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Lesson_7-1

# Lesson_7-1 - EEL 3135 Signals and Systems Dr Fred J Taylor...

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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Topic: Aliasing Lesson Number: 7 (Section 4-2) Sampling Modalities In Section 4-1, Shannon’s sampling theorem was introduced. The sampling theorem established the minimum rate at which a continuous-time 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 analog-to- digital-converter (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

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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. Consider the lowly cosine wave: ( 29 ( 29 t t x 0 cos ϖ = 1. sampled at a rate f s Sa/s. Assume that the sinusoid's frequency is known to be ϖ 0 =2 π f 0 = 2 π∆ 0 + m 2 π f s where 0 is a baseband frequency 0 [- f s /2, f s /2], and m is an integer. Then at the sample instances t = kT s : [ ] [ ] ( 29 ( 29 ( 29 ( 29 s s s kT mf kT k k x π π ϖ φ 2 2 cos cos cos 0 0 + = = = 2.
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Lesson_7-1 - EEL 3135 Signals and Systems Dr Fred J Taylor...

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