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Sparse signal recovery finds use in a variety of practical applications, such as signal and image restoration and the recovery of signals acquired by compressive sensing. In this paper, we present two generic very-large-scale integration (VLSI) architectures that implement the approximate message passing (AMP) algorithm for sparse signal recovery. The first architecture, referred to as AMP-M, employs parallel multiply-accumulate units and is suitable for recovery problems based on unstructured (e.g., random) matrices. The second architecture, referred to as AMP-T, takes advantage of fast linear transforms, which arise in many real-world applications. To demonstrate the effectiveness of both architectures, we present corresponding VLSI and field-programmable gate array implementation results for an audio restoration application. We show that AMP-T is superior to AMP-M with respect to silicon area, throughput, and power consumption, whereas AMP-M offers more flexibility.