Hardware impairment is inevitable in many wireless systems. It is particularly severe in low-cost applications due to the imperfect components used. In this paper, the channel estimation and non-coherent detection problems of hardware impaired signals are studied for a single-carrier, single-antenna and single-hop system. Specifically, three different cases are investigated: signals with additive distortion only, signals with in-phase and quadrature imbalance only, and signals with both impairments. The maximum likelihood and Gaussian approximation methods are used to derive the new non-coherent detectors for amplitude modulated signals, while the maximum likelihood and moment-based methods are employed to design the new channel estimators for all signals. Numerical results show that the new non-coherent detectors outperform the existing non-coherent detectors in the presence of hardware impairment. The performance gain can be as high as 8 dB. They also show that the new channel estimators have much higher accuracy than the existing estimator. In some conditions, the accuracy of the new estimator is about 100 times that of the existing estimator.