In a time-varying transmission channel, the received signals are subject to frequency shifts due to the Doppler effect. The Doppler frequency is dependent on the carrier frequency and channel variation rate. In a fixed wireless channel, the channel variations are caused by scatterer motion. In this paper, we investigate analytically the Doppler effects generated by scatterer motion under different scatterer velocity distributions using the ring-of-scatterers geometric model. The proposed model considers Doppler frequency components caused by scatterer mobility to both received and reflected signals at each scatterer, and therefore is called the double Doppler model. The analytical curves are compared and statistically tested with several measurement results published in the literature. At low scatterer speeds, e.g., generated by moving foliage, the exponential velocity distribution is an appropriate model to describe the time-varying nature of the fixed wireless channels. The curve fitting results also show that our analytical model better approaches the empirical curves than the single Doppler model does. However, further investigation is still needed to find a suitable scatterer velocity distribution that closely describes the double Doppler effect in fast-variation fixed wireless channels, e.g., caused by passing vehicles.

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In this paper, a new optimal watermarking scheme based on singular value decomposition (SVD) and lifting wavelet transform (LWT) using multi-objective genetic algorithm optimization (MOGAO) is presented. The singular values of the watermark is embedded in a detail subband of host image. To achieve the highest possible robustness without losing watermark transparency, multiple scaling factors (MSF) are used instead of single scaling factor (SSF). Determining the optimal values of the MSFs is a difficult problem. However, to find this values a multi-objective genetic algorithm optimization is used. Experimental results show a much improved performance in term of transparency and robustness of the proposed method compared to others methods.

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