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A microprocessor incorporates the functions of a computer's central processing unit (CPU) on a single or few integrated circuits (IC). The purpose of a microprocessor is to accept digital data as input, process it as per the instructions, and then provide the output. This is known as sequential digital logic. The microprocessor has internal memory and operates basically on the binary system.
Most general purpose microprocessors are present in personal computers. They are often used for computation, text editing, multimedia display, and communication over a network. Other microprocessors are part of embedded systems. These provide digital control over practically any technology, such as appliances, automobiles, cell phones, industrial process control, etc.
The DSP processor, on the other hand, is a particular type of microprocessor. DSP stands for digital signal processing. It is basically any signal processing that is done on a digital signal or information signal. A DSP processor is a specialized microprocessor that has an architecture optimized for the operational needs of digital signal processing.