basics ofbasic the ideas underlying theory areTechnical deferred details to the the small-signal The EESof and of the underlying theory are deferred to this function.analysis. However, this is Agilent a feature that shall goes support beyond appendices. small-signal analysis. The tool Agilent EESof shall and basics of the underlying theory are deferred to the this function. function. However, However, this is is a feature that that goes beyond beyond appendices. this appendices. the standard. this function. However, this this is aa feature feature that goes goes beyond appendices. the standard. the the standard. standard. VHDL-AMS Linearisation of nonlinear equations at the VHDL-AMS Linearisation Linearisation of of nonlinear nonlinear equations equations at at the the Work was supported by the German Federal Ministry of Education VHDL-AMS operating pointLinearisation solution as well as providing small-signal VHDL-AMS of nonlinear equations at the Work was supported by the German Federal Ministry of Education operating point solution as well as providing small-signal Work operating point solution as well small-signal frequency sources is done as in Verilog-AMS. and Research (BMBF, by Funding number: 16N12440). was the Federal Ministry operating solution as similarly well as as providing providing small-signal Work was supported supported the German German Federal Ministry of of Education Education frequency point sources is done done similarly as in in Verilog-AMS. Verilog-AMS. and Research (BMBF, by Funding number: 16N12440). frequency sources is similarly as and Research (BMBF, Funding number: 16N12440). frequency sources is done similarly as in Verilog-AMS. and Research (BMBF, Funding number: 16N12440). Copyright © 2015, 2015,IFAC IFAC (International Federation of Automatic Control) 556Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © Copyright © 2015, IFAC 556 Peer review© of International Federation of Automatic Copyright 2015, IFAC 556 Copyright ©under 2015,responsibility IFAC 556Control. 10.1016/j.ifacol.2015.05.066
MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Joachim Haase et al. / IFAC-PapersOnLine 48-1 (2015) 556–561
VHDL-AMS provides a function FREQUENCY that supports access to the frequency during small-signal analyses. However, according to the current VHDL-AMS standard (IEEE Computer Society, 2009), this function cannot be used in simultaneous statements to express the constitutive relations of a model. There are efforts to make this possible in a revised version of the standard (IEEE P1076.1 WG, 2014). Beyond the current version of the standard, application of the function FREQUENCY in simultaneous statements is provided at the moment in a few commercial tools as an undocumented feature. SystemC AMS For linear signal flow (LSF) modules, electrical linear networks (ELN), and timed data flow (TDF) modules, SystemC AMS supports smallsignal frequency analyses (Banerjee and Sur, 2014). For LSF and ELN clusters, the system of equations is automatically transformed from time-domain to smallsignal frequency-domain. For TDF modules, the user can specify complex-valued transfer functions. The function sca ac analysis::sca ac f returns the current frequency in Hertz. Access to time-domain and frequencydomain values at ports is permitted. Complex values at output ports can be processed using a function ac processing. The user is responsible for time and frequency consistency checks. Modelica The Modelica language does not provide specific features to describe small-signal frequency behaviour. Proposals to overcome this shortcoming have been developed (Urquia and Dormido, 2002; Enge et al., 2006; Cellier et al., 2007). A special connector that carries real and imaginary parts of current and voltage phasors has been proposed. 2.2 Principle of Extended Small-signal Frequency Analysis By application of a function that provides access to the frequency f , we can apply an approach to extend frequencydomain analysis as described by Haase et al. (2009) for VHDL-AMS. We briefly sketch the main idea. The following relation is given in the frequency-domain Y (f ) = (a(f ) + j · b(f )) · X(f ), (1)
where j is the imaginary unit, f is the frequency, X(f ) and Y (f ) are the input and the output resp. in the frequencydomain of an element described by the complex scalar transfer characteristic a(f ) + j · b(f ). The equation is equivalent to Y (f ) = a(f ) · X(f ) + bs(f ) · jω · X(f )
(2)
Applying this approach is particularly advantageous when a(f ) + j · b(f ) cannot be easily approximated by a complex transfer function composed of numerator and denominator polynomials depending on j2πf with constant real coefficients. 2.3 Example We use an example from reference (FAT, 2013) to outline the principle. Fig. 1 shows the real and imaginary part of an impedance.
Fig. 1. Real and imaginary part of an impedance described by data points The voltage current constitutive relation is described in VHDL-AMS in Listing 2. ... I == LOOKUP_1D ( FREQUENCY , F , RE ) * V + LOOKUP_1D ( FREQUENCY , F , IM ) / MATH_2_PI / FREQUENCY * V ’ DOT ; ...
Listing 2. Principle of extended small-signal modelling in VHDL-AMS LOOKUP 1D is a lookup table function. The vector F provides frequency points, and the vectors RE and IM provide the corresponding real and imaginary parts resp. of the impedance. The lookup table functions provide values at the current FREQUENCY. Fig. 2 shows a measurement arrangement where Z is characterized by the previously described impedance. The frequency dependency of the scattering parameters of the boardnet replacement (electrical system) is given. The voltage vmeas is measured, and the frequency dependency of the voltage vsource is unknown.
b(f ) 2πf
for with ω = 2πf . The coefficient bs(f ) equals frequencies f > 0. It can be set to zero for the frequency f = 0. During a DC or transient analyses, f equals zero. The following Verilog-AMS description is in the frequencydomain equivalent to equation (2) ... real x ; real y ; ... y = a ( $realfreq )* x + bs ( $realfreq )* ddt ( x );
Listing 1. Principle of extended small-signal modelling in Verilog-AMS 557
557
Fig. 2. Measurement arrangement
MATHMOD 2015 558 February 18 - 20, 2015. Vienna, Austria Joachim Haase et al. / IFAC-PapersOnLine 48-1 (2015) 556–561
This voltage can now easily be determined by simulation using a combination of a fixator that connects the node N1 and the reference node with voltage vmeas and current zero, and a norator that replaces the unknown voltage source.
behavioural model, we can propose a general approach that can always be applied. Correlated noise sources are defined by a power spectral density matrix Sxx (f ), see Appendix A.2. Because every correct power spectral density matrix is a complex positive semi-definite Hermitian (Haus and Adler, 1959) a diagonalisation is always possible as follows below T
(3) Sxx (f ) = H(f ) · Λ(f ) · H(f ) The matrix H(f ) is a unitary matrix. Its conjugate transT
pose is H(f ) . The diagonal matrix Λ(f ) consists of only real non-negative values. Thus, correlated noise signals can be provided at the output of a linear system with the transfer function H(f ) and uncorrelated inputs with spectral densities based on Λ(f ). This is in accordance with equation (A.13) in Appendix A.2. We can model the behaviour described by H(f ) as shown in Section 2. The diagonal elements λi of Λ(f ) may not be constant. In this case, the diagonal elements correspond to uncorrelated coloured noise sources. According to equation (A.8) in Appendix (A.1) they can be provided by uncorrelated white noise sources f Swhite noise (f ) = 1 = white noise(1) with the power spectral density one for all frequencies and amplifiers with the gain λi (f ) (cf. also Mierzwinski et al., 2010, slide 16). That means 2 λi (f ) = λi (f ) · white noise(1) (4)
Fig. 3. Network to determine vsource if vmeas is known
because the λi (f ) must be real and always greater equal zero. Thus, this approach is valid for all frequencies and all valid power spectral densities. Fig. 4. Simulation results
white_noise(1)
2( f )
n( f )
X1(f)
X2(f)
H(f)
...
white_noise(1)
...
3. EXTENDED SMALL-SIGNAL NOISE ANALYSIS
1 ( f )
...
The results of the simulation for the network from Figure 3 can be found in Figure 4. The voltage between node N2 and the reference node delivers the voltage vsource.
white_noise(1)
Xn(f)
3.1 Handling of Correlated Noise Sources Noise sources can be described in Verilog-AMS (see Accelera, 2014, Section 4.6.4) and VHDL-AMS (see IEEE Computer Society, 2009, Section 4.3.1.6). Due to the standard procedure supported in network analysis programs (see Appendix A.3), the basic Verilog-AMS and VHDLAMS statements only support uncorrelated noise sources. Correlated noise can be described combining uncorrelated noise sources. Correlated noise sources appear typically when splitting of a model into a noiseless part and corresponding noise sources is carried out (see for instance Blum, 1996, Chapter 9). Several proposals have been suggested to handle analyses with correlated noise sources (see for instance McAndrew et al., 2005; Mierzwinski et al., 2010; Rothe and Dahlke, 1956; Paasschens et al., 2003; Sakalas et al., 2006). They are more or less restricted with respect to some special conditions. Using access to the current frequency in a 558
Fig. 5. Principle of modelling correlated noise based on (3) Only when the elements of H(f ) can be expressed by broken rational expressions that depend on f with constant coefficients and frequency dependent coloured noise sources are provided to model the spectral densities given by the λi (f ), then the arrangement shown in Fig. 5 can be realized by controlled voltage and current sources and lumped passive network elements. Hardware description languages can also be applied in these and all other cases. If the matrices Sxx (f ) are positive definite a Cholesky factorization can be applied (Freund and Hoppe, 2007, Section 4.3). Then, Λ(f ) is the identity matrix of the corresponding dimension and H(f ) is a lower-triangular matrix. Another alternative algorithm in this case is the application of an LDL (also known as LDU) decomposition
MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Joachim Haase et al. / IFAC-PapersOnLine 48-1 (2015) 556–561
where L is a lower triangular and D a diagonal matrix (Watkins, 2004; Krishnamoorthy and Menon, 2011). The matrices Λ(f ) are diagonal matrices. The H(f ) are lowertriangular matrices with diagonal elements that equal one. An LDL decomposition was applied by (Sakalas et al., 2006) for modelling correlated noise in transistor models. For positive definite Hermitian matrices Sxx (f ), the diagonal elements of Λ(f ) must be real and greater than zero. Instead of an LDL decomposition, a decomposition can be carried out where the Λ(f ) are diagonal matrices and the H(f ) are upper-triangular matrices with diagonal elements that equal one. This approach was used by (Herricht et al., 2012). Diagonal elements of Λ(f ) must be real and greater than zero if the corresponding Sxx (f ) is a power spectral density matrix. In a Verilog-AMS model, we can branch to a special description of the small-signal noise behaviour by evaluating the Verilog-AMS analysis type "noise". The function call analysis("noise") returns one (1) if the current analysis matches small-signal .NOISE analysis (Accelera, 2014, Section 4.6). 3.2 Examples for Small-signal Noise Analysis Noise in Systems Haus and Adler (1959) describe how correlated noise sources can be handled in system analyses by using additional subnetworks. Domizioli et al. (2010) apply this approach to investigate compact diversity receivers. Using the current frequency in a Verilog-AMS model, arbitrary behaviour of the additional subnetworks can be described in the frequency-domain. Compact Transistor Modelling Based on Ziel’s investigations (van der Ziel, 1955, 1958; van der Ziel and Bosman, 1984), correlations between base and collector noise currents x1 = ib and x2 = ic resp. can be considered in transistor models in particular for high frequencies (Paasschens et al., 2003; Sakalas et al., 2006; Niu et al., 2001; Herricht et al., 2012). We start with the power density matrix (Herricht et al., 2012) 2 Sib ib (f ) = 2qIB · 1 + 2αqf Bf (ωτBf ) (5) Sib ic (f ) = jωτBf αit · 2qIC
(6)
Sic ic (f ) = 2qIC (7) with the imaginary unit j and ω = 2πf . For noise analyses the DC base and collector currents IB and IC resp. as well as the transistor parameters αqf = alqf, αit = alit, τBf = Tf, and the current gain Bf = betadc and the electron charge q = ‘P Q are constants. Furthermore, we mention that IC = Bf · IB . Applying the modified LDL decomposition (cp. Herricht et al., 2012) we get an upper-tridiagonal matrix H(f ) 1 jωτBf αit H(f ) = 01 2 λib (f ) = 2qIB · 1 + Bf (ωτBf ) · (2αqf − αit 2 )
λic (f ) = 2qIC
(8) (9) (10) 559
559
We could implement these relations straightforwardly in a Verilog-AMS model using a $realfreq equivalent task ... real l_1 , pwr_1 ; real pwr_2 ; ... l_1 = max (1.0 + betadc *( ‘M_TWO_P * $realfreq * Tf )**2) *(2.0* alqf - alit **2) ,0); pwr_1 = sqrt ( l_1 )* white_noise (2.0* ‘P_Q * abs ( I_B )); pwr_2 = white_noise (2.0* ‘P_Q * abs ( I_C )); I ( b_noise ) <+ pwr_1 + Tf * alit * ddt ( pwr_2 ); I ( c_noise ) <+ pwr_2 ; ...
Listing 3. Principle of correlated noise in Verilog-AMS The base noise current flows through the branch b noise. The collector noise current flows through c noise. In a simple way, the presented approach allows to check other approaches in order to handle correlated noise (see for instance Vitale and van der Toorn, 2010). The functions abs and max in Listing 3 shall avoid problems with improper values. These problems can only occur if the power density matrix given by (5), (6), and (7) is not positive (semi-)definite as it is required for power spectral density matrices (see Appendix A.2). 4. CONCLUSION The benefits of small-signal frequency analysis using behavioural modelling languages can be strongly extended if access to the current frequency is supported during .AC and .NOISE analyses. In particular, modelling of the EMC behaviour at higher frequencies and noise descriptions in compact device models can be simplified in this way. We obtain special advantages when modelling correlated noise. We have shown an approach that can be used in simulation engines that support the current Verilog-AMS standard. From the standardization point we believe it would make sense to provide a system task $realfreq that is equivalent to the VPI routine vpi get analog freq(). It is also recommended to handle the problem in a similar way in VHDL-AMS. Last but not least, we would like to encourage EDA vendors to implement the proposed features in their tools. At least, the current standard feature should be supported. This concerns, for instance, the VPI routine vpi get analog freq() (see Accelera, 2014, Section 12.8). REFERENCES Accelera (2014). Verilog-AMS Language Reference Manual Version 2.4.0. Accellera Organization, Inc. URL http://www.accellera.org/downloads/ standards/v-ams/VAMS-LRM-2-4.pdf. Banerjee, A. and Sur, B. (2014). SystemC and SystemCAMS in Practice - SystemC 2.3, 2.2 and SystemC-AMS 1.0. Springer, Cham, Heidelberg et al. Blum, A. (1996). Elektronisches Rauschen. B. G. Teubner, Stuttgart. Cellier, F.E., Clauß, C., and Urqua, A. (2007). Electronic Circuit Modeling and Simulation in Modelica. In Proc.
MATHMOD 2015 560 February 18 - 20, 2015. Vienna, Austria Joachim Haase et al. / IFAC-PapersOnLine 48-1 (2015) 556–561
6th Eurosim Congress on Modelling and Simulation (EUROSIM 2007), 1–10. URL http://www.inf.ethz. ch/personal/cellier/Pubs/Elect/eurosim 07.pdf. Davenport Jr., W.B. and Root, W.L. (1987). An Introduction to the Theory of Random Signals and Noise. WileyIEEE Press. IEEE Press edition of a book published by McGraw Hill Book Company in 1958 under the same title. Domizioli, C.P., Hughes, B.L., Gard, K.G., and Lazzi, G. (2010). Noise Correlation in Compact Diversity Receivers. IEEE Trans. Commun., 58(5), 1426–1436. Enge, O., Clauß, C., Schneider, P., Schwarz, P., Vetter, M., and Schwunk, S. (2006). Quasi-stationary AC Analysis Using Phasor Description With Modelica. In Proc. Modelica Conference 2006 (Modelica 2006), 579–588. The Modelica Association. URL https://www.modelica.org/events/modelica2006/ Proceedings/sessions/Session5d2.pdf. FAT (2013). Modellierung des dynamischen Verhaltens von Komponenten im Bordnetz unter Ber¨ ucksichtigung des EMV-Verhaltens im Hochvoltbereich. FATSchriftenreihe 248. FAT, Berlin. URL http://www.vda. de/de/publikationen/publikationen downloads/ detail.php?id=1142. Freund, R.W. and Hoppe, R.H. (2007). Stoer/Bulirsch: Numerische Mathematik 1. Springer-Verlag, Berlin Heidelberg New York, 10 edition. Haase, J., Hessel, E., and Mammen, H.T. (2009). Proposal to Extend Frequency Domain Analysis in VHDL-AMS. In Proc. Forum on specification Design Languages (FDL 2009), 1–4. Haus, H.A. and Adler, R.B. (1959). Circuit Theory of Linear Noisy Networks. Technology Press of Massachusetts Institute of Technology, Cambridge (MA). Herricht, J., Sakalas, P., Ramonas, M., Schroter, M., Jungemann, C., Mukherjee, A., and Moebus, K. (2012). Systematic Compact Modeling of Correlated Noise in Bipolar Transistors. IEEE Trans. Microw. Theory Tech., 60(11), 3403–3412. doi:10.1109/TMTT.2012. 2216284. IEEE Computer Society (2009). Behavioural Languages - Part 6: VHDL Analog and Mixed-Signal Extensions. IEEE SA-Standards Board, 1.0 2009-12 edition. doi: 10.1109/IEEESTD.2009.5465882. IEC 61691-6, IEEE Std 1076.1. IEEE P1076.1 WG (2014). Frequency-Domain Modeling. URL http://www.eda.org/ twiki/bin/view.cgi/P10761/ProjectsArea# Frequency Domain Modeling Champi. IEEE P1076.1 WG Project Area. Krishnamoorthy, A. and Menon, D. (2011). Matrix Inversion Using Cholesky Decomposition. Computing Research Repository CoRR, abs/1111.4144. McAndrew, C.C., Coram, G., Blaum, A., and Pilloud, O. (2005). Correlated Noise Modeling and Simulation. In Technical Proceedings of the 2005 NSTI Nanotechnology Conference and Trade Show (Nanotech). Mierzwinski, M., O’Halloran, P., Troyanovsky, B., and Sharrit, D. (2010). Verilog-A/MS for RF Simulation. URL http://www.mos-ak.org/california/ talks/03 Mierzwinsk MOS-AK SF2010.pdf. Presentation at MOS-AK/GSA Workshop.
560
Niu, G., Cressler, J.D., Zhang, S., Ansley, W.E., Webster, C.S., and Harame, D.L. (2001). A Unified Approach to RF and Microwave Noise Parameter Modeling in Bipolar Transistors. IEEE Trans. Electron Devices, 48(11), 2568–2574. Paasschens, J., Havens, R., and Tiemeijer, L. (2003). Modelling the Correlation in the High-Frequency Noise of (Hetero-junction) Bipolar Transistors using ChargePartitioning. In Proc. of the Bipolar/BiCMOS Circuits and Technology Meeting (BCTM 2003), 221–224. doi: 10.1109/BIPOL.2003.1274970. Rohrer, R., Nagel, L., Meyer, R., and Weber, L. (1971). Computationally Efficient Electronic-Circuit Noise Calculations. IEEE J. Solid-State Circuits, 6(4), 204–213. Rothe, H. and Dahlke, W. (1956). Theory of Noisy Fourpoles. Proc. of the IRE, 44(6), 811–818. Sakalas, P., Herricht, J., Chakravorty, A., and Schroter, M. (2006). Compact Modeling of High Frequency Correlated Noise in HBTs. In Proc. of the Bipolar/BiCMOS Circuits and Technology Meeting (BCTM 2006), 1–4. Urquia, A. and Dormido, S. (2002). DC, AC Small-Signal and Transient Analysis of Level 1 N-Channel MOSFET with Modelica. In Proc. of the 2nd International Modelica Conference, 99–108. URL https://modelica.org/ events/Conference2002/papers/p13 Urquia.pdf. van der Ziel, A. (1955). Theory of Shot Noise in Junction Diodes and Junction Transistors. Proc. of the IRE, 43(11), 1639–1646. van der Ziel, A. (1958). Noise in Junction Transistors. Proc. of the IRE, 46(6), 1019–1038. van der Ziel, A. and Bosman, G. (1984). Accurate Expression for the Noise Temperature of Common Emitter Microwave Transistors. IEEE Trans. Electron Devices, 31(9), 1280–1283. Vitale, F. and van der Toorn, R. (2010). A Verilog-A Implementation for Correlated Noise in HBT’s. URL http://ectm.ewi.tudelft.nl/publications pdf/ 110111 SAFE2010.pdf. Watkins, D.S. (2004). Fundamentals of Matrix Computation. John Wiley & Sons, Inc., New York, 2 edition. Appendix A. BASICS A.1 Basics of Small-signal Noise Analysis We assume a zero mean random ergodic process. Based on the Wiener-Khintchin theorem, the power spectral density (PSD) Sxx of a wide-sense stationary random process can be defined to be the Fourier transform of the autocorrelation function cxx (cf. Davenport Jr. and Root, 1987, Section 6-6). If the time series x describes an ergodic process, we obtain the frequency dependency of the PSD as follows ∞ cxx (τ ) exp(−j2πf τ ) dτ (A.1) Sxx (f ) = −∞ cxx (τ ) = E x(t)x(t − τ ) (A.2) T 1 = lim x(t)x(t − τ ) dt (A.3) T →∞ 2T −T E returns the expectation value of the random process, and x ¯ is the complex conjugate of x. The inverse Fourier transform
MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Joachim Haase et al. / IFAC-PapersOnLine 48-1 (2015) 556–561
cxx (τ ) =
∞
Sxx (f ) exp(j2πf τ ) df
(A.4)
−∞
yields the average power of x for τ = 0, T ∞ 1 Sxx (f ) df = lim |x(t)|2 dt (A.5) cxx (0) = T →∞ 2T −∞ −T xT equals x within the interval [−T , T ]. Outside the interval xT equals zero. We assume that the Fourier transform XT of xT exists. Then, we find that the power spectral density is proportional to the product of the Fourier transform and its complex conjugate 1 Sxx (f ) = lim XT (f ) · XT (f ) (A.6) T →∞ 2T Considering a linear SISO system with the transfer function H(f ) and a random input x and output y we can determine the output’s power spectral density 1 H(f ) XT (f ) · H(f ) XT (f ) (A.7) 2T 2 (A.8) = |H(f )| Sxx (f )
Syy (f ) = lim
T →∞
Similarly, we can define a time cross-correlation function cx1 x2 corresponding to the time series x1 and x2 and a cross-spectral density 1 Sx1 x2 (f ) = lim X1T (f ) · X2T (f ) (A.9) T →∞ 2T It is obvious that Sx1 x2 (f ) = Sx2 x1 (f ).
(A.10)
A.2 Power Spectral Density Matrix for System Analysis Let x and X be column vectors that represent the time series of a real vector-valued process and its Fourier transform resp. The outer product T
X ⊗ X = (X1 , X2 , . . . , Xn ) · (X1 , X2 , . . . , Xn ) defines the power spectral density matrix
(A.11)
1 XT (f ) ⊗ XT (f ) Sxx (f ) = lim T →∞ 2T Sx1 x1 (f ) Sx1 x2 (f ) . . . Sx1 xn (f ) Sx2 x1 (f ) Sx2 x2 (f ) . . . Sx2 xn (f ) = .. . . .. ... . . . Sxn x1 (f ) Sxn x2 (f ) . . . Sxn xn (f ) Sx1 x1 (f ) Sx1 x2 (f ) . . . Sx1 xn (f ) Sx1 x2 (f ) Sx2 x2 (f ) . . . Sx2 xn (f ) = .. .. . . .. . . . .
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series combined in the n dimensional vector x corresponding to the n-dimensional vector of Fourier transforms X. In a similar manner, we can describe the m-dimensional outputs y and Y resp. The analogue to the equation (A.8) for the SISO system is T 1 Syy (f ) = lim H(f ) XT (f ) · H(f ) XT (f ) (A.12) T →∞ 2T We finally get Syy (f ) = H(f ) · Sxx (f ) · H(f )
T
(A.13)
If the random inputs of a linear system are uncorrelated, the matrix Sxx is a diagonal matrix with positive real diagonal elements. Because of (A.13) we have to consider - even in the case of uncorrelated inputs - that in most cases random outputs of a linear system are correlated. The power spectral density of the k-th output can be determined in this case by n 2 |Hk,i (f )| · Sxi xi (f ) (A.14) Syk yk (f ) = i=1
We also can determine cross-spectral densities using the results of simple small-signal analyses. Details can be found in the paper of (Blum, 1996), Chapter 9. A.3 Noise Analysis with Uncorrelated Sources
In general, network analysis programs that use Spice-like algorithms support small-signal noise analysis with uncorrelated sources. The noise analysis algorithm determines one output signal. An arbitrary number of inputs described by their spectral densities Sxx is possible. That means, a MISO system’s behaviour is investigated. The basic idea of the efficient algorithm that is applied can be found in the paper of (Rohrer et al., 1971). We shortly figure out this basic idea. Starting point is a complex linear system of network equations in the frequency-domain n ci Xi (f ) (A.15) A(f ) · Y(f ) = i=1
Sxx (f ) equals its conjugate transpose Sxx (f ) . Thus, the power spectral density matrix is a Hermitian. Moreover, it is also positive semi-definite (cf. Haus and Adler, 1959, Section 2.1).
with a vector ci . Xi (f ) characterizes a scalar random process with the power spectral density Sxi xi . To determine the power spectral density of the k-th component of Y, we solve the linear complex system of equations A(f )T · ak (f ) = ek (A.16) where ek is vector. All components of the vector are zero, only the k-th component is 1. ak (f )T equals the k-th row of A(f )−1 . Only if all random processes Xi (f ) with i = 1, . . . , n are uncorrelated, the required power spectral density can be determined in accordance with equation (A.14) using n 2 Syk yk (f ) = |ak (f ) • ci | · Sxi xi (f ) (A.17)
We can use power spectral density matrices in order to investigate the noise behaviour of a linear time-invariant MIMO system. Let H(f ) be the transfer matrix in the frequency-domain of a system with n inputs and m outputs. The inputs are characterized by n real-valued time
The spectral densities are even functions. In general, the network program implementations use twice of the Fourier transforms S(f ) that is defined for positive f .
Sx1 xn (f ) Sx2 xn (f ) . . . Sxn xn (f ) T
561
i=1
The sign • denotes the inner product of the vectors ak (f ) and ci .
ScienceDirect
IFAC-PapersOnLine 48-1 (2015) 556–561
Correlated Correlated Correlated
Noise Noise Noise
Description Description Description
Using Using Using
HDLs HDLs HDLs
Joachim Haase and Andr´ e Lange Joachim Haase and Andr´ e Lange Joachim e Joachim Haase Haase and and Andr´ Andr´ e Lange Lange Fraunhofer IIS/EAS, Dresden, Germany Fraunhofer Dresden, Fraunhofer IIS/EAS, IIS/EAS, Dresden, Germany Germany e-mail: (joachim.haase | andre.lange)@eas.iis.fraunhofer.de Fraunhofer IIS/EAS, Dresden, Germany e-mail: (joachim.haase | andre.lange)@eas.iis.fraunhofer.de e-mail: (joachim.haase || andre.lange)@eas.iis.fraunhofer.de andre.lange)@eas.iis.fraunhofer.de e-mail: (joachim.haase
Abstract: There exist several problems where only a frequency-domain characteristic is given, Abstract: There exist several problems where only a frequency-domain characteristic is given, Abstract: exist problems where only characteristic is for instanceThere the description the terminal by frequency-dependent parameters, Abstract: There exist several severalof problems wherebehaviour only aa frequency-domain frequency-domain characteristic is given, given, for instance the description of the terminal behaviour by frequency-dependent parameters, for instance of the behaviour by frequency-dependent parameters, and where wethe aredescription only interested interminal the results of a small-signal frequency-domain analysis. for instance the description of the terminal behaviour by frequency-dependent parameters, and where we are only interested in the results of aa small-signal frequency-domain analysis. and we are interested the results frequency-domain Therefore, how to make usein an extended modelling approach to analysis. describe and where where we weshow are only only interested inof the results of ofsmall-signal a small-signal small-signal frequency-domain analysis. Therefore, we show how to make use of an extended small-signal modelling approach to Therefore, we show show how to to make make use of of an an extended extended small-signal modelling approach description to describe describe any small-signal behaviour in particular correlatedsmall-signal noise sources using hardware Therefore, we how use modelling approach to describe any small-signal behaviour in particular correlated noise sources using hardware description any small-signal in particular correlated noise sources using hardware languages (HDLs)behaviour such as the Verilog-AMS and VHDL-AMS behavioural modelling description languages. any small-signal behaviour in particular correlated noise sources using hardware description languages (HDLs) such as the Verilog-AMS and VHDL-AMS behavioural modelling languages. languages languages (HDLs) (HDLs) such such as as the the Verilog-AMS Verilog-AMS and and VHDL-AMS VHDL-AMS behavioural behavioural modelling modelling languages. languages. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Simulation languages, Circuit models, Noise analysis, Spectral density, Correlation Keywords: Simulation Simulation languages, Circuit Circuit models, Noise Noise analysis, Spectral Spectral density, Correlation Correlation Keywords: Keywords: Simulation languages, languages, Circuit models, models, Noise analysis, analysis, Spectral density, density, Correlation 2. EXTENDED SMALL-SIGNAL FREQUENCY 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 2. SMALL-SIGNAL FREQUENCY 1. 2. EXTENDED EXTENDED SMALL-SIGNAL FREQUENCY ANALYSIS USING HDLS 1. INTRODUCTION 2. EXTENDED SMALL-SIGNAL FREQUENCY ANALYSIS USING HDLS ANALYSIS USING HDLS ANALYSIS USING HDLS When a description in the small-signal frequency-domain When aa description description in in the the small-signal frequency-domain frequency-domain 2.1 Determination of the Current Frequency When is available, the standard approach that is well-known 2.1 Determination of the Current Frequency When a description in the small-signal small-signal frequency-domain is available, the standard approach that is well-known well-known 2.1 is available, the standard approach that is Determination of of the the Current Current Frequency Frequency and widely used in Spice-like simulation engines for the 2.1 Determination is available, the standard approach that is well-known and widely used in Spice-like simulation engines for the and widely used in Spice-like simulation engines for the We summarize how access to the frequency in analysis of electrical networks is to translate the frequencyand widely used in networks Spice-likeissimulation for the We summarize how access to the current current frequency in analysis of electrical electrical to translate translateengines the frequencyfrequencysummarize how access to the current frequency in analysis of networks to the small-signal analysis is supported in different hardware domain descriptions into linearis lumped element circuits or We We summarize how access to the current frequency in analysis of electrical networks is to translate the frequencysmall-signal analysis is supported in different hardware domain descriptions into linear lumped element circuits or small-signal analysis is supported in different hardware domain descriptions into linear lumped element circuits or description languages. state space equations. This procedure may not circuits be simple small-signal analysis is supported in different hardware domain descriptions into linear lumped element or state space space equations. equations. This This procedure procedure may may not not be be simple simple description languages. state and isspace oftenequations. inaccurate, particularmay when description languages. languages. state Thisin procedure notfrequencybe simple description and is often inaccurate, in particular when frequencyand is often inaccurate, in particular when frequencyVerilog-AMS Small-signal analyses use a complex sysdomain characteristics based on measurements are used. and is often inaccurate, in particular when frequencyVerilog-AMS Small-signal analyses use sysdomain characteristics based on measurements are used. Verilog-AMS Small-signal analyses use aaa complex complex sysdomain characteristics on are tem of equations obtained by linearisation around systhe In particular, problems based occur when the frequency-domain Small-signal analyses use complex domain characteristics based on measurements measurements are used. used. Verilog-AMS tem of equations obtained by linearisation around the In particular, problems occur when the frequency-domain of equations by linearisation around In particular, problems occur when the frequency-domain operating point. A obtained small-signal sinusoidal stimulus is the debehaviour cannot be described by broken rational transfer tem tem of equations obtained by linearisation around the In particular, problems occur when the frequency-domain operating point. A small-signal sinusoidal stimulus is debehaviour cannot be described by broken rational transfer operating A small-signal sinusoidal stimulus is debehaviour cannot described by broken rational transfer scribed bypoint. magnitude and phase. Uncorrelated smallfunctions. It is notbe reasonable to apply such a transformaoperating point. A small-signal sinusoidal stimulus is debehaviour cannot be described by broken rational transfer scribed by magnitude and phase. Uncorrelated smallfunctions. It is not reasonable to apply such a transformascribed by magnitude and phase. Uncorrelated smallfunctions. It is not reasonable to apply such a transformasignal noise sources can be described by their spectral tion if only a frequency-domain analysis is required. Therescribed by magnitude and phase. Uncorrelated smallfunctions. Ita is not reasonable toanalysis apply such a transformasignal noise sources can be described by their spectral tion if only frequency-domain is required. Therenoise can by spectral tion ifitonly aa frequency-domain analysis is Theredensities. Wesources can branch in described a Verilog-AMS model to a fore, makes sense to provide features in behavioural signal noise sources can be be described by their their spectral tion frequency-domain analysis is required. required. There- signal densities. We can branch in aa Verilog-AMS model to a fore, ifit itonly makes sense to to provide provide features in behavioural behavioural densities. We can branch in Verilog-AMS model fore, makes sense features in specific description of the small-signal frequency behaviour modelling languages that allow to features provide models that are densities. We can branch in a Verilog-AMS model to to a a fore, it makes sense to provide in behavioural specific description of the small-signal frequency behaviour modelling languages that allow to provide models that are specific description of the small-signal frequency behaviour modelling languages that allow to provide models that are by evaluating the Verilog-AMS analysis type "ac". The only used for frequency analysis and avoid the transformaspecific description of the small-signal frequency behaviour modelling languages that allow to provide models that are the analysis type "ac". The only used used for for frequency frequency analysis analysis and and avoid the the transformatransforma- by by evaluating evaluating the Verilog-AMS Verilog-AMSreturns analysis type "ac". The only function call analysis("ac") onetype (1) "ac". if the curtion from frequency to time-domain models. instance, by evaluating the Verilog-AMS analysis The only used for frequency analysis and avoid avoid theFor transformafunction call analysis("ac") returns one (1) if the curtion from frequency to time-domain models. For instance, function call analysis("ac") returns one (1) if the curtion from frequency to time-domain models. For instance, rent analysis matches small-signal .AC analysis (Accelera, these problems in particular occur when electromagnetic function call analysis("ac") returns one (1) if the curtion from frequency to time-domain models. For instance, analysis matches small-signal .AC analysis (Accelera, these problems problems in in particular particular occur occur when when electromagnetic electromagnetic rent small-signal these 2014,analysis Section matches 4.6). compatibility problems in when complex systems are rent rent analysis matches small-signal .AC .AC analysis analysis (Accelera, (Accelera, these problems(EMC) in particular occur electromagnetic 2014, Section 4.6). compatibility (EMC) problems in complex systems are 2014, Section 4.6). compatibility (EMC) problems in complex systems are under investigation. 2014, Section 4.6). compatibility (EMC) problems in complex systems are under investigation. investigation. The routine vpi get analog freq() is a Verilog Proceunder The routine vpi get analog analog freq() freq() is is aa Verilog Verilog ProceProceunder The routine vpi get Interface routine freq() that is part the VerilogIn thisinvestigation. paper, we show how the problem can be handled dural The routine vpi(VPI) get analog is a of Verilog Procedural Interface (VPI) routine that is part of the VerilogIn this paper, we show how the problem can be handled Interface (VPI) routine is part of the In this paper, we show how the problem can be handled AMS standard (Accelera, 2014,that Section 12.8) thatVerilogcan be with special emphasis on small signal-noise analysis. We dural dural Interface (VPI) routine that is part of the VerilogIn this paper, we show how the problem can be handled AMS standard (Accelera, 2014, Section 12.8) that can be with special special emphasis emphasis on on small small signal-noise signal-noise analysis. analysis. We We AMS standard (Accelera, 2014, Section 12.8) that can be with used to determine the current frequency used in can smalldescribe how emphasis an arbitrary standard (Accelera, 2014, Section 12.8) that be with special on frequency-domain small signal-noisebehaviour analysis. can We AMS used to determine the current frequency used in smalldescribe how an arbitrary frequency-domain behaviour can used determine current frequency in smalldescribe how an arbitrary frequency-domain behaviour can signalto analysis. Thethe function returns zero used during DC or be described using a system task that makes the current used to determine the current frequency used in smalldescribe how an arbitrary frequency-domain behaviour can signal analysis. analysis. The The function function returns returns zero zero during during DC DC or or be described described using using aa system system task task that that makes makes the the current current signal be analyses. Thus, access to the frequency is, or in simulation frequency available. This approach is in current partic- transient signal analysis. The function returns zero during DC be described using a system task that makes the transient analyses. Thus, access to the frequency is, in simulation frequency available. This approach is in partictransient analyses. Thus, access to the frequency is, in simulation frequency available. This approach is in particprinciple, possible. The user can define a (tool-dependent) ular interesting when the behaviour at higher frequencies transient possible. analyses. The Thus, access to the frequency is, in simulation frequency available. This approach isfrequencies in partic- principle, user can define a (tool-dependent) ular interesting when the behaviour at higher The can define ular when the at frequencies task that possible. allows access to the and that can be is of interesting interest. Such problems occur when investigating the principle, principle, The user user canfrequency define aa (tool-dependent) (tool-dependent) ular when the behaviour behaviour at higher higher frequencies task that that possible. allows access access to the the frequency and that that can can be be is of of interesting interest. Such Such problems occur when when investigating the task to frequency and is interest. problems occur investigating the used that in theallows description of the model behaviour. However, EMC behaviour. This approach also allows for handling task allows access to the frequency and that can be is of interest. Such problems occur when investigating the used in the description of the model behaviour. However, EMC behaviour. This approach also allows for handling in description of the behaviour. However, EMC behaviour. approach allows the VPI routine vpi get freq() is currently not small-signal noise This straightforwardly. we can used used in the the description of analog the model model behaviour. However, EMC behaviour. approach also alsoFurthermore, allows for for handling handling the VPI VPI routine vpi get get analog freq() is currently currently not small-signal noise This straightforwardly. Furthermore, we can can the routine vpi analog freq() is not small-signal noise straightforwardly. Furthermore, we supported in some widely used commercial simulation establish a general approach to handle correlated noise the VPI routine vpi widely get analog freq() is currently not small-signal noise straightforwardly. Furthermore, wenoise can supported in some used commercial simulation establish a general approach to handle correlated some used simulation establish aa general approach to noise tools. The in Verilog-AMS language reference manual does sources. This is important in modelling thecorrelated noise behaviour supported in some widely widely used commercial commercial simulation establish general approach to handle handle correlated noise supported tools. The Verilog-AMS language reference manual does sources. This is important in modelling the noise behaviour The reference manual does sources. This is in behaviour not define correspondinglanguage system task. Mierzwinski al. in systems (Domizioli et al., 2010) andthe thenoise noise of tran- tools. tools. The aaVerilog-AMS Verilog-AMS reference manual et does sources. This is important important in modelling modelling behaviour not define define correspondinglanguage system task. task. Mierzwinski et al. in systems systems (Domizioli et al., al., 2010) and andthe thenoise noise of trantran- not a corresponding system Mierzwinski et al. in (Domizioli et 2010) the noise of (2010) report on a non-standard Verilog-A(MS) system sistors at higher frequencies (McAndrew et al., 2005). We not define a corresponding system task. Mierzwinski et al. in systems (Domizioli et al., 2010) and the noise of tran(2010) report on a non-standard Verilog-A(MS) system sistors at higher frequencies (McAndrew et al., 2005). We report aa non-standard Verilog-A(MS) sistors higher et We task $realfreq that provides the current frequency system during describe basicfrequencies ideas in the(McAndrew main text. Technical details (2010) report on on non-standard Verilog-A(MS) system sistors at atthe higher frequencies (McAndrew et al., al., 2005). 2005). We (2010) task $realfreq that provides the current frequency during describe the basic ideas in the main text. Technical details $realfreq that current frequency during describe the basic in text. small-signal analysis. The toolthe Agilent EESof shall support and basics the ideas underlying theory areTechnical deferred details to the task task $realfreq that provides provides current frequency during describe theof in the the main main text. small-signal analysis. The tool toolthe Agilent EESof shall support support and basics basics ofbasic the ideas underlying theory areTechnical deferred details to the the small-signal The EESof and of the underlying theory are deferred to this function.analysis. However, this is Agilent a feature that shall goes support beyond appendices. small-signal analysis. The tool Agilent EESof shall and basics of the underlying theory are deferred to the this function. function. However, However, this is is a feature that that goes beyond beyond appendices. this appendices. the standard. this function. However, this this is aa feature feature that goes goes beyond appendices. the standard. the the standard. standard. VHDL-AMS Linearisation of nonlinear equations at the VHDL-AMS Linearisation Linearisation of of nonlinear nonlinear equations equations at at the the Work was supported by the German Federal Ministry of Education VHDL-AMS operating pointLinearisation solution as well as providing small-signal VHDL-AMS of nonlinear equations at the Work was supported by the German Federal Ministry of Education operating point solution as well as providing small-signal Work operating point solution as well small-signal frequency sources is done as in Verilog-AMS. and Research (BMBF, by Funding number: 16N12440). was the Federal Ministry operating solution as similarly well as as providing providing small-signal Work was supported supported the German German Federal Ministry of of Education Education frequency point sources is done done similarly as in in Verilog-AMS. Verilog-AMS. and Research (BMBF, by Funding number: 16N12440). frequency sources is similarly as and Research (BMBF, Funding number: 16N12440). frequency sources is done similarly as in Verilog-AMS. and Research (BMBF, Funding number: 16N12440). Copyright © 2015, 2015,IFAC IFAC (International Federation of Automatic Control) 556Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © Copyright © 2015, IFAC 556 Peer review© of International Federation of Automatic Copyright 2015, IFAC 556 Copyright ©under 2015,responsibility IFAC 556Control. 10.1016/j.ifacol.2015.05.066
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VHDL-AMS provides a function FREQUENCY that supports access to the frequency during small-signal analyses. However, according to the current VHDL-AMS standard (IEEE Computer Society, 2009), this function cannot be used in simultaneous statements to express the constitutive relations of a model. There are efforts to make this possible in a revised version of the standard (IEEE P1076.1 WG, 2014). Beyond the current version of the standard, application of the function FREQUENCY in simultaneous statements is provided at the moment in a few commercial tools as an undocumented feature. SystemC AMS For linear signal flow (LSF) modules, electrical linear networks (ELN), and timed data flow (TDF) modules, SystemC AMS supports smallsignal frequency analyses (Banerjee and Sur, 2014). For LSF and ELN clusters, the system of equations is automatically transformed from time-domain to smallsignal frequency-domain. For TDF modules, the user can specify complex-valued transfer functions. The function sca ac analysis::sca ac f returns the current frequency in Hertz. Access to time-domain and frequencydomain values at ports is permitted. Complex values at output ports can be processed using a function ac processing. The user is responsible for time and frequency consistency checks. Modelica The Modelica language does not provide specific features to describe small-signal frequency behaviour. Proposals to overcome this shortcoming have been developed (Urquia and Dormido, 2002; Enge et al., 2006; Cellier et al., 2007). A special connector that carries real and imaginary parts of current and voltage phasors has been proposed. 2.2 Principle of Extended Small-signal Frequency Analysis By application of a function that provides access to the frequency f , we can apply an approach to extend frequencydomain analysis as described by Haase et al. (2009) for VHDL-AMS. We briefly sketch the main idea. The following relation is given in the frequency-domain Y (f ) = (a(f ) + j · b(f )) · X(f ), (1)
where j is the imaginary unit, f is the frequency, X(f ) and Y (f ) are the input and the output resp. in the frequencydomain of an element described by the complex scalar transfer characteristic a(f ) + j · b(f ). The equation is equivalent to Y (f ) = a(f ) · X(f ) + bs(f ) · jω · X(f )
(2)
Applying this approach is particularly advantageous when a(f ) + j · b(f ) cannot be easily approximated by a complex transfer function composed of numerator and denominator polynomials depending on j2πf with constant real coefficients. 2.3 Example We use an example from reference (FAT, 2013) to outline the principle. Fig. 1 shows the real and imaginary part of an impedance.
Fig. 1. Real and imaginary part of an impedance described by data points The voltage current constitutive relation is described in VHDL-AMS in Listing 2. ... I == LOOKUP_1D ( FREQUENCY , F , RE ) * V + LOOKUP_1D ( FREQUENCY , F , IM ) / MATH_2_PI / FREQUENCY * V ’ DOT ; ...
Listing 2. Principle of extended small-signal modelling in VHDL-AMS LOOKUP 1D is a lookup table function. The vector F provides frequency points, and the vectors RE and IM provide the corresponding real and imaginary parts resp. of the impedance. The lookup table functions provide values at the current FREQUENCY. Fig. 2 shows a measurement arrangement where Z is characterized by the previously described impedance. The frequency dependency of the scattering parameters of the boardnet replacement (electrical system) is given. The voltage vmeas is measured, and the frequency dependency of the voltage vsource is unknown.
b(f ) 2πf
for with ω = 2πf . The coefficient bs(f ) equals frequencies f > 0. It can be set to zero for the frequency f = 0. During a DC or transient analyses, f equals zero. The following Verilog-AMS description is in the frequencydomain equivalent to equation (2) ... real x ; real y ; ... y = a ( $realfreq )* x + bs ( $realfreq )* ddt ( x );
Listing 1. Principle of extended small-signal modelling in Verilog-AMS 557
557
Fig. 2. Measurement arrangement
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This voltage can now easily be determined by simulation using a combination of a fixator that connects the node N1 and the reference node with voltage vmeas and current zero, and a norator that replaces the unknown voltage source.
behavioural model, we can propose a general approach that can always be applied. Correlated noise sources are defined by a power spectral density matrix Sxx (f ), see Appendix A.2. Because every correct power spectral density matrix is a complex positive semi-definite Hermitian (Haus and Adler, 1959) a diagonalisation is always possible as follows below T
(3) Sxx (f ) = H(f ) · Λ(f ) · H(f ) The matrix H(f ) is a unitary matrix. Its conjugate transT
pose is H(f ) . The diagonal matrix Λ(f ) consists of only real non-negative values. Thus, correlated noise signals can be provided at the output of a linear system with the transfer function H(f ) and uncorrelated inputs with spectral densities based on Λ(f ). This is in accordance with equation (A.13) in Appendix A.2. We can model the behaviour described by H(f ) as shown in Section 2. The diagonal elements λi of Λ(f ) may not be constant. In this case, the diagonal elements correspond to uncorrelated coloured noise sources. According to equation (A.8) in Appendix (A.1) they can be provided by uncorrelated white noise sources f Swhite noise (f ) = 1 = white noise(1) with the power spectral density one for all frequencies and amplifiers with the gain λi (f ) (cf. also Mierzwinski et al., 2010, slide 16). That means 2 λi (f ) = λi (f ) · white noise(1) (4)
Fig. 3. Network to determine vsource if vmeas is known
because the λi (f ) must be real and always greater equal zero. Thus, this approach is valid for all frequencies and all valid power spectral densities. Fig. 4. Simulation results
white_noise(1)
2( f )
n( f )
X1(f)
X2(f)
H(f)
...
white_noise(1)
...
3. EXTENDED SMALL-SIGNAL NOISE ANALYSIS
1 ( f )
...
The results of the simulation for the network from Figure 3 can be found in Figure 4. The voltage between node N2 and the reference node delivers the voltage vsource.
white_noise(1)
Xn(f)
3.1 Handling of Correlated Noise Sources Noise sources can be described in Verilog-AMS (see Accelera, 2014, Section 4.6.4) and VHDL-AMS (see IEEE Computer Society, 2009, Section 4.3.1.6). Due to the standard procedure supported in network analysis programs (see Appendix A.3), the basic Verilog-AMS and VHDLAMS statements only support uncorrelated noise sources. Correlated noise can be described combining uncorrelated noise sources. Correlated noise sources appear typically when splitting of a model into a noiseless part and corresponding noise sources is carried out (see for instance Blum, 1996, Chapter 9). Several proposals have been suggested to handle analyses with correlated noise sources (see for instance McAndrew et al., 2005; Mierzwinski et al., 2010; Rothe and Dahlke, 1956; Paasschens et al., 2003; Sakalas et al., 2006). They are more or less restricted with respect to some special conditions. Using access to the current frequency in a 558
Fig. 5. Principle of modelling correlated noise based on (3) Only when the elements of H(f ) can be expressed by broken rational expressions that depend on f with constant coefficients and frequency dependent coloured noise sources are provided to model the spectral densities given by the λi (f ), then the arrangement shown in Fig. 5 can be realized by controlled voltage and current sources and lumped passive network elements. Hardware description languages can also be applied in these and all other cases. If the matrices Sxx (f ) are positive definite a Cholesky factorization can be applied (Freund and Hoppe, 2007, Section 4.3). Then, Λ(f ) is the identity matrix of the corresponding dimension and H(f ) is a lower-triangular matrix. Another alternative algorithm in this case is the application of an LDL (also known as LDU) decomposition
MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Joachim Haase et al. / IFAC-PapersOnLine 48-1 (2015) 556–561
where L is a lower triangular and D a diagonal matrix (Watkins, 2004; Krishnamoorthy and Menon, 2011). The matrices Λ(f ) are diagonal matrices. The H(f ) are lowertriangular matrices with diagonal elements that equal one. An LDL decomposition was applied by (Sakalas et al., 2006) for modelling correlated noise in transistor models. For positive definite Hermitian matrices Sxx (f ), the diagonal elements of Λ(f ) must be real and greater than zero. Instead of an LDL decomposition, a decomposition can be carried out where the Λ(f ) are diagonal matrices and the H(f ) are upper-triangular matrices with diagonal elements that equal one. This approach was used by (Herricht et al., 2012). Diagonal elements of Λ(f ) must be real and greater than zero if the corresponding Sxx (f ) is a power spectral density matrix. In a Verilog-AMS model, we can branch to a special description of the small-signal noise behaviour by evaluating the Verilog-AMS analysis type "noise". The function call analysis("noise") returns one (1) if the current analysis matches small-signal .NOISE analysis (Accelera, 2014, Section 4.6). 3.2 Examples for Small-signal Noise Analysis Noise in Systems Haus and Adler (1959) describe how correlated noise sources can be handled in system analyses by using additional subnetworks. Domizioli et al. (2010) apply this approach to investigate compact diversity receivers. Using the current frequency in a Verilog-AMS model, arbitrary behaviour of the additional subnetworks can be described in the frequency-domain. Compact Transistor Modelling Based on Ziel’s investigations (van der Ziel, 1955, 1958; van der Ziel and Bosman, 1984), correlations between base and collector noise currents x1 = ib and x2 = ic resp. can be considered in transistor models in particular for high frequencies (Paasschens et al., 2003; Sakalas et al., 2006; Niu et al., 2001; Herricht et al., 2012). We start with the power density matrix (Herricht et al., 2012) 2 Sib ib (f ) = 2qIB · 1 + 2αqf Bf (ωτBf ) (5) Sib ic (f ) = jωτBf αit · 2qIC
(6)
Sic ic (f ) = 2qIC (7) with the imaginary unit j and ω = 2πf . For noise analyses the DC base and collector currents IB and IC resp. as well as the transistor parameters αqf = alqf, αit = alit, τBf = Tf, and the current gain Bf = betadc and the electron charge q = ‘P Q are constants. Furthermore, we mention that IC = Bf · IB . Applying the modified LDL decomposition (cp. Herricht et al., 2012) we get an upper-tridiagonal matrix H(f ) 1 jωτBf αit H(f ) = 01 2 λib (f ) = 2qIB · 1 + Bf (ωτBf ) · (2αqf − αit 2 )
λic (f ) = 2qIC
(8) (9) (10) 559
559
We could implement these relations straightforwardly in a Verilog-AMS model using a $realfreq equivalent task ... real l_1 , pwr_1 ; real pwr_2 ; ... l_1 = max (1.0 + betadc *( ‘M_TWO_P * $realfreq * Tf )**2) *(2.0* alqf - alit **2) ,0); pwr_1 = sqrt ( l_1 )* white_noise (2.0* ‘P_Q * abs ( I_B )); pwr_2 = white_noise (2.0* ‘P_Q * abs ( I_C )); I ( b_noise ) <+ pwr_1 + Tf * alit * ddt ( pwr_2 ); I ( c_noise ) <+ pwr_2 ; ...
Listing 3. Principle of correlated noise in Verilog-AMS The base noise current flows through the branch b noise. The collector noise current flows through c noise. In a simple way, the presented approach allows to check other approaches in order to handle correlated noise (see for instance Vitale and van der Toorn, 2010). The functions abs and max in Listing 3 shall avoid problems with improper values. These problems can only occur if the power density matrix given by (5), (6), and (7) is not positive (semi-)definite as it is required for power spectral density matrices (see Appendix A.2). 4. CONCLUSION The benefits of small-signal frequency analysis using behavioural modelling languages can be strongly extended if access to the current frequency is supported during .AC and .NOISE analyses. In particular, modelling of the EMC behaviour at higher frequencies and noise descriptions in compact device models can be simplified in this way. We obtain special advantages when modelling correlated noise. We have shown an approach that can be used in simulation engines that support the current Verilog-AMS standard. From the standardization point we believe it would make sense to provide a system task $realfreq that is equivalent to the VPI routine vpi get analog freq(). It is also recommended to handle the problem in a similar way in VHDL-AMS. Last but not least, we would like to encourage EDA vendors to implement the proposed features in their tools. At least, the current standard feature should be supported. This concerns, for instance, the VPI routine vpi get analog freq() (see Accelera, 2014, Section 12.8). REFERENCES Accelera (2014). Verilog-AMS Language Reference Manual Version 2.4.0. Accellera Organization, Inc. URL http://www.accellera.org/downloads/ standards/v-ams/VAMS-LRM-2-4.pdf. Banerjee, A. and Sur, B. (2014). SystemC and SystemCAMS in Practice - SystemC 2.3, 2.2 and SystemC-AMS 1.0. Springer, Cham, Heidelberg et al. Blum, A. (1996). Elektronisches Rauschen. B. G. Teubner, Stuttgart. Cellier, F.E., Clauß, C., and Urqua, A. (2007). Electronic Circuit Modeling and Simulation in Modelica. In Proc.
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6th Eurosim Congress on Modelling and Simulation (EUROSIM 2007), 1–10. URL http://www.inf.ethz. ch/personal/cellier/Pubs/Elect/eurosim 07.pdf. Davenport Jr., W.B. and Root, W.L. (1987). An Introduction to the Theory of Random Signals and Noise. WileyIEEE Press. IEEE Press edition of a book published by McGraw Hill Book Company in 1958 under the same title. Domizioli, C.P., Hughes, B.L., Gard, K.G., and Lazzi, G. (2010). Noise Correlation in Compact Diversity Receivers. IEEE Trans. Commun., 58(5), 1426–1436. Enge, O., Clauß, C., Schneider, P., Schwarz, P., Vetter, M., and Schwunk, S. (2006). Quasi-stationary AC Analysis Using Phasor Description With Modelica. In Proc. Modelica Conference 2006 (Modelica 2006), 579–588. The Modelica Association. URL https://www.modelica.org/events/modelica2006/ Proceedings/sessions/Session5d2.pdf. FAT (2013). Modellierung des dynamischen Verhaltens von Komponenten im Bordnetz unter Ber¨ ucksichtigung des EMV-Verhaltens im Hochvoltbereich. FATSchriftenreihe 248. FAT, Berlin. URL http://www.vda. de/de/publikationen/publikationen downloads/ detail.php?id=1142. Freund, R.W. and Hoppe, R.H. (2007). Stoer/Bulirsch: Numerische Mathematik 1. Springer-Verlag, Berlin Heidelberg New York, 10 edition. Haase, J., Hessel, E., and Mammen, H.T. (2009). Proposal to Extend Frequency Domain Analysis in VHDL-AMS. In Proc. Forum on specification Design Languages (FDL 2009), 1–4. Haus, H.A. and Adler, R.B. (1959). Circuit Theory of Linear Noisy Networks. Technology Press of Massachusetts Institute of Technology, Cambridge (MA). Herricht, J., Sakalas, P., Ramonas, M., Schroter, M., Jungemann, C., Mukherjee, A., and Moebus, K. (2012). Systematic Compact Modeling of Correlated Noise in Bipolar Transistors. IEEE Trans. Microw. Theory Tech., 60(11), 3403–3412. doi:10.1109/TMTT.2012. 2216284. IEEE Computer Society (2009). Behavioural Languages - Part 6: VHDL Analog and Mixed-Signal Extensions. IEEE SA-Standards Board, 1.0 2009-12 edition. doi: 10.1109/IEEESTD.2009.5465882. IEC 61691-6, IEEE Std 1076.1. IEEE P1076.1 WG (2014). Frequency-Domain Modeling. URL http://www.eda.org/ twiki/bin/view.cgi/P10761/ProjectsArea# Frequency Domain Modeling Champi. IEEE P1076.1 WG Project Area. Krishnamoorthy, A. and Menon, D. (2011). Matrix Inversion Using Cholesky Decomposition. Computing Research Repository CoRR, abs/1111.4144. McAndrew, C.C., Coram, G., Blaum, A., and Pilloud, O. (2005). Correlated Noise Modeling and Simulation. In Technical Proceedings of the 2005 NSTI Nanotechnology Conference and Trade Show (Nanotech). Mierzwinski, M., O’Halloran, P., Troyanovsky, B., and Sharrit, D. (2010). Verilog-A/MS for RF Simulation. URL http://www.mos-ak.org/california/ talks/03 Mierzwinsk MOS-AK SF2010.pdf. Presentation at MOS-AK/GSA Workshop.
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Niu, G., Cressler, J.D., Zhang, S., Ansley, W.E., Webster, C.S., and Harame, D.L. (2001). A Unified Approach to RF and Microwave Noise Parameter Modeling in Bipolar Transistors. IEEE Trans. Electron Devices, 48(11), 2568–2574. Paasschens, J., Havens, R., and Tiemeijer, L. (2003). Modelling the Correlation in the High-Frequency Noise of (Hetero-junction) Bipolar Transistors using ChargePartitioning. In Proc. of the Bipolar/BiCMOS Circuits and Technology Meeting (BCTM 2003), 221–224. doi: 10.1109/BIPOL.2003.1274970. Rohrer, R., Nagel, L., Meyer, R., and Weber, L. (1971). Computationally Efficient Electronic-Circuit Noise Calculations. IEEE J. Solid-State Circuits, 6(4), 204–213. Rothe, H. and Dahlke, W. (1956). Theory of Noisy Fourpoles. Proc. of the IRE, 44(6), 811–818. Sakalas, P., Herricht, J., Chakravorty, A., and Schroter, M. (2006). Compact Modeling of High Frequency Correlated Noise in HBTs. In Proc. of the Bipolar/BiCMOS Circuits and Technology Meeting (BCTM 2006), 1–4. Urquia, A. and Dormido, S. (2002). DC, AC Small-Signal and Transient Analysis of Level 1 N-Channel MOSFET with Modelica. In Proc. of the 2nd International Modelica Conference, 99–108. URL https://modelica.org/ events/Conference2002/papers/p13 Urquia.pdf. van der Ziel, A. (1955). Theory of Shot Noise in Junction Diodes and Junction Transistors. Proc. of the IRE, 43(11), 1639–1646. van der Ziel, A. (1958). Noise in Junction Transistors. Proc. of the IRE, 46(6), 1019–1038. van der Ziel, A. and Bosman, G. (1984). Accurate Expression for the Noise Temperature of Common Emitter Microwave Transistors. IEEE Trans. Electron Devices, 31(9), 1280–1283. Vitale, F. and van der Toorn, R. (2010). A Verilog-A Implementation for Correlated Noise in HBT’s. URL http://ectm.ewi.tudelft.nl/publications pdf/ 110111 SAFE2010.pdf. Watkins, D.S. (2004). Fundamentals of Matrix Computation. John Wiley & Sons, Inc., New York, 2 edition. Appendix A. BASICS A.1 Basics of Small-signal Noise Analysis We assume a zero mean random ergodic process. Based on the Wiener-Khintchin theorem, the power spectral density (PSD) Sxx of a wide-sense stationary random process can be defined to be the Fourier transform of the autocorrelation function cxx (cf. Davenport Jr. and Root, 1987, Section 6-6). If the time series x describes an ergodic process, we obtain the frequency dependency of the PSD as follows ∞ cxx (τ ) exp(−j2πf τ ) dτ (A.1) Sxx (f ) = −∞ cxx (τ ) = E x(t)x(t − τ ) (A.2) T 1 = lim x(t)x(t − τ ) dt (A.3) T →∞ 2T −T E returns the expectation value of the random process, and x ¯ is the complex conjugate of x. The inverse Fourier transform
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cxx (τ ) =
∞
Sxx (f ) exp(j2πf τ ) df
(A.4)
−∞
yields the average power of x for τ = 0, T ∞ 1 Sxx (f ) df = lim |x(t)|2 dt (A.5) cxx (0) = T →∞ 2T −∞ −T xT equals x within the interval [−T , T ]. Outside the interval xT equals zero. We assume that the Fourier transform XT of xT exists. Then, we find that the power spectral density is proportional to the product of the Fourier transform and its complex conjugate 1 Sxx (f ) = lim XT (f ) · XT (f ) (A.6) T →∞ 2T Considering a linear SISO system with the transfer function H(f ) and a random input x and output y we can determine the output’s power spectral density 1 H(f ) XT (f ) · H(f ) XT (f ) (A.7) 2T 2 (A.8) = |H(f )| Sxx (f )
Syy (f ) = lim
T →∞
Similarly, we can define a time cross-correlation function cx1 x2 corresponding to the time series x1 and x2 and a cross-spectral density 1 Sx1 x2 (f ) = lim X1T (f ) · X2T (f ) (A.9) T →∞ 2T It is obvious that Sx1 x2 (f ) = Sx2 x1 (f ).
(A.10)
A.2 Power Spectral Density Matrix for System Analysis Let x and X be column vectors that represent the time series of a real vector-valued process and its Fourier transform resp. The outer product T
X ⊗ X = (X1 , X2 , . . . , Xn ) · (X1 , X2 , . . . , Xn ) defines the power spectral density matrix
(A.11)
1 XT (f ) ⊗ XT (f ) Sxx (f ) = lim T →∞ 2T Sx1 x1 (f ) Sx1 x2 (f ) . . . Sx1 xn (f ) Sx2 x1 (f ) Sx2 x2 (f ) . . . Sx2 xn (f ) = .. . . .. ... . . . Sxn x1 (f ) Sxn x2 (f ) . . . Sxn xn (f ) Sx1 x1 (f ) Sx1 x2 (f ) . . . Sx1 xn (f ) Sx1 x2 (f ) Sx2 x2 (f ) . . . Sx2 xn (f ) = .. .. . . .. . . . .
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series combined in the n dimensional vector x corresponding to the n-dimensional vector of Fourier transforms X. In a similar manner, we can describe the m-dimensional outputs y and Y resp. The analogue to the equation (A.8) for the SISO system is T 1 Syy (f ) = lim H(f ) XT (f ) · H(f ) XT (f ) (A.12) T →∞ 2T We finally get Syy (f ) = H(f ) · Sxx (f ) · H(f )
T
(A.13)
If the random inputs of a linear system are uncorrelated, the matrix Sxx is a diagonal matrix with positive real diagonal elements. Because of (A.13) we have to consider - even in the case of uncorrelated inputs - that in most cases random outputs of a linear system are correlated. The power spectral density of the k-th output can be determined in this case by n 2 |Hk,i (f )| · Sxi xi (f ) (A.14) Syk yk (f ) = i=1
We also can determine cross-spectral densities using the results of simple small-signal analyses. Details can be found in the paper of (Blum, 1996), Chapter 9. A.3 Noise Analysis with Uncorrelated Sources
In general, network analysis programs that use Spice-like algorithms support small-signal noise analysis with uncorrelated sources. The noise analysis algorithm determines one output signal. An arbitrary number of inputs described by their spectral densities Sxx is possible. That means, a MISO system’s behaviour is investigated. The basic idea of the efficient algorithm that is applied can be found in the paper of (Rohrer et al., 1971). We shortly figure out this basic idea. Starting point is a complex linear system of network equations in the frequency-domain n ci Xi (f ) (A.15) A(f ) · Y(f ) = i=1
Sxx (f ) equals its conjugate transpose Sxx (f ) . Thus, the power spectral density matrix is a Hermitian. Moreover, it is also positive semi-definite (cf. Haus and Adler, 1959, Section 2.1).
with a vector ci . Xi (f ) characterizes a scalar random process with the power spectral density Sxi xi . To determine the power spectral density of the k-th component of Y, we solve the linear complex system of equations A(f )T · ak (f ) = ek (A.16) where ek is vector. All components of the vector are zero, only the k-th component is 1. ak (f )T equals the k-th row of A(f )−1 . Only if all random processes Xi (f ) with i = 1, . . . , n are uncorrelated, the required power spectral density can be determined in accordance with equation (A.14) using n 2 Syk yk (f ) = |ak (f ) • ci | · Sxi xi (f ) (A.17)
We can use power spectral density matrices in order to investigate the noise behaviour of a linear time-invariant MIMO system. Let H(f ) be the transfer matrix in the frequency-domain of a system with n inputs and m outputs. The inputs are characterized by n real-valued time
The spectral densities are even functions. In general, the network program implementations use twice of the Fourier transforms S(f ) that is defined for positive f .
Sx1 xn (f ) Sx2 xn (f ) . . . Sxn xn (f ) T
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i=1
The sign • denotes the inner product of the vectors ak (f ) and ci .