Vishwani D. Agrawal Auburn University, Dept. of Elec. & Comp. Engg. Auburn, AL 36849, U.S.A. Nitin Yogi NVIDIA Corporation, Santa Clara, CA 95050 20 th.

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Vishwani D. Agrawal Auburn University, Dept. of Elec. & Comp. Engg. Auburn, AL 36849, U.S.A. Nitin Yogi NVIDIA Corporation, Santa Clara, CA th April th IEEE VLSI Test Symposium Santa Cruz, CA

Introduction Propose an information and noise analysis method for digital signals using Hadamard transform. Analysis identifies information (spectrally structured) and noise (random) contents of the signal in relative measures. The analysis is useful in a variety of applications like test generation, BIST, test compression, etc. We illustrate its application to test generation. 4/20/20102VTS'10, Santa Cruz, CA

June 12, 2009Nitin Yogi - Doctoral Defense3 Test Vectors and Bit-streams Circuit Under Test (CUT) Input 1 Input 2 Input 3 Input 4 Input 5 Input J Vector 1 → Vector 2 → Vector 3 → Vector 4 → Vector 5 → Vector 6 → Vector 7 → Vector 8 → Outputs Time in clocks A digital binary bit- stream signal vector

Hadamard Transform H(8) = Hadamard transform transforms a digital signal from time domain to frequency-related domain. Uses Walsh functions, which are a complete orthogonal set of basis functions that can represent any arbitrary bit-stream. Can be used for binary signals by using the representation {0,1} -> {-1,1} Example of Hadamard matrix of order 8 w0w0 w1w1 w2w2 w3w3 w4w4 w5w5 w6w6 w7w7 Walsh functions (order 3) time 4/20/20104VTS'10, Santa Cruz, CA

Hadamard transformation where: X Time domain digital signal vector of stream of N bits H(N) Hadamard transform matrix of order N S Hadamard transform (Walsh spectrum) of X 4/20/20105VTS'10, Santa Cruz, CA Forward transformation (time domain to spectral domain): Reverse transformation (spectral domain to time domain):

Properties of Hadamard Transform Orthogonality and symmetry Energy conservation where:  H(N) Hadamard transform matrix of order N  X Binary bit-stream vector in time domain  S Hadamard transform in spectral domain (Walsh spectrum) 4/20/20106VTS'10, Santa Cruz, CA

Energy Analysis Total energy in a {-1,+1} binary signal of length N clocks in time domain: Total energy in spectrum: 4/20/20107VTS'10, Santa Cruz, CA

Analysis of random binary bit-streams Values of spectral components of random binary bit- streams can be approximated as Gaussian distribution  Mean (µ) = 0  Standard deviation (σ) where:  S r (j) j th spectral component of a random binary bit-stream of length N  Square of the mean of S r (j) 4/20/20108VTS'10, Santa Cruz, CA Equal to N (by energy conservation) Equal to 0 (since mean = 0)

Spectral coefficients of random bit-stream 500 samples of random binary bit-streams of length 64 were generated Distribution of values of spectral components analyzed Mean = ≈ 0 Standard deviation = ≈ 1 1σ (68.27%) 2σ (95.45%) 3σ (99.73%) 4/20/20109VTS'10, Santa Cruz, CA Spectral components below a magnitude of 2σ or 3σ can be treated as noise components

Generating spectral bit-streams 1. Perform Hadamard transform on binary bit-stream. 2. Filter out noise-like spectral components having magnitudes less than a spectral threshold (Energy conservation of the transfom transfers the energy of filtered components to noise). 3. Perform reverse Hadamard transform to obtain time- domain values in the range (-1,+1) for bits in the bit- stream. 4. Time-domain values are normalized to range (0,1) and used as probabilities of logic 1 in new random bit-streams. 4/20/201010VTS'10, Santa Cruz, CA

Generating spectral bit-streams Hadamard Matrix H(3) {-1,+1} binary bit stream (X) Hadamard transform (S) = Example {0,1} binary bit-stream 1 √ {0,1} converted to {-1,+1} 4/20/201011VTS'10, Santa Cruz, CA

Example of generating spectral bit-streams 0 6/√ Hadamard transform (S) of 8-bit binary bit- stream Spectral threshold = 2 σ = 2 0 6/√ Probabilities for generating bit-streams 4/20/201012VTS'10, Santa Cruz, CA 0’s are filtered spectral coefficients Time-domain Reverse transformation Unfiltered spectral component 1 √8 1 Normalization for probabilities [X(k)+1] 2

Generated spectral bit-streams Original {0,1} binary bit- stream Unfiltered spectral component Probabilities for generating bit-stream Randomly generated bit-streams from probabilities 1 bit difference between original bit- stream & unfiltered spectral component Generated bit-streams exhibit similar correlation with the unfiltered spectral component as the original bit-stream. Few bits changed by noise are shown in red. 4/20/201013VTS'10, Santa Cruz, CA

Application of analysis Application of spectral information analysis: Test generation, BIST, Test data compression, etc. Illustration of effectiveness of analysis using test generation Test vectors generated for RTL faults (PIOs & flip- flops) Generate spectral vectors & fault grade on circuit Compare with random, weighted random & randomly perturbed vectors Analysis applied to ISCAS’89 benchmark circuits s1488, s5378 & s /20/201014VTS'10, Santa Cruz, CA

Spectral coefficients & power analysis for s1488 InputsSpectral CoefficientAmplitudePowerNoise Power Input 1w0w Input 2 w1w w Input 3 w1w w Input 4 w0w w Input 5 w1w w5w w w Input 6 w4w w Input 7 w1w w5w w Input 8 w4w w w /20/201015VTS'10, Santa Cruz, CA 32 vectors were generated to detect RTL faults (PIOs & FFs) & analyzed using H(32)

Gate level coverage for s1488 4/20/201016VTS'10, Santa Cruz, CA

Gate level coverage for s5378 4/20/201017VTS'10, Santa Cruz, CA

Gate level coverage for for s /20/201018VTS'10, Santa Cruz, CA

Conclusion Proposed an information analysis framework to distinguish noise from signal content Illustrated effectiveness of method for the application of test generation The method can easily be extended to other applications like BIST and test compression. See, N. Yogi and V. D. Agrawal, “BIST/Test-Compressor Design using Combinational Test Spectrum,” Proc. 13 th IEEE VLSI Design & Test Symp. (VDAT), July 2009, pp N. Yogi and V. D. Agrawal, “Sequential Circuit BIST Synthesis using Spectrum and Noise from ATPG Patterns” Proc. 17 th IEEE Asian Test Symp. (ATS), Nov 2008, pp There is potential for further applications. 4/20/201019VTS'10, Santa Cruz, CA

Thank you 4/20/2010VTS'10, Santa Cruz, CA20 Questions please?

Test generation comparison with commercial tool 4/20/2010VTS'10, Santa Cruz, CA21 Circuit name RTL-ATPG spectral testsFlexTest gate-level ATPG Coverage (%) No. of vectors CPU* (secs) Coverage (%) No. of vectors CPU* (secs) s s N. Yogi and V. D. Agrawal, “Spectral RTL Test Generation for Gate-Level Stuck-at Faults,” Proc. 15 th IEEE Asian Test Symp. (ATS), Nov 2006, pp * Sun Ultra 5, 256MB RAM Circuit name FlexTest gate-level ATPGBIST gate-level fault coverage (%) Coverage (%) No. of vectors 64k random vectors 64k weighted random vectors Spectral BIST (64k vectors) s s s N. Yogi and V. D. Agrawal, “Sequential Circuit BIST Synthesis using Spectrum and Noise from ATPG Patterns,” Proc. 17 th IEEE Asian Test Symp. (ATS), Nov 2008, pp Test generation: BIST: