Concerning that the Wiener filtering algorithm leads signal distortion with low Signal-to-Noise Ratio (SNR) when dealing with the noise of non-stationary speech signal, a new speech denoising algorithm named SSA-WF was proposed combining with Singular Spectrum Analysis (SSA) and Wiener Filtering (WF). To obtain the speech signal as smooth as possible, SSA was used to denoise the nonlinear and non-stationary speech signal to improve the SNR of the noisy speech. Then the processed signal was put into WF to further eliminate the high frequency noise that still existed in the speech signal. The simulation results from different intensity of background noise show that the proposed algorithm is superior to the traditional methods in SNR and Root-Mean-Square Error (RMSE). The results also demonstrate that the new algorithm can not only remove the background noise efficiently, but also reserve the details of the original signal, it is suitable for the denoising of nonlinear and non-stationary speech signal.