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Unformatted text preview: Lesson 4 Lesson 4: Spectral Analysis Challenge 03: Lesson 04 (Sect. 3.13.4) Spectrum Sum and product spectrum Periodicity Introduction to Fourier Series Challenge 04 Lesson 4 Challenge 03 You where asked to analyze the RC circuit shown under the assumption that V(j ϖ ) = 8 ∠ ° at ϖ = 1000 r/s. It was established that V R (j ϖ )= 4.44 ∠ 56.3 ° . Using Kichoff’s Law, the capacitor voltage can be computed using phasor algebra as shown below. 20 + i(t)j30 v C (t) V =8 ∠ ° = V R + V C V R =4.44 ∠ 56.3 ° v C (t)=6.66 cos(1000t – 33.7 ° ) V C = V V R =6.66 ∠33.7 ° V C Phasor Lesson 4 Chapter 3 Spectral representation Lesson 4 Spectrum Suppose a signal has be representation given by: Euler says that the signal representation is complex. The signal consists of complex components residing at positive, negative, and zero (DC) frequencies. How can this information be graphically represented? ( 29 C a e a t x k N N k t kf j k ∈ = ∑ = ; 2 π Lesson 4 Spectrum Complex spectrum representation choices: Magnitude and phase response (phasor) Real and imaginary spectrum (complex) Each displays the frequency spectrum of the signal under study. Variations on this theme are: Log (dB) magnitude vs. linear frequency (semilog) Log (dB) magnitude vs. log frequency (loglog or Bode plot) Lesson 4 Spectrum Example: x(t)=cos( ϖ t) = (1/2)e j ϖ t + (1/2)ej ϖ t for some ϖ . Magnitude 0 Phase Real 0 Imaginary Lesson 4 Spectrum Example: x(t)=sin( ϖ t) = (j1/2)e j ϖ t + (j1/2)ej ϖ t for some ϖ . Magnitude 0 Phase 0 Real 0 Imaginary 0 90 °90 °1/2 ° 1/2 ° Lesson 4 Product Spectrum For multiplication (product modulation) opérations...
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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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