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Linear canonical transform (LCT) is a powerful tool for improving the detection accuracy of the conventional Wigner distribution (WD). However, the LCT free parameters embedded increase computational complexity. Recently, the instantaneous cross-correlation function type of WD (ICFWD), a specific WD relevant to the LCT, has shown to be an outcome of the tradeoff between detection accuracy and computational complexity. In this paper, the ICFWD is applied to detect noisy single component and bi-component linear frequency-modulated (LFM) signals through the output signal-to-noise ratio (SNR) inequality modeling and solving with respect to the ICFWD and WD. The expectation-based output SNR inequality model between the ICFWD and WD on a pure deterministic signal added with a zero-mean random noise is proposed. The solutions of the inequality model in regard to single component and bi-component LFM signals corrupted with additive zero-mean stationary noise are obtained respectively. The detection accuracy of ICFWD with that of the closed-form ICFWD (CICFWD), the affine characteristic Wigner distribution (ACWD), the kernel function Wigner distribution (KFWD), the convolution representation Wigner distribution (CRWD) and the classical WD is compared. It also compares the computing speed of ICFWD with that of CICFWD, ACWD, KFWD and CRWD.

Recently, Wigner distribution (WD) associated with linear canonical transforms (LCTs) is quickly becoming a promising technique for detecting linear frequency-modulated (LFM) signals corrupted with noises by establishing output signal-to-noise ratio (SNR) inequality model or optimization model. Particularly, the closed-form instantaneous cross-correlation function type of WD (CICFWD), a unified linear canonical Wigner distribution, has shown to be competitive in detecting noisy LFM signals under an extremely low SNR. However, the CICFWD has up to nine LCT free parameters so that it requires a heavy computational load. To improve the efficiency of real-time processing, this paper focuses on the instantaneous cross-correlation function type of WD (ICFWD), which has only six LCT free parameters but is not a special case of the CICFWD. The main advantage of ICFWD is that it could be expected to reduce the computational complexity while maintaining detection performance. This paper first proposes an optimization model to the ICFWD’s output SNR with respect to deterministic signals embedded in additive zero-mean noises. It then deduces the model’s solution to a single component LFM signal added with white noise, leading to the optimal selection strategy on LCT free parameters. Simulation results demonstrate that the ICFWD improves almost a doubling of computing speed in comparison with the CICFWD while sharing the same level of detection performance. To be specific, the computing time of ICFWD in sampling frequencies 5 Hz, 10 Hz, 15 Hz, and 20 Hz is about 0.048 s, 0.111 s, 0.226 s, and 0.392 s, respectively, while 0.075 s, 0.233 s, 0.478 s, and 0.821 s for the computing time of CICFWD; the ICFWD and CICFWD have nearly the same output SNR higher than that of the WD.