Uplink integrated sensing and communication (ISAC) systems have recently emerged as a promising research direction, enabling simultaneous uplink signal detection and target sensing. {In this paper, we propose the flexible projection (FP)-type receiver that unifies the projection-type receiver and the successive interference cancellation (SIC)-type receiver by using a flexible tradeoff factor to adapt to dynamically changing uplink ISAC scenarios.} The FP-type receiver addresses the joint signal detection and target response estimation problem through two coordinated phases: 1) Communication signal detection using a reconstructed signal whose composition is controlled by the tradeoff factor, followed by 2) Target response estimation performed through subtraction of the detected communication signal from the received signal. With adjustable tradeoff factors, the FP-type receiver can balance the enhancement of the signal-to-interference-plus-noise ratio (SINR) with the reduction of correlation in the reconstructed signal for communication signal detection. The pairwise error probability (PEP) expressions are analyzed for both the maximum likelihood (ML) and the zero-forcing (ZF) detectors, revealing that the optimal tradeoff factor should be determined based on the adopted detection algorithm and the relative power of the sensing and communication (S\&C) signals. A homotopy optimization framework is first applied for the FP-type receiver with a fixed tradeoff factor. This framework is then extended to develop the dynamic flexible projection (DFP)-type receiver, which iteratively adjusts the tradeoff factor for improved algorithm performance and environmental adaptability. Finally, we show that the length of the jointly processed signal should scale with the antenna size to fully unleash the potential of the uplink ISAC receiver.

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