# CH5 - CHAPTER 5 Linear Systems Most DSP techniques are...

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87 CHAPTER 5 Linear Systems Most DSP techniques are based on a divide-and-conquer strategy called superposition . The signal being processed is broken into simple components, each component is processed individually, and the results reunited. This approach has the tremendous power of breaking a single complicated problem into many easy ones. Superposition can only be used with linear systems , a term meaning that certain mathematical rules apply. Fortunately, most of the applications encountered in science and engineering fall into this category. This chapter presents the foundation of DSP: what it means for a system to be linear, various ways for breaking signals into simpler components, and how superposition provides a variety of signal processing techniques. Signals and Systems A signal is a description of how one parameter varies with another parameter. For instance, voltage changing over time in an electronic circuit, or brightness varying with distance in an image. A system is any process that produces an output signal in response to an input signal . This is illustrated by the block diagram in Fig. 5-1. Continuous systems input and output continuous signals, such as in analog electronics. Discrete systems input and output discrete signals, such as computer programs that manipulate the values stored in arrays. Several rules are used for naming signals. These aren't always followed in DSP, but they are very common and you should memorize them. The mathematics is difficult enough without a clear notation. First, continuous signals use parentheses, such as: and , while discrete signals use x ( t ) y ( t ) brackets, as in: and . Second, signals use lower case letters. Upper x [ n ] y [ n ] case letters are reserved for the frequency domain, discussed in later chapters. Third, the name given to a signal is usually descriptive of the parameters it represents. For example, a voltage depending on time might be called: , or v ( t ) a stock market price measured each day could be: . p [ d ]

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The Scientist and Engineer's Guide to Digital Signal Processing 88 Continuous System Discrete System x(t) y(t) x[n] y[n] FIGURE 5-1 Terminology for signals and systems. A system is any process that generates an output signal in response to an input signal. Continuous signals are usually represented with parentheses, while discrete signals use brackets. All signals use lower case letters, reserving the upper case for the frequency domain (presented in later chapters). Unless there is a better name available, the input signal is called: x ( t ) or x [ n ], while the output is called: y ( t ) or y [ n ]. Signals and systems are frequently discussed without knowing the exact parameters being represented. This is the same as using x and y in algebra, without assigning a physical meaning to the variables. This brings in a fourth rule for naming signals. If a more descriptive name is not available, the input signal to a discrete system is usually called: , and the output signal: . x [ n ] y [ n ] For continuous systems, the signals: and are used. x ( t ) y ( t ) There are many reasons for wanting to understand a
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CH5 - CHAPTER 5 Linear Systems Most DSP techniques are...

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