Digital Processing Systems 2.8 2000/07/14 2004/08/05 10:44:07.245 GMT-5 Don Johnson dhj@rice.edu CJ Ganier seejaie@rice.edu Don Johnson dhj@rice.edu John Austin Cottrell jac3@rice.edu bit representation clocked device computer digital processing systems digital signal processing discrete-time word length Discrete-time systems can operate on both analog and discrete-time signals. Computers are discrete-time systems because they perform operations according to a clock. Discrete-time systems can act on discrete-time signals in ways similar to those found in analog signals and systems. Because of the role of software in discrete-time systems, many more different systems can be envisioned and constructed with programs than can be with analog signals. In fact, a special class of analog signals can be converted into discrete-time signals, processed with software, and converted back into an analog signal, all without the incursion of error. For such signals, systems can be easily produced in software, with equivalent analog realizations difficult, if not impossible, to design. Digital systems are more general than analog systems because they can also deal with symbolic signals. How do computers perform signal processing? We will discover that digital signals have their own Fourier transform definition and that digital filters can also be designed. We will also discover that digital systems enjoy an algorithmic advantage that contributes to rapid processing speeds: Computations can be restructured in non-obvious ways to speed the processing. Before we explore these ideas, we need to understand what a computer is and how it computes. A simple computer operates fundamentally in discrete time. Computers are clocked devices, in which computational steps occur periodically according to ticks of a clock. This description belies clock speed: When you say "I have a 400MHz computer," you mean that your computer takes 250 nanoseconds to perform each step. That is incredibly fast! A step does not, unfortunately, necessarily mean a computation like an addition; computers break such computations down into several stages, which means that the clock speed need not express the computational speed. Computational speed is expressed in units of millions of instructions/second (Mips). Your 400MHz computer (clock speed) may have a computational speed of 100Mips. Computers perform integer (discrete-valued) computations. Each computer instruction that performs an elementary calculation—an addition, a multiplication, or a division—does so only for integers. The sum or product of two integers is also an integer, but the quotient of two integers is likely to not be an integer. How does a computer deal with numbers that have digits to the right of the decimal point? This problem is addressed by using the so-called floating-point representation of real numbers. At its heart, however, this representation relies on integer-valued computations.