Unified 2-D X-Alignment for Improving the Observability of Response Compactors

Despite the advantages of performing response compaction in integrated-circuit testing, unknown response bits (x's) inevitably reflect into loss in test quality. The distribution of these x's within the captured response, which varies for each test pattern, directly impacts the number of scan cells observed through the response compactor. In this paper, we propose a unified 2-D x-alignment technique in order to judiciously manipulate the distribution of x's in the test response prior to its compaction. The controlled response manipulation is performed on a per pattern basis, in the form of scan chain delay and intra-slice rotate operations, and with the objective that x's are aligned within as few scan slices and chains as possible. Consequently, a larger number of scan cells are observed after compaction for any test pattern. In an effort to tackle the unified 2-D x-alignment problem and to achieve maximum overall observability, we first decipher the interaction between 1-D x-alignment operations, and formulate 1-D and 2-D x-alignment operations all as maximum satisfiability (MAX-SAT) problems; a weighted MAX-SAT formulation is necessitated in the 2-D case to identify the best possible 2-D x-alignment, which may differ from back to back application of the individual best possible 1-D alignments in two dimensions. The proposed technique is test set independent, leading to a generic, simple, and cost-effective hardware implementation. While we show in this paper that x-alignment improves horizontal and vertical compactors, covering a wide spectrum of compactors, it is expected to improve other types of compactors as well by manipulating the x-distribution properly.