A novel two step adaptive identification framework is proposed for sinusoidal signals to estimate the unknown offset, amplitude, frequency and phase, where only the output measurements are used. After representing the sinusoidal signal as a linearly parameterized form, several adaptive laws are developed. The proposed adaptive laws are driven by parameter estimation error information that is derived by applying filter operations on the output measurements, so that globally exponential convergence of the parameter estimation is proved. By using the sliding mode technique, we further improve the design of adaptations to achieve finite-time (FT) parameter estimation. The proposed approaches are independent of any observer/predictor design and robust to bounded measurement noises. The developed estimation methods are finally extended to the full parameter estimation of multi-sinusoids with only output measurements. Comparative simulation results are provided to illustrate the efficacy of the proposed methods. (C) 2015 Elsevier Ltd. All rights reserved.