TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P14

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P14:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | SPECTRAL LEAKAGE AND ALIASING 51 regions may not be well known it is usually difficult to predict the exact behavior in the alias region. We will look at ways to deal with these overlaps so that aliasing is reduced if not to zero then at least to sufficiently low levels that it becomes unimportant. In many cases a small amount of aliasing can be tolerated. We emphasize that Fig. 3-3 deals with phasors. Mathematically physical sine and cosine waves as defined explicitly in Eq. 2-1 do not exist as separate entities at negative frequencies despite occasional rumors to the contrary. Instruments such as spectrum analyzers oscilloscopes etc. can be used to create certain visual illusions the Wells-Fargo stage coach wheels in old-time Westerns often appear to be turning in the wrong direction. For example as compared to phasors that can rotate at positive or negative angular frequencies cos rnt cos t 3-3 sin -rnf sin t sin t Z 180 3-4 We note also that the average power in any single phasor of any constant amplitude is zero the resolution of the computer. So the phasor at frequency k must never be thought of as a true signal that can light a light bulb or communicate. The sine or cosine wave requires two phasors one at positive frequency and one at negative frequency and the result is at positive frequency. As an electrical signal the individual phasor is a mathematical concept only and not a physical entity we often lose sight of this . Therefore aliasing for sine and cosine spectra requires special attention which we consider in this chapter. Adjacent segments of the eternal steady-state positive-frequency sinecosine spectrum can overlap at frequencies greater than zero and it is a common problem. In Fig. 3-4a the positive-frequency eternal steady-state spectrum is centered at 5 15 25 35 45 and so on. Each side of each spectrum segment collides with an adjacent segment producing alias regions. This spectrum pattern repeats every 10 frequency units. The plots are shown .

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