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.