EE3210M2a_0910SemA - City University of Hong Kong...

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City University of Hong Kong Department of Electronic Engineering EE3210 Lab 2: Introduction to Complex Exponentials Prelab: Read the Background section. Verification: The Warm-Up section must be completed during your assigned lab time. The steps marked Instructor Verification must also be signed off during the lab time. When you have completed a step that requires verification, simply demonstrate the step to the instructor. Lab Report: It is only necessary to turn in a report on the Experiment section with graphs and explanations. 1. Background Recalled that sinusoids may be expressed in the form: ( ) { } t f j j o o e Ae t f A t x π φ 2 Re 2 cos ) ( = + = . Now consider the sum of cosine waves given by ( = + = N k k k t f A t x 1 0 2 cos ) ( ) . ( 1 ) Note that x ( t ) is the sum of N cosine waves whose frequencies are all equal to f 0 . Although this formula is expressed by using trigonometric identities, it can also be concisely expressed as the sum of complex exponentials. By using complex exponential representation of cosine functions we have {} () s s t f j s t f j N k k N k t f j k t f A e Z e Z e Z t x π π π + = = = = = = 0 2 2 1 1 2 2 cos Re Re Re ) ( where k j k k A Z φ = , and . s j s N k k s e Z Z φ = = = 1 We see that the signal x ( t ) itself is a single sinusoid and it is also periodic with period T 0 = 1 / f 0 . Harmonic function is an important extension of sinusoidal function. For harmonic function, x ( t ) is the sum of N cosine waves whose frequencies f k are multiples of one single frequency f 0 : Frequency) (Harmonic 0 kf
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The harmonic function can be expressed as () = + = = = N k t kf j k N k k k e Z t kf A t x 1 2 1 0 0 Re 2 cos ) ( π φ . This signal x ( t ) contains harmonics of f 0 and is also periodic with period T 0 = 1 / f 0 . The frequency f 0 is called the fundamental frequency , and T 0 is called the fundamental period . 2
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This note was uploaded on 11/22/2009 for the course ELECTRONIC EE3210 taught by Professor Sungchiwan during the Spring '09 term at École Normale Supérieure.

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EE3210M2a_0910SemA - City University of Hong Kong...

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