CN1645378A - Ripple analyzing method for non-linear circuit signal continuous frequency spectrum - Google Patents

Abstract

An analysing method of wavelet includes building up nonlinear system equation, mapping actual analog zone onto wavelet analog zone, carrying out series expansion for system variable and carrying out system discretization, obtaining wavelet coefficient by solving the equation and calculating out signal value of system variable on discrete time point, utilizing Fourier series expansion formula in frequency domain to obtain continuous requency spectrum.

Description

The wavelet analysis method of signal continuous frequency spectrum in the non-linear circuit

Technical field

The invention belongs to the breadboardin technical field, be specifically related to the wavelet analysis method of signal continuous frequency spectrum in a kind of non-linear circuit.

Technical background

Integrated circuit has developed into and the electronic system that comprises 1,000,000,000 above devices can be integrated on the chip piece, i.e. System on Chip/SoC SOC.At millions of large-scale circuits, how in rational time, quick and precisely the correctness of simulation and its design of checking has become the bottleneck problem of System on Chip/SoC SOC design.According to statistics, the time of SOC chip simplation verification has accounted for 70% of whole design time.

At present increasing SOC chip is that digital-to-analogue is mixed, and along with the continuous progress of mixed signal circuit, wherein the circuit structure of artificial circuit part is complicated day by day, and presents diverse trends.Because the mimic channel overwhelming majority is a non-linear circuit, the time number that its transient state, stable state and spectrum analysis are spent is in the moon, and this makes the Design of Simulating Circuits time that only accounts for chip area 20% will account for 80% of the whole SOC chip design time.Therefore invention can the simulation of fast lifting mimic channel and the gordian technique of verifying speed, and to improving the efficient of SOC chip design, the Time To Market that shortens chip product has important use value.

Exist some to have the signal of continuous frequency spectrum in the actual mixed signal circuit, for evaluation circuits performance effectively, must calculate the continuous frequency spectrum of these signals, electrical noise promptly is wherein a kind of modal signal with continuous frequency spectrum form.Along with reducing and the reduction of supply voltage of process, the electrical noise signal becomes more and more important for the Circuits System Effect on Performance.In order to realize the simulation of mixed signal circuit rapidly and accurately, need noise analysis technology efficiently, therefore, the computing method of non-linear circuit signal continuous frequency spectrum are essential efficiently.

Up to now, the noise analysis approach of most of non-linear circuit [1] [2] [3] [4] nonlinear characteristic of treatment circuit well all.These method supposition noises are enough little to the disturbance that system produces, thereby can be with the response approximately linearization of system to noise.Yet, in the circuit of reality, the usually discontented whole enough little conditions of the system responses that interference noise causes, the linearization of system responses at this moment just can not be satisfied, and the analysis by Linearization method is no longer suitable.The method overwhelming majority of the nonlinear characteristic of disposal system is based on harmonic balance thought [5] [6] [7] effectively, but the harmonic wave equilibrium method algorithm complex is higher, and discrete spectrum that only can picked up signal, and can not handle the signal with continuous frequency spectrum.For the effective nonlinear characteristic of disposal system, continuous frequency spectrum that simultaneously can picked up signal, frequency domain fourier progression expanding method algorithm [8] is suggested, but this algorithm still exists error bigger, finds the solution the high deficiency of complexity.

The deficiencies in the prior art part

Be without loss of generality, a non-linear circuit can be described in order to following equation:

$\frac{\mathrm{dX}\left(t\right)}{\mathrm{dt}}=f(X\left(t\right),t)+\mathrm{Du}\left(t\right)---\left(1\right)$

X(0)＝X ₀

X (t)=[x wherein ₁(t) x ₂(t) ... x _N(t)] ^TRepresenting dimension is the known variables vector of N, f (X (t), t) be about state variable X and time t nonlinear function, the input stimulus of u (t) indication circuit, X (0) has provided the starting condition of circuit, D is a constant coefficient.

The basic thought of harmonic wave equilibrium method [5] [6] [7] is write circuit known variables X (t) as the fourier progression expanding method form in time domain

$X\left(t\right)=\underset{k=-N}{\overset{N}{\mathrm{\&Sigma;}}}{X}_k\&CenterDot;{\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="A20051002330400161.png"\; state="\{\{state\}\}">e</figure-callout>}}^{\frac{j}{}}---\left(2\right)$

Be the linear combination that X (t) can be expressed as direct current signal, fundamental frequency signal and higher hamonic wave signal.Usually, harmonic order N must obtain enough big, to guarantee that being higher than the N subharmonic can ignore for the influence of analog result.Simultaneously, harmonic wave equilibrium method is divided linear and non-linear two parts with entire circuit, and the linear circuit part is directly found the solution in frequency domain, and the non-linear circuit part is found the solution in time domain, need the repeated calculation discrete Fourier to change and inverse discrete Fourier transformer inverse-discrete in the simulation, cause algorithm complex higher.On the other hand, there is its intrinsic deficiency in the harmonic balance method, discrete spectrum that promptly can only picked up signal, and can not obtain the continuous frequency spectrum of signal.

For the nonlinear characteristic of treatment circuit effectively, the continuous frequency spectrum of picked up signal simultaneously, document [8] proposes to come based on fourier progression expanding method in the frequency domain continuous frequency spectrum of picked up signal, thereby realizes the noise analysis of non-linear circuit.

Variable signal and the input signal of supposing circuit are all bandlimited signal, and signal frequency range is [ω _M, ω _M], then the frequency spectrum of variable signal X (t) is deployable is following fourier series form.

$X\left(\mathrm{\&omega;}\right)=\frac{\mathrm{\&pi;}}{{\mathrm{\&omega;}}_M}\underset{k=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{X}_k{e}^{-\mathrm{jk\&pi;\&omega;}/{\mathrm{\&omega;}}_M}\left(\right|\mathrm{\&omega;}|\&le;{\mathrm{\&omega;}}_M)---\left(3\right)$

Formula (3) is carried out inverse fourier transform, can obtain the sampling function series expansion form of the X (t) shown in formula (4) and (5), wherein h=π/ω _M, t _k=kh, sinc (x)=sin (π x)/(π x).What formula (4) and (5) were described is famous sampling theorem just.

$X\left(t\right)=\underset{k=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{X}_k\sin c[(t-t_k)/h]---\left(4\right)$

X _k＝X(t _k) (5)

Get t=t _n, differential being carried out at formula (4) two ends, can get:

$\stackrel{\&CenterDot;}{X}\left(t_n\right)=\frac{1}{h}\underset{k=-\&infin;}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{X}_k\sin c^{\&prime;}[(t_n-t_k)/h]---\left(6\right)$

Because

${\sin c}^{\&prime;}[(t_n-t_k)/h]=\sin c^{\&prime;}(n-k)$

$=\left\{\begin{array}{cc}0,& n=k\\ \frac{{(-1)}^{n-k}}{n-k},& n\&NotEqual;k\end{array}\right.$

Therefore

$\stackrel{\&CenterDot;}{X}\left(t_n\right)=\frac{1}{h}\underset{k\&NotEqual;n}{\overset{\&infin;}{\underset{k=-\&infin;}{\mathrm{\&Sigma;}}}}{X}_k\frac{{(-1)}^{n-k}}{n-k}---\left(7\right)$

With formula (5) and (7) substitution equation (1), then get t=t _nShi Fangcheng (1) can be written as:

$\frac{1}{h}\underset{k\&NotEqual;n}{\overset{\&infin;}{\underset{k=-\&infin;}{\mathrm{\&Sigma;}}}}{X}_k\frac{{(-1)}^{n-k}}{n-k}=f(X_n,t_n)+\mathrm{Du}\left(t_n\right)---\left(8\right)$

Equation (8) is the infinite Nonlinear System of Equations of dimension, comprises infinite known variables X _n

In the practice, it is impossible handling infinite progression shown in (7) formula and (8) formula, need carry out finite term according to accuracy requirement and block.

Formula (7) carried out the 2M+1 item is limited to block (wherein the M size is decided by accuracy requirement), have

$\stackrel{\&CenterDot;}{X}\left(t_n\right)=\frac{1}{h}\underset{k\&NotEqual;n}{\overset{M}{\underset{k=-M}{\mathrm{\&Sigma;}}}}{X}_k\frac{{(-1)}^{n-k}}{n-k}---\left(9\right)$

If choose limited discrete time point t _-M, t _-M+1..., t _MEquation (8) is dispersed, and then equation (8) can be similar to and be write as the system of equations that comprises 2M+1 discrete equation shown in (10), wherein known variables number 2M+1, i.e. X _-M, X _-M+1..., X _M

$\frac{1}{h}\underset{k\&NotEqual;n}{\overset{M}{\underset{k=-M}{\mathrm{\&Sigma;}}}}{X}_k\frac{{(-1)}^{n-k}}{n-k}\&ap;f(X_n,t_n)+\mathrm{Du}\left(t_n\right)---\left(10\right)$

Using iterative algorithm solving equation group (10) can obtain known variables X _-M, X _-M+1..., X _MThe finite term clipped form of modus ponens (4), the time solution X (t) of original system can calculate by following formula.

$X\left(t\right)\&ap;\underset{k=-M}{\overset{M}{\mathrm{\&Sigma;}}}{X}_k\sin c[(t-t_k)/h]---\left(11\right)$

At last, the continuous frequency spectrum of signal can obtain by the finite term clipped form (12) of (3) formula.

$X\left(\mathrm{\&omega;}\right)\&ap;\frac{\mathrm{\&pi;}}{{\mathrm{\&omega;}}_M}\underset{k=-M}{\overset{M}{\mathrm{\&Sigma;}}}{X}_k{e}^{-\mathrm{jk\&pi;\&omega;}/{\mathrm{\&omega;}}_M}\left(\right|\mathrm{\&omega;}|\&le;{\mathrm{\&omega;}}_M)---\left(12\right)$

Though but the continuous frequency spectrum of picked up signal, still there are two big deficiencies in above-mentioned frequency domain fourier progression expanding method algorithm [8].

At first, calculate the signal continuous frequency spectrum that obtains based on (12) formula and exist bigger error.Error comprises two parts, and a part is to block the truncation error that the formula of obtaining (12) is introduced to formula (3) is limited, and this part truncation error can cause Gibbs phenomenon, but can eliminate by the window technique that proposes in the document [9].Another part error is the expansion coefficient X of formula (12) _-M, X _-M+1..., X _MThe error of calculation.X _-M, X _-M+1..., X _MObtain by finding the solution the Nonlinear System of Equations that dimension is 2M+1 (10) and since system of equations (10) is the finite term of infinite dimension system of equations (8) block approximate, the X that obtains by solving equation group (10) _-M, X _-M+1..., X _MHave bigger error, thereby the signal continuous frequency spectrum that causes finally obtaining exists than mistake.

Secondly, the complexity of algorithm is very high.For one group of expansion coefficient X in the acquisition formula (12) _-M, X _-M+1..., X _M, need find the solution the Nonlinear System of Equations that dimension is 2M+1 (10).Based on sampling theorem, signal must be with nyquist frequency 2 ω _MUniform sampling.For given high frequency frequencies omega _M, in order to obtain enough precision, need a large amount of discrete sampling time points, promptly the M value can be very big, thereby cause the dimension of system of equations (10) huge, finds the solution the complexity height.

List of references

[1]A.Demir，E.W.Y.Liu，and?A.L.Sangiovanni?Vincentelli，“Time-domain?non-Monte?Carlonoise?simulation?for?nonlinear?dynamic?circuits?with?arbitrary?excitations，”IEEE?Trans.on?Computer-Aided?Design，vol.15，pp.493-505，May?1996.

[2]M.Okumura，H.Tanimoto，T.Itakura，and?T.Sugawara，“Numerical?noise?analysis?fornonlinear?circuits?with?a?periodic?large?signal?excitation?including?cyclostationary?noisesources，”IEEE?Trans.on?Circuits?and?Systems.Part?I，vol.40，pp.581-590，Sept.1993.

[3]J.Roychowdhury，D.Long，and?P.Feldmann，“Cyclostationary?noise?analysis?of?large?RFcircuits?with?multitone?excitations，”IEEE?J.Solid-State?Circuits，vol.33，pp.324-336，Mar.1998.

[4]R.Telichevesky，K.S.Kundert?and?J.White，“Efficient?AC?and?noise?analysis?of?two-toneRF?circuits，”in?Proc.33rd?Design?Automation?Conf.，pp.292-297，Jun.1996.

[5]Kenneth?S.Kundert?and?Alberto?Sangiovanni-Vincentelli，“Simulation?of?nonlinear?circuitsin?the?frequency?domain”，IEEE?Transactions?on?Computer-Aided?Design，vol.CAD-5，no.4，pp.521-535，Oct.1986.

[6]Kenneth?S.Kundert，Jacob?White，and?Alberto?Sangiovanni-Vincentelli，Steady-StateMethods?for?Simulating?Analog?and?Microwave?Circuits，Kluwer?Academic?Publishers，Norwell，MA，1990.

[7]V.Rizzoli?and?A.Neri，“State?of?the?art?and?present?trends?in?nonlinear?microwave?CADtechniques，”IEEE?Trans.on?Microwave?Theory?Tech.，vol.36，pp.343-365，Feb.1988.

[8]Giorgio?Casinovi，“An?algorithm?for?frequency-domain?noise?analysis?in?nonlinear?systems，”Proceedings?of?the?2002?Design?Automation?Conference，New?Orleans，LA，pp.514-517，Jun.2002.

[9]Giorgio?Casinovi，“Windowing?technique?in?frequency-domain?simulation”，Proceedings?ofthe?2002?IEEE?International?Workshop?on?Behavioral?Modeling?and?Simulation，pp.54-60，Oct.2002.

Summary of the invention

At above problem, we have invented the wavelet analysis method of signal continuous frequency spectrum in a kind of non-linear circuit, can obtain the continuous frequency spectrum of non-linear circuit signal in high-speed, high precision ground, so can realize the noise analysis of non-linear circuit efficiently.

The wavelet analysis method of signal continuous frequency spectrum the steps include: at first to construct the nonlinear system equation for a non-linear circuit, and the realistic simulation interval is mapped between the small echo simulation region in the non-linear circuit that the present invention proposes; Then system variable being carried out wavelet series launches.And carry out system's discretize; At last, after solving equation obtains wavelet coefficient, calculate the signal value of the system variable on the discrete time point, utilize the finite term fourier progression expanding method formula of signal in frequency domain again, obtain the continuous frequency spectrum of signal.Below each step is specifically introduced respectively:

Step 1: structure nonlinear system

For a non-linear circuit, the nonlinear system equation of structure shown in (1) formula, and suppose that the variable signal and the input signal of circuit are all bandlimited signal, signal frequency range is [ω _M, ω _M]

$\frac{\mathrm{dX}\left(t\right)}{\mathrm{dt}}=f(X\left(t\right),t)+\mathrm{Du}\left(t\right)---\left(1\right)$

X(0)＝X ₀

Wherein, the implication of various functions and parameter is the same.

Step 2: spatial mappings

Choose an integer M (M is big, and I is adjusted by accuracy requirement), definition t _k=kh (M≤k≤M), T=Mh, wherein h=π/ω _M

Because algorithm adopts efficient spline wavelets basis function to realize simulation, we need be mapped to realistic simulation time interval [T, T] [0, L] between the small echo simulation region

l＝K·(t+T) (13)

T ∈ [T, T] wherein, l ∈ [0, L], K=L/2T.

With (13) formula substitution equation (1), can get:

$K\frac{\mathrm{dX}\left(l\right)}{\mathrm{dl}}=f(X\left(l\right),\frac{l}{K}-T)+\mathrm{Du}(\frac{l}{K}-T)---\left(14\right)$

X(KT)＝X ₀

Step 3: wavelet series is launched and system's discretize

For given exponent number J 〉=0, in analog domain, variable X (l) is carried out wavelet series and launches:

$X\left(l\right)=\left[\begin{array}{c}{X}_1\left(l\right)\\ X_2\left(l\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ X_N\left(l\right)\end{array}\right]=\left[\begin{array}{cccc}{C}_{11}& C_{12}& \&CenterDot;\&CenterDot;\&CenterDot;& C_{1P}\\ C_{21}& C_{22}& \&CenterDot;\&CenterDot;\&CenterDot;& C_{2P}\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ C_{N1}& {\mathrm{<figure-callout\; id="2"\; label="C\; N"\; filenames="A20051002330400161.png"\; state="\{\{state\}\}">C</figure-callout>}}_N& \&CenterDot;\&CenterDot;\&CenterDot;& C_{\mathrm{NP}}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{B}_1\left(l\right)\\ B_2\left(l\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ B_P\left(l\right)\end{array}\right]=C\&CenterDot;B\left(l\right)---\left(15\right)$

Wherein, C ∈ R ^{N * P}Be small echo expansion coefficient matrix, B (l) represents wavelet basis function, and P is total wavelet basis number, P=2 ^JL+3.

With (15) substitution equation (14), can get:

$K\&CenterDot;C\&CenterDot;\frac{\mathrm{dB}\left(l\right)}{\mathrm{dl}}=f(\mathrm{CB}\left(l\right),\frac{l}{K}-T)+\mathrm{Du}(\frac{l}{K}-T)---\left(16\right)$

For each basis function B _i(l), a sampled point l all can be arranged _iCorresponding with it, make B _i(l) at l=l _iThe place reaches maximal value.Obtain wavelet coefficient Matrix C ∈ R in order to find the solution ^{N * P}, with circuit equation (16) at sampled point { l ₁, l ₂... l _MLocating discretize, can obtain:

$K\&CenterDot;C\&CenterDot;\left[\begin{array}{cccc}\frac{\mathrm{dB}\left(l_1\right)}{\mathrm{dl}}& \frac{\mathrm{dB}\left(l_2\right)}{\mathrm{dl}}& \&CenterDot;\&CenterDot;\&CenterDot;& \frac{\mathrm{db}\left(l_P\right)}{\mathrm{dl}}\end{array}\right]$

$=\left[\begin{array}{cccc}f(\mathrm{CB}\left(l_1\right),\frac{l_1}{K}-T)& f(\mathrm{CB}\left(l_2\right),\frac{l_2}{K}-T)& \&CenterDot;\&CenterDot;\&CenterDot;& f(\mathrm{CB}\left(t_P\right),\frac{l_P}{K}-T)\end{array}\right]---\left(17\right)$

$+\left[\begin{array}{cccc}u(\frac{l_1}{K}-T)& u(\frac{l_2}{K}-T)& \&CenterDot;\&CenterDot;\&CenterDot;& u(\frac{l_P}{K}-T)\end{array}\right]$

Step 4: system is found the solution

Comprise P * N equation in the system of equations (17), known variables is C ∈ R ^{N * P}Consider circuit starting condition X (KT)=X ₀, we can obtain another equation about C:

CB(KT)＝X ₀ (18)

Comprehensively (17) and (18), we can get (P+1) * N nonlinear equation, and wherein known variables Matrix C size is P * N.We utilize the Levenberg-Marquardt method to find the solution this (P+1) * N nonlinear equation and obtain the known variables Matrix C.

Can obtain the interior signal response X (t) of realistic simulation time interval [T, T] with finding the solution the matrix of coefficients C substitution formula (19) that obtains.

$X\left(t\right)=C\&CenterDot;B(\mathrm{Kt}+\mathrm{KT})=C\&CenterDot;B(\frac{L}{2T}t+\frac{L}{2})---\left(19\right)$

Step 5: signal calculated continuous frequency spectrum

After obtaining the signal response X (t) in [T, T], can obtain 2M+1 discrete time point t by Direct Sampling _-M, t _-M+1..., t _MOn signal value X _-M, X _-M+1..., X _M, be shown below

$X_k=X\left(t_k\right)=C\&CenterDot;B(\frac{L}{2T}{t}_k+\frac{L}{2})---\left(20\right)$

Be similar to the frequency domain series expansion method in [8], signal carried out in frequency domain shown in the finite term fourier progression expanding method (suc as formula (12)):

$X\left(\mathrm{\&omega;}\right)\&ap;\frac{\mathrm{\&pi;}}{{\mathrm{\&omega;}}_M}\underset{k=-M}{\overset{M}{\mathrm{\&Sigma;}}}{X}_k{e}^{-\mathrm{jk\&pi;\&omega;}/{\mathrm{\&omega;}}_M}\left(\right|\mathrm{\&omega;}|\&le;{\mathrm{\&omega;}}_M)---\left(12\right)$

Because coefficient X _-M,

, X _MAll calculate by (20), we can find the solution the continuous frequency spectrum that obtains signal by following formula.Finite term progression for following formula blocks the error of being introduced, and can eliminate by window technique.

There is a kind of adaptive algorithm in the wavelet analysis method of signal continuous frequency spectrum in the non-linear circuit.Based on this adaptive algorithm, can choose the small echo exponent number and the wavelet basis function number that need under the given simulation precision automatically, thereby it is minimum that the wavelet basis number is reduced to, promptly farthest reduce P, required nonlinear equation exponent number of finding the solution is reduced to minimum, reduce the computation complexity of algorithm.

The main thought of adaptive algorithm is as follows: owing to the rising of wavelet coefficient amplitude with small echo exponent number J descends, can judge whether that therefore needs increase the small echo exponent number according to the size of J rank wavelet coefficient.The maximal phase of definition J rank wavelet coefficient is to amplitude

$R_J=\frac{\mathrm{MAX}\left|C_i^J\right|}{\mathrm{MAX}\left|C_i\right|}---\left(21\right)$

Wherein, MAX|C _i ^J| be the maximal value of J rank wavelet coefficient, MAX|C _i| be the maximal value of all wavelet coefficients in 0 to the J rank.

If R _JGreater than a certain threshold epsilon, then increase small echo exponent number J to J '=J+1; Otherwise, if R _JLess than threshold epsilon, then keep original exponent number J.Because the compact support of wavelet basis function, therefore need be between whole simulation region [0, L] go up and use unified exponent number J, adopt the high-order basis function and only change the zone of very fast (promptly containing the high frequency composition) at X (l), changing zone employing low order basis function more slowly, thereby making the basis function minimum number of employing.

The present invention has following advantage:

But the continuous frequency spectrum of 1 picked up signal

The present invention has versatility for the general nonlinearity circuit, can effectively obtain continuous frequency 0 spectrum of non-linear circuit signal.

2, high precision

The signal continuous frequency spectrum error that the present invention obtained comes from the truncation error that formula (12) is introduced basically, and this part error can be eliminated by the window technique that proposes in the document [9].With respect to finding the solution based on having sampling function of overall importance in the frequency domain fourier progression expanding method algorithm [8], the present invention has effectively utilized tight the characteristic and the adaptive algorithm of small echo, can change fast zone in signal waveform and adopt the high-order wavelet basis function to approach automatically, thereby accurately obtain expansion coefficient X _-M, X _-M+1..., X _M, guaranteed high computational accuracy.

3, high-speed

The present invention adopts efficient spline wavelets to find the solution and obtains frequency domain series expansion coefficient.With respect to the sampling function that adopts in the frequency domain fourier progression expanding method algorithm [8], wavelet basis function has speed of convergence faster, so the present invention utilizes less wavelet basis function just can obtain corresponding precision.Have characteristics such as multiresolution owing to wavelet basis function simultaneously, the present invention can adopt adaptive algorithm to choose wavelet basis function automatically, thereby it is minimum that the wavelet basis function number is reduced to, and there is not such characteristic in sampling function.Therefore, the relative frequency domain fourier progression expanding method of the Nonlinear System of Equations number algorithm [8] that the present invention need find the solution has bigger minimizing, thereby has improved computing velocity greatly.

Description of drawings

Fig. 1 metal-oxide-semiconductor amplifier circuit.

The time domain waveform that Fig. 2 the present invention obtains and the comparison of SPICE precision waveform.

The time domain waveform that Fig. 3 frequency domain fourier progression expanding method algorithm [8] obtains and the comparison of SPICE precision waveform.

Discrete time sampled point signal value that Fig. 4 the present invention and frequency domain fourier progression expanding method algorithm [8] obtain and SPICE precise results are relatively.

The signal continuous frequency spectrum that Fig. 5 the present invention and frequency domain fourier progression expanding method algorithm [8] obtain relatively.

Embodiment

Be example with the noise analysis below, further specify the present invention by specific embodiment.

To metal-oxide-semiconductor amplifier circuit shown in Figure 1, wherein the input stimulus of circuit is the white noise signal (amplitude peak is 1) of 10Khz, utilizes the present invention (Wavelet based method) to simulate, and concrete steps are as follows.

Step 1: the nonlinear system of structure amplifier circuit is described:

$\mathrm{RC}\frac{d{V}_{\mathrm{out}}}{\mathrm{dt}}=-V_{\mathrm{out}}-I_{\mathrm{DS}}\&CenterDot;R+5$

V _out| _t＝0＝-3.2266v

Here R=1000 ohm, C=0.1 microfarad, I _DSBe defined as follows λ=0.0953 wherein, k=9.2 * 10 ^-4, V _T=0.8 volt, V _InBe input stimulus, be the white noise signal (amplitude peak is 1) of 10Khz.

(1) V _In+ 5 〉=V _TAnd V _In〉=V _Out+ V _TThe time

$I_{\mathrm{DS}}=k\&CenterDot;(V_{\mathrm{out}}+5)(V_{\mathrm{in}}+5-V_T-\frac{V_{\mathrm{out}}+5}{2})[1+\mathrm{\&lambda;}(V_{\mathrm{out}}+5)]$

(2) V _In+ 5 〉=V _TAnd V _In＜V _Out+ V _TThe time

$I_{\mathrm{DS}}=\frac{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="A20051002330400161.png"\; state="\{\{state\}\}">k</figure-callout>}}{2}\&CenterDot;{(V_{\mathrm{in}}+5-V_T)}^2[1+\mathrm{\&lambda;}(V_{\mathrm{out}}+5)]$

(3) V _In+ 5＜V _TThe time

I _DS＝0

The variable signal V of circuit _OutBe all bandlimited signal with input signal, signal frequency range is [20kHz, 20kHz].

Step 2: choose M=40, definition sampled point t _k=kh (M≤k≤M), wherein h=5 * 10 ^-5Therefore the realistic simulation time interval is [T, T], wherein T=1 * 10 ^-3S.

Choose L=10, the realistic simulation time interval be mapped to [0, L] between the small echo simulation region:

l＝K·(t+T)

K=5 * 10 wherein ³With the non-linear description of following formula substitution circuit, the small echo simulation avd interval that obtains shown in (14) formula is described.

Step 3: choose J=3, in analog domain to variable V _OutCarry out wavelet series and launch, as the formula (15), wherein total wavelet basis number is P=83, and known variables is small echo expansion coefficient C.

With small echo expansion substitution equation shown in (14) formula, obtain equation shown in (16) formula.Discrete to (16) formula 83 sample point, can obtain the discrete equation shown in (17), the equation number is 83.

Step 4: consider the circuit starting condition, can obtain the equation of extra shape as (18).Comprehensive (17) and (18) formula, we have 84 nonlinear equations, known variables wherein, promptly small echo expansion coefficient number is 83.We utilize the Levenberg-Marquardt method to find the solution these 84 nonlinear equations and obtain the small echo expansion coefficient.

Can obtain the interior output voltage values V of realistic simulation time interval [T, T] with finding the solution small echo expansion coefficient substitution (19) formula that obtains _Out

Step 5: get the output voltage V in [T, T] _OutAfter, can obtain 81 discrete time point t by Direct Sampling _k=kh (output voltage values on M≤k≤M), as the formula (20).

With the output voltage values substitution formula (12) on 81 discrete time points of being tried to achieve, calculate the continuous frequency spectrum of output voltage.

Provided among Fig. 2 utilize the present invention obtain the comparison of time domain simulation result and SPICE precise results, can see that the present invention can obtain the very high result in time domain of precision, guaranteed the high precision that the expansion coefficient of continuous frequency spectrum calculating formula (12) calculates.

In order to compare, the frequency domain fourier progression expanding method method (Frequency domain Fourierexpansion method) [8] that we utilize document [8] to propose is simulated same circuit.Provided the comparison of time domain simulation result and SPICE precise results among Fig. 3, can see that there are bigger deviation in time domain waveform and SPICE precise results that this algorithm obtains, expansion coefficient error of calculation of this explanation signal continuous frequency spectrum calculating formula (12) is bigger, causes the signal continuous frequency spectrum precision that obtains at last relatively poor.We have provided the present invention and the signal value on the discrete time sampled point of frequency domain fourier progression expanding method method [8] acquisition and the comparison of SPICE precise results in Fig. 4, the signal value on these discrete time sampled points is corresponding to the expansion coefficient value of continuous frequency spectrum calculating formula (12).Can see that the expansion coefficient value that the present invention obtains overlaps with actual value basically, and the expansion coefficient value that frequency domain fourier progression expanding method method [8] calculates has departed from actual value out and away.The signal continuous frequency spectrum that has provided the acquisition of the present invention and frequency domain fourier progression expanding method method [8] among Fig. 5 compares.There are some deviations between the signal continuous frequency spectrum waveform that two methods obtain, main cause is that frequency domain fourier progression expanding method method (seeing formula (12)) when calculating continuous frequency spectrum is big owing to the expansion coefficient error, therefore there is bigger error in the continuous frequency spectrum that obtains, and the present invention has obtained accurate expansion coefficient, can obtain the accurate signal continuous frequency spectrum.In addition, all there is part vibration in two waveforms, and this blocks institute mainly due to limited fourier series and causes, and this part error can be eliminated by window technique.

In addition, we have also compared two kinds of basis function number, simulated time and relative errors that method is utilized when finding the solution the expansion coefficient of continuous frequency spectrum calculating formula (12), as shown in the table:

Basis function number simulated time relative error

Algorithm [8] 81 1.547 5.151e-002

The present invention 83 0.719 3.547e-003

Be not difficult to find that under the prerequisite that adopts the same base function, the relative algorithm of the present invention [8] has high precision and low time complexity.

This practical circuit shows that the present invention has the high and low characteristics of time complexity of precision in the continuous frequency spectrum computing method of non-linear circuit signal.

Claims (1)

1, the wavelet analysis method of signal continuous frequency spectrum in a kind of non-linear circuit is characterized in that concrete steps are as follows:

Step 1: structure nonlinear system

For a non-linear circuit, the nonlinear system equation of structure shown in (1) formula described, and supposes that the variable signal and the input signal of circuit are all bandlimited signal, and signal frequency range is [ω _M, ω _M]

$\frac{\mathrm{dX}\left(t\right)}{\mathrm{dt}}=f(X\left(t\right),t)+\mathrm{Du}\left(t\right)---\left(1\right)$

X(0)＝X ₀

Step 2: spatial mappings

Choose an integer M, definition t _k=kh (M≤k≤M), T=Mh, wherein h=π/ω _M, realistic simulation time interval [T, T] is mapped to [0, L] between the small echo simulation region:

l＝K·(t+T) (13)

T ∈ [T, T] wherein, l ∈ [0, L], K= ^L/ _2T.

With (13) formula substitution equation (1):

$K\frac{\mathrm{dX}\left(l\right)}{\mathrm{dl}}=f(X\left(l\right),\frac{l}{K}-T)+\mathrm{Du}(\frac{l}{K}-T)---\left(14\right)$

X(KT)＝X ₀

Step 3: wavelet series is launched and system's discretize

For given exponent number J 〉=0, in analog domain, variable X (l) is carried out wavelet series and launches:

$X\left(l\right)=\left[\begin{array}{c}{X}_1\left(l\right)\\ X_2\left(l\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ X_N\left(l\right)\end{array}\right]=\left[\begin{array}{cccc}{C}_{11}& C_{12}& \&CenterDot;\&CenterDot;\&CenterDot;& C_{1P}\\ C_{21}& C_{22}& \&CenterDot;\&CenterDot;\&CenterDot;& C_{2P}\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ C_{N1}& C_{N2}& \&CenterDot;\&CenterDot;\&CenterDot;& C_{\mathrm{NP}}\end{array}\right]\left[\begin{array}{c}{B}_1\left(l\right)\\ B_2\left(l\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ B_P\left(l\right)\end{array}\right]=C\&CenterDot;B\left(l\right)---\left(15\right)$

Wherein, C ∈ R ^{N * P}Be small echo expansion coefficient matrix, B (l) represents wavelet basis function, and P is total wavelet basis number, P=2 ^JL+3.

With (15) substitution equation (14):

$K\&CenterDot;C\&CenterDot;\frac{\mathrm{dB}\left(l\right)}{\mathrm{dl}}=f(\mathrm{CB}\left(l\right),\frac{l}{K}-T)+\mathrm{Du}(\frac{l}{K}-T)---\left(16\right)$

For each basis function B _i(l), a sampled point l all can be arranged _iCorresponding with it, make B _i(l) at l=l _iThe place reaches maximal value; Obtain wavelet coefficient Matrix C ∈ R in order to find the solution ^{N * P}, with circuit equation (16) at sampled point { l ₁, l ₂... l _M) locate discretize, obtain:

$K\&CenterDot;C\&CenterDot;[\frac{\mathrm{dB}\left(l_1\right)}{\mathrm{dl}}\frac{\mathrm{dB}\left(l_2\right)}{\mathrm{dl}}\&CenterDot;\&CenterDot;\&CenterDot;\frac{\mathrm{dB}\left(l_P\right)}{\mathrm{dl}}]$

$=[f(\mathrm{CB}\left(l_1\right),\frac{l_1}{K}-T)$ $(\mathrm{CB}\left(l_2\right),\frac{l_2}{K}-T)\&CenterDot;\&CenterDot;\&CenterDot;f(\mathrm{CB}\left(t_P\right),\frac{l_P}{K}-T)$ $---$ $\left(17\right)$

$+[u(\frac{l_1}{K}-T)u(\frac{l_2}{K}-T)\&CenterDot;\&CenterDot;\&CenterDot;u(\frac{l_P}{K}-T)]$

Step 4: system is found the solution

Because circuit starting condition X (KT)=X ₀, obtain another equation about C:

CB(KT)＝X ₀ (18)

Comprehensively (17) and (18) get (P+1) * N nonlinear equation, and wherein known variables Matrix C size is P * N; Utilize the Levenberg-Marquardt method to find the solution this (P+1) * N nonlinear equation and obtain the known variables Matrix C;

To find the solution the matrix of coefficients C substitution formula (19) that obtains, obtain the signal response X (t) in the realistic simulation time interval [T, T]:

$X\left(t\right)=C\&CenterDot;B(\mathrm{Kt}+\mathrm{KT})=C\&CenterDot;B(\frac{L}{2T}t+\frac{L}{2})---\left(19\right)$

Step 5: signal calculated continuous frequency spectrum

After obtaining the signal response X (t) in [T, T], can obtain 2M+1 discrete time point t by Direct Sampling _-M, t _-M+1..., t _MOn signal value X _-M, X _-M+1..., X _M, be shown below:

$X_k=X\left(t_k\right)=C\&CenterDot;B(\frac{L}{2T}{t}_k+\frac{L}{2})---\left(20\right)$

Signal is carried out the finite term fourier progression expanding method in frequency domain, as the formula (12):

$X\left(\mathrm{\&omega;}\right)\&ap;\frac{\mathrm{\&pi;}}{{\mathrm{\&omega;}}_M}\underset{k=-M}{\overset{M}{\mathrm{\&Sigma;}}}{X}_k{e}^{-\mathrm{jk\&pi;\&omega;}/{\mathrm{\&omega;}}_M}\left(\right|\mathrm{\&omega;}|\&le;{\mathrm{\&omega;}}_M)$

Because coefficient X _-M, X _-M+1..., X _MAll calculate by (20), can find the solution the continuous frequency spectrum that obtains signal by following formula.

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