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High efficiency and low power consumption are among the main topics in embedded systems today. For complex applications, off-the-shelf processor cores might not provide the desired goals in terms of power consumption. By optimizing the processor for the application, one can improve the computing power by introducing special purpose hardware units. In this paper, we present a case study with a possible design methodology for an ultra low power application specific instruction-set processor. A cardiac beat detector algorithm based on the Continuous Wavelet Transform is implemented in the C language. This application is further optimized using several software power optimization techniques. The resulting application is mapped on a basic processor architecture provided by Target Compiler Technologies, and the processor is further optimized for ultra low power consumption by applying application specific hardware, and by using several hardware optimization techniques. The optimized processor is compared with the unoptimized version, resulting in a 55% reduction in power consumption. The reduction in the total execution cycle count is 81%. Power gating, and dynamic voltage and frequency scaling, are investigated for further power optimization. For a given case, the reduction in the already optimized power consumption is estimated to be 62% and 35%, respectively.