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Chapter-09-Filtering - Chapter 9 Filtering Linear time...

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Chapter 9 Filtering Linear time invariant systems have the property that if the input is described as a sum of sinusoids, then the output is a sum of sinusoids of the same frequency. Each sinusoidal component will typ- ically be scaled differently, and each will be subjected to a phase change, but the output will not contain any sinusoidal components that are not also present in the input. For this reason, an LTI system is often called a filter . It can filter out frequency components of the input, and also enhance other components, but it cannot introduce components that are not already present in the input. It merely changes the relative amplitudes and phases of the frequency components that are present in the inputs. LTI systems arise in two circumstances in an engineering context. First, they may be used as a model of a physical system. Many physical systems are accurately modeled as LTI systems. Second, they may present an ideal for an engineered system. For example, they may specify the behavior that an electronic system is expected to exhibit. Consider for example an audio system. The human ear hears frequencies in the range of about 30 to 20,000 Hz, so a specification for a high fidelity audio system typically requires that the frequency response be constant (in magnitude) over this range. The human ear is relatively insensitive to phase, so the same specification may say nothing about the phase response (the argument, or angle of the frequency response). An audio system is free to filter out frequencies outside this range. Consider an acoustic environment, a physical context such as a lecture hall where sounds are heard. The hall itself alters the sound. The sound heard by your ear is not identical to the sound cre- ated by the lecturer. The room introduces echoes, caused by reflections of the sound by the walls. These echoes tend to occur more for the lower frequency components in the sound than the higher frequency components because the walls and objects in the room tend to absorb higher frequency sounds better. Thus, the lower frequency sounds bounce around in the room, reinforcing each other, while the higher frequency sounds, which are more quickly absorbed, become relatively less pro- nounced. In an extreme circumstance, in a room where the walls are lined with shag carpeting, for example, the higher frequency sounds are absorbed so effectively that the sound gets muffled by the room. The room can be modeled by an LTI system where the frequency response H ( ω ) is smaller in 287
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288 CHAPTER 9. FILTERING magnitude for large ω than for small ω . This is a simple form of distortion introduced by a channel (the room), which in this case carries a sound from its transmitter (the lecturer) to its receiver (the listener). This form of distortion is called linear distortion , a shorthand for linear, time-invariant distortion (the time invariance is left implicit).
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