CH13 - CHAPTER 13 Continuous Signal Processing Continuous...

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243 CHAPTER 13 Continuous Signal Processing Continuous signal processing is a parallel field to DSP, and most of the techniques are nearly identical. For example, both DSP and continuous signal processing are based on linearity, decomposition, convolution and Fourier analysis. Since continuous signals cannot be directly represented in digital computers, don't expect to find computer programs in this chapter. Continuous signal processing is based on mathematics ; signals are represented as equations, and systems change one equation into another. Just as the digital computer is the primary tool used in DSP, calculus is the primary tool used in continuous signal processing. These techniques have been used for centuries, long before computers were developed. The Delta Function Continuous signals can be decomposed into scaled and shifted delta functions , just as done with discrete signals. The difference is that the continuous delta function is much more complicated and mathematically abstract than its discrete counterpart. Instead of defining the continuous delta function by what it is , we will define it by the characteristics it has . A thought experiment will show how this works. Imagine an electronic circuit composed of linear components, such as resistors, capacitors and inductors. Connected to the input is a signal generator that produces various shapes of short pulses The output of the circuit is connected to an oscilloscope, displaying the waveform produced by the circuit in response to each input pulse. The question we want to answer is: how is the shape of the output pulse related to the characteristics of the input pulse ? To simplify the investigation, we will only use input pulses that are much shorter than the output. For instance, if the system responds in milliseconds, we might use input pulses only a few microseconds in length. After taking many measurement, we come to three conclusions: First, the shape of the input pulse does not affect the shape of the output signal. This
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The Scientist and Engineer's Guide to Digital Signal Processing 244 is illustrated in Fig. 13-1, where various shapes of short input pulses produce exactly the same shape of output pulse. Second, the shape of the output waveform is totally determined by the characteristics of the system, i.e., the value and configuration of the resistors, capacitors and inductors. Third, the amplitude of the output pulse is directly proportional to the area of the input pulse. For example, the output will have the same amplitude for inputs of: 1 volt for 1 microsecond, 10 volts for 0.1 microseconds, 1,000 volts for 1 nanosecond, etc. This relationship also allows for input pulses with negative areas. For instance, imagine the combination of a 2 volt pulse lasting 2 microseconds being quickly followed by a -1 volt pulse lasting 4 microseconds. The total area of the input signal is zero , resulting in the output doing nothing . Input signals that are brief enough to have these three properties are called impulses
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CH13 - CHAPTER 13 Continuous Signal Processing Continuous...

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