In recent years, adaptive beamformers have been researched more extensively,to be able to use it in the application of medical ultrasound imaging.The adaptive beamformers can provide a higher resolution and better contrastsin the resulting images than non-adaptive, and most commonly used,delay-and-sum beamformer. The difficulties of applying adaptive beamformersto ultrasound imaging can e.g. be numerical complexity, stabilityof statistics, coherent sources, or the robustness of the beamformer. Methodsto handle these types of difficulties have been researched and successfullyapplied to utilize the performance advantages provided by adaptivebeamformers.Why is it important to prevent these types of errors? The beamformeris used in the image formation stage, and errors that occur at this stage aredifficult to get rid of even for the most sophisticated imaging software. Wehave in this thesis investigated methods that attempt to force the estimateof the covariance matrix to become a Toeplitz matrix. A Toeplitz matrixhas equal elements along its diagonals, and has several useful propertiesthat are desirable in array processing. Assuming the Toeplitz structure canbe achieved, this is because of the spatial stationarity in the received data,it can be applied in medical ultrasound imaging. Three different methodsare proposed to reach the desired Toeplitz structure in this thesis; IAAAPES,Adaptive Spatial Averaging and Spatial Convolution. The threemethods for making the covariance matrix Toeplitz will be compared andanalyzed to other known adaptive beamformers to detect their strengthsand weaknesses.