Path Deay Estimation using Power Suppy Transient Signas: A Comparative Study using Fourier and Waveet Anaysis Abhishek Singh, Jitin Tharian and Jim Pusqueic VLSI Research Laboratory Department of Computer Engineering University of Maryand, Batimore County Hitop Circe, Batimore MD-225 Abstract Transient Signa Anaysis (TSA) is a parametric device testing technique based on the anaysis of dynamic (transient) current (i DDT ) drawn by the core ogic from the power suppy pads in a CMOS digita circuit. In previous work, we deveop a test procedure that can be used both to detect signa variations caused by defects and to obtain deay information in defect free chi. Phase spectra of transient signas obtained using discrete Fourier transform are shown to track path deays of defect-free chi under a wide range of process variations. However, in recent work, we were abe to demonstrate through simuation experiments incorporating deep submicron transistor modes, a circuit design and path sensitization scenario in which our existing TSA method is not abe to yied accurate predictions of path deays. More specificay, a circuit composed of two inverter chains constructed with widey varying transistor sizes was shown to produce path deays that were weaky correated across a set of worst case process modes. In this paper, an aternative waveet-based anaysis of i DDT waveforms is shown to improve the accuracy of predicting mutipe path deays under these conditions. Abbreviations: i DDT : A time varying (ac) signa representing the dynamic current sourced by set of connected gates (path) under an input transition. Introduction The unique attribute of power suppy transient (i DDT ) signas to capture the parametric characteristics of the underying ogic circuit enabes the deveopment of aternative defect-oriented testing methods. The most important information contained in the i DDT signas is the functiona and deay characteristics of sensitized ogic paths. The use of power suppy transient signas as a means of estimating path deay characteristics has severa advantages. First, these signas can be used to detect deay fauts introduced by resistive shorting and open defects that are traditionay not targeted by stuck faut based methods. Second, the goba observabiity provided by the power suppy transient signas permits deay to be estimated without the need to sensitize paths to observation points such as primary outputs or scan-atches. Third, the suppy transients potentiay provide a rich source of parametric information about the chip. Transient Signa Anaysis (TSA) is a defect oriented testing technique that expoits the device information contained in the power suppy transient signas. Defect detection is accompished by anayzing the i DDT s measured simutaneousy at mutipe power suppy ports (Pads) on a chip. Linear regression anaysis is appied to time and frequency domain representation of the i DDT s to detect outiers [][2]. Fourier phase anaysis of the i DDT s provides a means of estimating path deays in defect-free chi []. In recent work, the TSA technique was evauated under deep submicron process conditions using a circuit design that incorporates gates with widey varying transistor sizes (W/L ratios) [4]. It was discovered that deep submicron variations in process parameters weaken the correation of deay across ogic paths on the same chip. Given the reationship between the deay characteristics of a ogic path and i DDT is cause-effect, the reduction in path correation on the same chip reduces the correation of i DDT s across chi. Since our deay prediction strategy is based on correation anaysis of i DDT s across chi, it is not abe to accuratey track a path deays under test sequences that sensitize mutipe paths. The imitation of our Fourier phase-based method derives from its focus on the anaysis of a singe attribute of the i DDT waveform -- its width. However, embedded within the i DDT waveform are other features that can be anayzed as a means of improving the accuracy of predicting mutipe independent path deays. The additiona dimension of signa decomposition provided by the waveet transform makes it better suited than the Fourier transform for the extraction and processing of these aternative i DDT waveform features. In this paper, a set of circuit modes incorporating deep submicron process variations and widey varying transistor sizes are simuated to demonstrate the imitations of Fourier anaysis for estimating mutipe path deays. A waveet-based anaysis is shown to overcome these imitations by providing an extra dimension of time based information. The objective of this paper is to compare the computationa compexity and accuracy of deay estimation methods based on a discrete Fourier transform (DFT) and a waveet transform (WT). The rest of this paper is organized as foows. Section 2 describes reated work. Section presents detais of the simuation experiments. Section 4 evauates the characteristics of the i DDT signas under various process modes and defines the probem. Sections 5 and describe the path deay estimation procedures and resuts using Fourier and waveet anaysis, respectivey. Section 7 anayzes the computationa compexity of the two approaches. Section 8 presents our concusions. 2 Background The iterature contains a wide range of pubications on the appication of waveet transform ranging from seismic and biomedica image processing to eectronics. Papers reated to the atter topic incude the foowing. Santoso e a. [5][] appied the WT to detect and ocaize disturbances in eectric power ines. Since eectric power signas are ideay composed of a singe frequency component, any anomay in these signas can be detected by anayzing the high frequency (detais) components of the WT representation. Bhunia et a. [7] appied the WT on power suppy transient signas to detect short and open defects. The mean square differences between WT coefficients of a goden chip and
that of the test chip are used to identify defective chi. Defect ocaization is achieved by mapping the time at which the WT coefficient of the defective chip differs from that of the defect-free chip into ogic depth. Simuation Experiment Design Figure shows the ayout of the test circuit used in the simuation experiments. The ayout consists of two paths impemented using chains of inverters. The heads of the two paths are abeed Fast Path Input and Sow Path Input in Figure. The fast path is composed of transistors with W/L ratios ranging from 2 to 5 for n-mos and 4 to for p-mos whie the sow path is composed of minimum sized transistors with W/L ratios of.5. The inverters aong both paths fanout to as many as three other inverters in addition to the next inverter in the path. The purpose of using various transistor sizes and oading conditions is to introduce diversity in the i DDT. In fp In In sp I DSi I DSi i DDTfp i DDTsp V DD Out fp Out sp Fig. 2. Iustration of composite-i DDT formation from individua gate i DS curves. i DDT V DD Ony 2 meta ayers shown for 5 meta ayer power grid Fast Path Input Sow Path Input Fanout Region Fast Path Fanout Region Sow Path Fig.. Layout incorporating dua paths of inverters with a fast path (upper) and sow path (ower) 4 Mutipe Path Sensitization Chaenges In order to determine the reationship between path deays and the corresponding i DDT s under the different process modes, it is first necessary to evauate the signa propagation characteristics aong each of the two paths across the process modes. Figure shows the output waveforms from the ast inverters of the two paths superimposed in each row. One pairing is shown for each of the 4 process modes. Even though the transistor modes are identica for a transistors in each circuit mode, it is cear across many of these pairings that the deays between the fast and sow paths are not we correated. (The vertica dotted ine provides a reference point for comparison.) This is particuary noticeabe for the pairings abeed h, j and. Here, significant speed-up is observed in the fast path deay whie the sow path deay remains reativey constant and consistent with other sow path deays from other runs, e.g., g. vots a The suppy grid in this design is routed in 5 meta ayers (the figure shows ony the ower meta and meta 2 ayers for carity). The SPICE votage source representing the power suppy is connected to meta 5 at the point shown on the upper eft of Figure. The process modes are derived from a set of MOSIS specifications for TSMC s.25µm process [8]. Each of the specifications incude ot-averaged conductor RC parasitics and BSIM modeing parameters derived from test structure measurements. We had 4 such data sets avaiabe. These data sets were used to configure a set of technoogy fies for the SPACE extraction too [] and the corresponding SPICE simuation modes were extracted from the ayout. These parameter vaues represent worst case vaues because they were obtained from wafer ots fabricated over a period of severa years. The test stimuus drives both paths simutaneousy, as a means of representing the more common muti-path signa propagation mode. Since the suppy grid is unified, the i DS signas generated by the inverters aong both paths superimpose in a composite-i DDT curve. Figure 2 shows a simpified ogic eve representation of the inverter chains (without the fanout branches). The i DS curves shown beside each inverter represents the spatia distribution of transient current drawn by the corresponding inverter from the V DD rai. The curves abeed i DDTfp and i DDTsp represent the current transients generated by fast and sow paths respectivey, when these paths are sensitized separatey. The curve abeed i DDT on the right is the waveform that is generated when both paths are sensitized simutaneousy. -2.5Vots.5.5 2 2.5.5 4 4.5 5 ns Fig.. Path outputs under 4 TSMC s process modes with varying BSIM parameters. The ow eve of correation in the deays across the output waveform pairings is argey due to the V t dependency on transistor width (W) in the BSIM modeing equations. The variations in other passive eements of the simuation modes have ony a sma impact on the deay characteristics, as iustrated beow. Figure 4 pots the reative deays of the sow path versus the fast path. The reative deays are computed by subtracting the absoute sow and fast path deays under each of the modes from the corresponding absoute path deays of first (and sowest) process mode, a (given at (,) in the figure). It is cear in Figure 4 that correations between the two paths across the process modes is poor. For exampe, the data points spanning the region abeed Actua BSIM Parameters are poory approximated by a straight ine. In contrast, the data points spanning the region abeed Constant BSIM Parameters are the reative deays obtained when the b c d e f g h ij k m n
44.5 77. 4.7 2.28 2. 25.5 27. 2.72 8.8 5.4 2.5 8. 7.27 42.8 Micro 4.4 -.4-55.2 5.7 25.5 42.4 8. 874.2 8. 285. 4.8 7.7.5 2.4 25. actua BSIM modes are repaced with the BSIM mode from process a. These data points span a much smaer region and are neary co-inear. Sow Path -5 - -5-2 -25 - -5 Actua BSIM Parameters j h -5 - -25-2 -5 - -5 Constant BSIM Parameters Fast Path Fig. 4. Sow path vs. fast path reative deays. Since our method anayzes i DDT as a means of predicting deay, it is important to understand how weaky correated mutipe path deays affect the i DDT features. The waveforms abeed i DDTfp and i DDTsp in Figure 5 represent the transient signas generated under singe path sensitization using process mode i. The top-most signa represents the composite-i DDT generated under dua path sensitization. The i DDTsp waveform has a arger width but is smaer in ampitude in comparison to the i DDTfp waveform. This resuts from the smaer W/L ratio used for the transistors in the gates of this path. micro am 42.84 25 5 75 25 25 w-i DDTfp w-i DDTsp = w-i DDT V DD Wed Mar 22::44 2.2.4..8..2.4..8 2. ns The width of i DDTsp, denoted by w-i DDTsp in Figure 5, is simiar to the composite width w-i DDT. (The widths are shown measured above the zero baseine in the figure). This indicates that the width of the composite-i DDT is defined by the sow path deay. However, the arger ampitude i DDTfp introduces a sharp change in the composite-i DDT on its faing transition. The i DDTfp aso decays before the i DDTsp and eaves a Posterior-Bump (or PoB) in the composite-i DDT. The time of occurrence of the PoB coincides with the end of i DDTfp and is a feature that can be used to obtain w-i DDTfp. The above anaysis suggests that a time domain anaysis of the composite-i DDT is sufficient to estimate both w-i DDTfp and w- i DDTsp, and the corresponding fast and sow path deays. However, the environmenta noise (and EMI) and the parasitics present in a production test equipment make it difficut to obtain the i DDT waveform in its pure, core ogic generated form. The abiity to seect and anayze specific frequency bands using the frequency domain representation of i DDT makes it attractive as a means of overcoming the test environment imitations. Composite (i DDT ) Fast Path (i DDTfp ) Sow Path (i DDTsp ) PoB Fig. 5. i DDT waveforms of fast and sow path (bottom) and the composite (top) 5 Path Deay Estimation using Fourier Anaysis The Fourier transform decomposes a signa into a inear sum of sinusoids or cosinusoids with different frequencies. Eq. gives the expression for a discrete Fourier transform (DFT) of a signa x(t). Here ω and t are discrete variabes that represent frequency and time respectivey. X( ω) N --- 2 j2πtk -------------- N = --- x( t)e N t = Eq.. where k denotes a discrete frequency [k=,,2...n/2] and t denotes a discrete time sampe [t=,,2...n-] We have observed that process variations introduce two main types of variations in the i DDT signa, shift and scaing. The Fourier phase component of the frequency domain representation of i DDT naturay tracks time shift and scaing and is therefore the basis of our deay estimation technique. The Fourier shift property states that a time shift (deay) of d units causes a phase shift of ω*d in the frequency domain. This property is expressed formay by Eq. 2 for a signa x(t) with the x( t) X( ω) or x( t) M( ω) θ( ω) x( t d) X( ω)e jωd or x( t d) M( ω) ( θ( ω) ωd) where M(ω) represents magnitude response and θ(ω) represents phase response Eq. 2. frequency representation given by X(ω). An aternative magnitude and phase representation of this reationship is given on the right side. The Fourier time scaing property states that if a signa is scaed in time by a factor α, its frequency spectrum (both Magnitude and Phase) is scaed by a factor /α, proportiona to frequency, as given by Eq.. x( αt) -- X ω α ᾱ -- or x( αt) -- M ω α ᾱ -- θ ω ᾱ -- Eq.. This expression indicates that irrespective of the shape and start time of the signa, any variation (scaing) in the i DDT width can be tracked exacty by its phase spectrum. TSA s deay estimation procedure takes advantage of these properties by computing Phase Signature Waveforms (PSWs) from the Fourier phase spectra of i DDT s. A PSW represents the difference waveform obtained by subtracting the Phase spectrum of the i DDT measured in a test chip from that of a reference chip. Figure shows the PSWs of simuated chi, representing process modes b through n, computed with respect to reference mode a. degrees +/-8 b c d e f g h i j k m n 2 5 MHz Fig.. Phase SWs from 4 process simuations under sow path sensitization.
8 2 4 5 8 4 5 8 2 2 2 4 5 7 7 2 The area under the PSWs is computed within the desired frequency band (-MHz in Figure ) to obtain a singe quantity, referred to as PSWA. Figure 7 shows the cross correation of PSWAs against the reative sow path deay. The correation coefficient (CC) of.% indicates that the reationship between the PSWA of i DDTsp and the corresponding path deay is inear. PSWA (E+) 27 24 8 5 2 7-5 - -25-2 -5 - -5.% Reative sow path deay Fig. 7. Phase SWA vs. reative sow path deays. The inearity of this reationship is based on the property of the Phase spectrum to uniquey track the features in the i DDT s that are most sensitive to deay variations, such as their widths. This resut indicates that under the constraint of singe path sensitization (or correated mutipe path sensitizations), Fourier anaysis of i DDT is capabe of tracking a imited number of goba events, such as rising and faing edges, in the i DDT waveform. However, the i DDT s generated from chi with mutipe, weaky correated path deays require the tracking of severa features, as discussed above in reference to Figure 5. In this case, the accuracy of estimating both path deays using Fourier anaysis is degraded since our procedure produces one quantity, a PSWA, where two are needed. For exampe, Figure 8 pots the PSWAs of the compositei DDT against the deay of the fast (top) and sow (bottom) paths under dua path sensitization. The markings on the y axis correspond to the fast path PSWAs; the sow path data points have been shifted down to make it easier to distinguish between the two sets of points. The CC for each anaysis are given as 8.8% and 8.2%, respectivey. (The quantities inside the parenthesis represent the idea CCs obtained from simuation experiments in which each of the paths are sensitized individuay.) The singepath tracking imitation of Fourier anaysis is ceary refected in this anaysis, which shows that the PSWAs track the fast path deay more accuratey than the sow path deay. PSWA (E+) 27 24 8 5 2 j h 2 Sow Path Regression Line Fast Path Regression Line σ Prediction Limits -5 - -25-2 -5 - -5 The higher degree of correation between the PSWAs and the fast path deay can be attributed to the weighted average property of Fourier phase, given by Eq. 4 and Eq. 5. Eq. 4 indicates Fast Path CCs: 8.8% (.7%) Sow Path CCs: 8.2% (.%) Reative Deay Fig. 8. Phase SWAs vs. fast (top) and sow (bottom) reative path deays. ACos( ωt + Φ) = A Cos( ωt + Φ ) + A 2 Cos( ωt + Φ 2 )... = N A i Cos( ωt + Φ i ) Eq. 4. i = that the superposition of cosinusoida waveforms with the same frequency ω but with different ampitudes A i and phase-anges Φ i, generates a cosinusoid of the same frequency ω with ampitude A and phase-ange Φ. Eq. 5 indicates that the resutant phase ange Φ can be approximated as a weighted average of the individua phase anges Φ i. Since the ampitude of i DDTfp is much higher A i Φ i i Φ --------------------- Eq. 5. A i i than that of i DDTsp, the composite-i DDT phase spectrum is weighed towards the phase spectrum of the fast path, yieding a better estimation of its deay. In summary, Fourier anaysis is capabe of providing accurate estimates of path deays for cases in which the path deays remain correated across regions of the chip (or when it is possibe to sensitize one path at a time) []. If transistor dimensions vary widey, then the accuracy of estimating mutipe path deays is reduced under Fourier anaysis because of its inabiity to track mutipe features of a signa simutaneousy. Under such conditions, a waveet transform (WT) can be used to improve prediction accuracy. Path Deay Estimation using Waveets Time-frequency anaysis, unike Fourier anaysis, partitions the time domain signa into smaer sections using ocaized window functions. Each section is decomposed into an aternate representation using Fourier basis functions (sine/cosines) or waveets. The advantages of time-frequency anaysis over Fourier anaysis are two fod. First, the abiity to anayze frequency components in a specific time interva makes it an effective method for decomposing non-stationary signas whose frequency content change over time. However, the frequency content of the gate i DS waveforms vary over a narrow frequency band, which suggests that i DDT waveforms are we characterized as stationary. Therefore this advantage of time-frequency anaysis cannot be everaged for deay estimation purposes in defect-free chi. (It is noted that the frequency content of a defective gate s i DS typicay occupies a ower and/or wider frequency band and consequenty, this feature of time-frequency anaysis may be usefu for defect detection purposes). A second advantage of time-frequency anaysis is the abiity to ocaize events in time. This means it can provide frequency information ocaized to corresponding time intervas in a signa. The abiity of time-frequency anaysis to ocaize more than one feature in a signa makes it suitabe for tracking mutipe independent deays in i DDT waveforms. This capabiity comes at the cost of increased computationa compexity, however, as described in Section 7. Time-frequency anaysis based on the waveet transform (WT) uses waveet basis functions derived from a mother waveet. Waveets are finite in ength and have a characteristic shape that dictate the distribution of their frequency content. The frequency and time resoution of waveets is varied by appying shifting and scaing operations to the mother waveet. Eq. gives the expres-
W ( s, τ) = x( t)ψ s, τ ( t) t Ψ τ, s ( t) d Eq.. = -----Ψ t --------- τ s = scae τ = shift (or transation) s s sion for a continuous waveet transform (CWT) of a signa x(t). The s parameter is the scaing factor and the τ parameter is the transation factor (shift) that are appied to the mother waveet, Ψ τ,s (t). The factor s -/2 normaizes energy across the different scaes. As s changes, the waveet covers different frequency ranges, with arger vaues corresponding to sma frequencies. The time ocaization center of the waveet in the signa is changed through parameter τ. Proper seection of the mother waveet pays an important roe in the abiity of a WT to extract signa features. Waveets are generay categorized based on their compactness in the time and frequency domain and their smoothness. Waveets that are characterized by sharp changes (ess smooth) are more capabe of tracking discontinuities or abrupt changes in a signa, whereas, waveets with smooth curves are more capabe of tracking the goba features of the signa. In this paper, we determine the i DDT feature extraction capabiities of two mother waveets, Haar and Mexican-Hat, shown in Figure. The Haar waveet, denoted by h(t), is characterized by a sharp change in its shape. In contrast, the Mexican-Hat waveet, denoted as m(t), is fairy smooth. The Haar waveet inherenty has good time ocaization, which suggests that it is better at ocating sharp changes in a signa, such as the starting and ending transitions of the i DDTfp in the composite-i DDT shown in Figure 5. h(t) -.5 m(t).8..4.2 -.2-8 - -4-2 2 4 8 Fig.. Haar (eft) and Mexican-Hat waveets (right). Figure (b) shows the CWT of the i DDT waveform shown in Figure (a), obtained using the Haar mother waveet. Figure (b) is essentiay a D-pot, showing the absoute vaues of the waveet coefficients as a grayscae gradient, with scaes (s) potted aong the y-axis and time-shift (τ) aong the x-axis. The darker regions in the pot represent coefficients with arger absoute magnitudes. The ower portion of this figure shows the coefficient vaues at finer scaes (high frequencies) for a possibe time-shifts. The abiity of the Haar CWT to ocate sharp changes in the i DDT waveform is exempified by the inward tapering of the two funneshaped dark regions towards the bottom of the pot. These time positions correspond to the starting and ending transitions of the fast path in the composite-i DDT (Figure (a)). At higher scaes, the dark region starts to broaden, and begins to track the more sowy changing features in the i DDT waveform. Figure (c) pots the coefficients for s = 4, iustrating that the dark regions in the CWT actua correspond to the both oca minima and maxima in the coefficients. From these observations, it foows that the time interva between the minima and maxima at ower scaes, denoted as t maxmin in Figure (c), shoud be we correated with the fast path deay. Bear in mind that very ow scae vaues correspond to fre- (a) (b) (c) t max-min Composite-i DDT for process mode CWT Coefficients Y-axis: Scaes (s) X-axis: Time (τ) grayscae: Coef. Vaue Coefficient Curve at Scae s = 4 Fig.. CWT (b) of process mode i DDT waveform (a) using Haar waveet. (c) gives the waveet coefficient curve at scae s=4. quencies that cannot be measured (at any significant ampitude) in the testing environment. For exampe, s=2 corresponds to a frequency band centered at 5GHz. Therefore, any practica appication must restrict scaes to s>=5 (~2GHz). In spite of this restriction, the s=82 scae yieds the highest eve of correation (.7%) with the fast path deays across the 4 process modes. Figure shows the scatterpot of t max-min versus the reative fast path deay. The scae s=82 corresponds to a frequency band centered around.2ghz, derived using Eq. 7. t max-min (E-) -5-2 -25 - -5 scae s=82-5 - -25-2 -5 - -5 Reative fast path deay.7% Fig.. t max-min vs. reative fast path deays from simuations under simutaneous dua path sensitization. f F c = ---------- s The Haar waveet aso tracks the sow path deay, but does so at higher scaes. The scae s=24 (~47MHz) gives the highest CC in this case. Figure 2 shows the scatterpot of t max-min vs. fast path deay at this scae. The computed CC of 5.4% is significanty arger than corresponding vaue of 8.7% obtained using Fourier anaysis. The CWT anaysis of the 4 composite-i DDT waveforms using the Mexican-Hat waveet yieds simiar resuts for the fast path anaysis but sighty worse resuts for the sow path anaysis. Figure (b) shows the CWT of the process mode compositei DDT waveform. The signa decomposition is noticeaby different under the Mexican-Hat and Haar waveet functions. For exampe, Eq. 7. where F c is the center frequency of the waveet spectrum (Hz) is the samping period and s is a scae
t max-min (E-) - -2 - -4-5 -5 - -25-2 -5 - -5 Reative sow path deay 5.4% Fig. 2. t max-min vs. reative sow path deays from simuations under simutaneous dua path sensitization. the coefficient curve shown in Figure (c) resembes a segment of a cosinusoid curve. The dark region encosed between the bright-white ines in Figure (b) represents the oca maxima of the coefficient curves. For scaes ower than s~, the dark region bifurcates into two portions (two maxima) that taper to time ocations corresponding to the sharp transitions of the i DDT waveform shown in Figure (a). This shows the basic capabiity of the CWT to track sharp features at ower scaes. The occurrence of a singe maxima at higher scaes refects its abiity to track goba features of the signa. (a) (b) (c) t zero-cross Fig.. CWT (b) of process mode i DDT waveform (a) using Mexican-Hat waveet. (c) gives the waveet coefficient curve at scae s=4. The time interva t zero-cross shown in Figure (c) deimits the time interva between the points defined when the coefficient curve becomes zero. The scaes that yied the highest CCs in the corresponding t zero-cross vs. fast path and t zero-cross vs. sow path deay pots (not shown) are s=22 (CC=.%) and s= (CC=.%), respectivey. For the Mexican-Hat CWT, these scaes correspond to a frequency band centered around.ghz and 42MHz respectivey. 7 Compexity Anaysis An anaysis of the computationa compexity of the DFT and CWT-based procedures competes their comparison. The computation of X(ω) defined by Eq. for a given ω requires 2N mutipications and 2N additions. There are N/2 frequencies in the compete spectrum yieding a compexity of O(N 2 ). However, our DFT-based deay estimation procedure anayzes ony a constant band of C frequencies (with C in the range of 5-) and therefore Composite-i DDT for process mode CWT Coefficients Y-axis: Scaes (s) X-axis: Time (τ) grayscae: Coef. Vaue Coefficient Curve at Scae s = 4 the compexity reduces to O(N). The computation of the CWT defined by Eq. 8 entais N 2 mutipications and N 2 additions, yieding a compexity of O(N 2 ). Note that a continuous waveet transform is required (as opposed to a discrete waveet transform (DWT)) because a time positions must be anayzed in order to adequatey ocaize time events. Aternative, more efficient agorithms for computing the DWT are not usefu because our anaysis shows that the best deay estimation is obtained at scaes that are not dyadic. Moreover, the deay estimation accuracy is sensitive to the incrementa shift in the waveet (τ). Therefore, the compexity of waveet transform for this procedure cannot be further reduced using efficient fitering agorithms appicabe to DWT. C( τ, s) = N x( n)ψ --------- τ s Eq. 8. n = Where s denotes a scae vaue [s=,2...n-] n denotes a discrete time sampe [n=,,2...n-] and t denotes a incrementa time shift [t=,,2...n-] 8 Concusions This paper compares the prediction accuracy and compexity of Fourier and waveet anaysis techniques for path deay estimation. Waveet anaysis of i DDT is more accurate than Fourier anaysis for predicting mutipe deays in custom circuits that incorporate paths constructed with transistors of widey varying widths. The cost of increased accuracy is increased computationa compexity, O(N 2 ) vs. O(N), respectivey. Two waveets, Haar and Mexican-Hat are investigated. Both waveets track the fast path deays more accuratey using ower scaes (corresponding to ~GHz), whie higher scaes (~4MHz) more accuratey track sower path deays. The Haar performs sighty better than the Mexican-Hat for sow path deay predictions. References []A. Germida, Z. Yan, J. Pusqueic and F. Muradai, Defect Detection using Power Suppy Transient Signa Anaysis, Internationa Test Conference, pp. 7-7,. [2]A. Singh, C. Pate, S. Liao, J. Pusqueic and A. Gattiker, Detecting Deay Fauts using Power Suppy Transient Signa Anaysis, Internationa Test Conference, pp.74-72, 2. []J. Pusqueic, A. Germida, J. Hudson, E. Staroswiecki and C. Pate, Predicting Device Performance From Pass/Fai Transient Signa Anaysis Data, Internationa Test Conference, pp.7-7, 2. [4]A. Singh, J. Pusqueic and A. Gattiker, Power Suppy Transient Signa Anaysis under Rea Process and Test Hardware Modes, VLSI Test Symposium, pp.57-2, 22. [5]S. Santoso, E.J. Powers, W.M. Grady and P. Hofmann, Power Quaity Assessment Via Waveet Transform Anaysis, Transactions on Power Deivery, Vo., No. 2,. []S. Santoso, E.J. Powers and W. M. Grady, Eectric Power Quaity Disturbance Detection using Waveet Transform Anaysis, IEEE-SP Internationa Symposium on Time-Frequency and Time-Scae Anaysis, pp.-, 4. [7]S. Bhunia and K. Roy, A Nove Waveet Transform Based Transient Current Anaysis for Faut Detection and Locaization, Design Automation Conference, pp. -, 22. [8]MOSIS at http://www.mosis.edu/technica/testdata/tsmc- 25-prm.htm. []Genderen et a., SPACE, Layout to Circuit Extraction Software Modue of the Nesis IC Design System, Deft University of Technoogy,.