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Unformatted text preview: EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Topic: Aliasing Lesson Number: 7 (Section 42) Sampling Modalities In Section 41, Shannon’s sampling theorem was introduced. The sampling theorem established the minimum rate at which a continuoustime signal can be sampled and reconstructed from its sample values. The lower bound on the sample rate was found to be f s >2f max where f max is the highest frequency in the signal being sampled (f max /2 called the Nyquist frequency). In reality, there are three possible sampling modalities. They are: • over sampling f s >> f max /2 (common case) • critical sampling f s = f max /2 • and undersampled f s < f max /2 The reconstructed output varies depending on the sampling strategy used. In addition, practical considerations, such as the minimum or maximum sample rates supported by an analogto digitalconverter (ADC), influence the sampling decisions. Aliasing Whenever Shannon’s Sampling Theorem is violated ( i.e ., under sampled case), a phenomenon called aliasing occurs. Aliasing is a phenomenon that manifests itself as a corruption of a reconstructed signal or image. Aliasing occurs when a reconstructed signal impersonates another signal or image. To illustrate, consider as a youth seeing your first Hollywood western. The camera was following a moving stage coach at a 30 frames/s rate. While the stagecoach wheel was moving at a leisurely clockwise rate of say 12 ° per frame, the viewer would see the wheel spin at a rate of one revolution per second in a clockwise direction as reported in Figure 1. Then a danger appears under a sea of arrows and the background music is translated to a minor key. The stagecoach was now rushing forward at full speed; wheel’s spinning at a rate near 30 revolutions per second which translates to a rotational rate is 348 ° degrees (348 ° =360 ° 12 ° ) per frame. From the viewer’s perspective it would appear to the wheel is turning at a rate of 12 ° per fame backwards as suggested in Figure 1! This is a physical instantiation of aliasing. ALIAS/V0.1  1 EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Figure 1: Image aliasing experiment. Slowly moving stagecoach (top) and rapidly moving stagecoach (bottom). In this supplement, aliasing will be studied using a number of strategies in this supplemental note. All are based on the requirement that a signal, reconstructed from its sample values, is a baseband signal. This means that the reconstructed signal’s frequency is assumed to be restricted to the baseband range f ∈ (f s /2, f s /2). The principal object of this note is to provide you with the tools needed to determine the frequency a reconstructed signal....
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.
 Spring '08
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 Aliasing

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