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24-26 September 2008, Rome, Italy*RiR = i1−exp( −t/ τ )(7)Obviously the excessive attenuation of the farther timeconstantsand the noise raises hard limits of this procedure.III. EVALUATION OF SHORT PULSE, SHORT RECORDINGMEASUREMENTSIt is a well-known fact that thermal measurements areexcessively time consuming. The usual waiting time to reachthe steady state is normally 5-30 minutes, which isprohibitive for production testing. At the same time, in anumber of cases we are interested only in the time constantsbelow 1 s. A good example is the die attach testing wherethe critical time constants lie at about 1-10 ms. The questionobviously arises: whether it is possible to extract correctlythe interesting range of the time constant spectrum from ashort fragment of the thermal response or not. Weinvestigated this question in a recent paper [6]. This sectionis dealing with the further possibilities and limits of this task.In the thermal transient method we measure the a(t) stepfunctionresponse. During the early processing steps wecalculate its da/dz derivative, where z = ln(t). Let us referthis as the "measured function" m ( z)= da / dz . This isrelated to the t(z) "true" time-constant spectrum through theconvolution equation:m( z)= t(z)⊗ w(z)(8)where the w(z) weight function is defined by (9):P( z − exp( ))iw( z)= exp z(9)function is wanted instead. This approximate function doesnot produce the Dirac pulse in the left hand side of (10) but asimilar function consisting only a narrow peak at z = 0. Thewidth of this peak determines the resolution of the truespectrum calculated by (11).Calculating the approximate inverse weight function wecan apply constraints concerning the nonzero part width ofthe function. This is very important since the length of thefragment of the measured function needed for exactreconstruction is (see the integral of (11))∆z = ∆z+ ( ζ − ) 1(12)mt2ζwhere ∆z t is the length of the true function that has to bereconstructed correctly.In the Appendix we present an algorithm, working in thez = ln(t) logarithmic time domain that meets ourrequirements. Especially, the length of the inverse weightfunction can be prescribed.Beside this, the inverse function is optimized for both thenoise and the spurious sidelobes of the resultant function.In case of RC (and R th ,C th ) networks identification theweight function is given by Eq. (9). The shape of thisfunction is shown in Fig. 4.Equation (8) describes a linear filtering in the z domain.This filtering acts as a "smearing" on the true function. Thedeconvolution step is devoted to undo this smearing. Thisdeconvolution is usually proceeded directly on the m(z)function. But there is other possibility as well. We can find afurther linear filter which is capable to undo the effect ofw(z). Let us call this function as "inverse weight function",i(z). The successive applications of w(z) and i(z) have toresult in "no effect" which is equivalent to the convolution ofa δ(z) Dirac pulse:δ ( z)= w(z)⊗ i(z)(10)This is the definition of the inverse weight function. Theprocedure using i(z) is referred as inverse filtering. Havingthe inverse function the true spectrum is calculated by asimple convolution step:ζ 2∫t ( z)= m(z)⊗ i(z)= i(ζ ) m(z − ζ ) dζ(11)ζ 1where the nonzero region of i(z) is bounded by ζ 1 and ζ 2 .We have to know that the exact inverse function can notbe constructed (excepting a few cases). An approximateFig.4. The shape of the weight function given by (9)As an example, we are looking for the inverse weightfunction of the above function with the followingparameters:• Sampling rate: 27 samples pro decade• Length of the inverse function: 50 samples• Signal-to-noise ratio of the measurement: 40• Autocorrelation factor between the i-th and j-th| i− j|measured samples: (0.5)• Required resolution on the reconstructed function: 10samplesHere we expected real measurement circumstances withexcessive noise. For this reason, the required resolutionparameter is a moderated one.The inverse weight function provided by the algorithmdescribed in the Appendix is shown in Fig. 5. At the firstglance it is a bit surprising that this array of scattered values©EDA Publishing/THERMINIC 2008 22ISBN: 978-2-35500-008-9