The focus of this thesis is on the design and selection of systolic architectures for ASIC implementation of the real-time digital signal processing task of Modi- fied Covariance spectral estimation. When used with pulsed Doppler ultrasound blood flow detectors, the Modified Covariance spectral estimator offers increased sensitivity in the detection of arterial disease over conventional Fourier transform based methods. The systolic model of computation is considered because through pipelining and parallel processing high levels of concurrency can be achieved to attain the nec- essary throughput for real-time operation. Systolic arrays of simple processing units are also well suited for implementation on VLSI. The versatility of the de- sign of systolic arrays using the rigorous data dependence graph methodology is demonstrated throughout the thesis by application to all sections of the spectral estimator design at both word and bit levels. Systolic array design for the model order 4 Modified Covariance spectral estima- tor, known to offer accurate estimation of blood flow mean velocity and d1stur- bance at an acceptable computational burden, is initially discussed. A variety of problem size dependent systolic arrays for real-time implementation of the fixed model order spectral estimator are designed using data dependence graph mapping methods. Optimal designs are chosen by comparison of hardware, com- munication and control costs, as well as efficiency, timing, data flow and accuracy considerations. A cost/benefit analysis, based on results from structural simula- tion of the arrays, allows the most suitable word-lengths to be chosen. Problem size independent systolic arrays are then discussed as means of coping with the huge increases in computational burden for a Modified Covariance spec- tral estimator which is programmable up to high model orders. This type of array can be used to reduce the number of PEs and increase efficiency when compared to the problem size dependent arrays and the research culminates in the proposal of a novel spiral systolic array for Cholesky decomposition.