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 flexible projection (FP)-type receivers that unify the projection-type receivers and the successive interference cancellation (SIC)-type receivers by using a flexible tradeoff factor to adapt to dynamically changing uplink ISAC scenarios. The FP-type receivers address 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 receivers 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 probabilities (PEPs) are analyzed for both maximum likelihood (ML) and 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) signal. A homotopy optimization framework is first applied for the FP-type receivers with a fixed trade-off factor. This framework is then extended to develop dynamic FP (DFP)-type receivers, which iteratively adjust the trade-off factor for improved algorithm performance and environmental adaptability. Subsequently, two extensions are explored to further enhance the receivers' performance: parallel DFP (PDFP)-type receivers and a block-structured receiver design. Finally, the effectiveness of the proposed receiver designs is verified via simulations.

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