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Programmable photonic integrated circuits (PICs), offering diverse signal processing functions within a single chip, are promising solutions for applications ranging from optical communications to artificial intelligence. While the scale and complexity of programmable PICs is increasing, the characterization, and thus calibration, of them becomes increasingly challenging. Here we demonstrate a phase retrieval method for programmable PICs using an on-chip fractional-delay reference path. The impulse response of the chip can be uniquely and precisely identified from only the insertion loss using a standard complex Fourier transform. We demonstrate our approach experimentally with a 4-tap finite-impulse-response chip. The results match well with expectations and verifies our approach as effective for individually determining the taps' weights without the need for additional ports and photodiodes.