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Unformatted text preview: 1 AKW Fall 2010 Chapter 345 ELEC 211 Signals and Systems Module 2 Fourier Analysis and Frequency Domain Representation of Signals 2 AKW Fall 2010 Chapter 345 • Complex Exponentials as Eigenfunction of LTI Systems • System Function and Frequency Response • Frequency Domain Representation of Signals Lecture 7 Chapter 3 – System Function and Frequency Response 3 AKW Fall 2010 Chapter 345 What is Analysis? • In science and engineering, the concept of analysis is decompose a complex subject into and represent it as a collection of simpler constituents. • We may decompose the same subject differently depending on the problem that we want to solve, or our domain of interest, Example: Decomposing a human being Spiritual Domain = Body + Soul Anatomy Domain = Head + torso + limbs + other body parts Chemical Domain = Carbon + hydrogen + oxygen + other minerals • For signals, what are the basic constituents of interest? • Generally, we want to decompose subjects into constituents that are independent so that we can deal with them individually, and complete so that they can represent the subject without any missing parts. 4 AKW Fall 2010 Chapter 345 Chapter 2  Impulses as Basic Signal • In Chapter 2, we defined the impulse response and derive the inputoutput relationship of LTI systems in the form of a convolution integral/sum by regarding any signal as a integral/summation of shifted impulses. • The basic constituents here are the shifted impulses δ ( tτ ) t ) ( t CT system ) ( t h t ) ( t h ( ) ( ) ( ) x t x t d ∞ −∞ = − ∫ LTICT system ( ) ( ) ( ) y t x h t d ∞ −∞ = − ∫ We can regard any input signal as a superposition of shifted & weighted impulses System is LTI, hence output must be the same superposition of shifted & weighted impulse responses Assume the input is an impulse Call the output the Impulse Response = 5 AKW Fall 2010 Chapter 345 Observation of the real world  Frequency of Signal • We observe that many real world systems appear to respond to and are characterized by frequencies. Our ears hear by detecting acoustic waves at different frequencies; our eyes see by detecting electromagnetic light waves at different frequencies. Frequencies in signals determine what we hear and see middle C D E F G A B C 65 Hz 130 Hz 261 Hz 523 Hz 1046 Hz 493.88 Hz Sound Sight Wavelength (speed of light/frequency in Hz) 6 AKW Fall 2010 Chapter 345 • We observe also that we can characterize the properties of many natural as well as manmade transmission media (e.g., the atmosphere for radio signals, optical fiber for optical signal) by their responses to signals at different frequencies (or equivalently, wavelengths) independently ; i.e., we can address each frequency on its own without worrying about interactions with other frequencies Wavelength (speed of light/frequency) μ m (106 m) At wavelength of 1.5 μ m, frequency is 2 x 10 14 Hz Frequency range used for radio communications Frequency range used...
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 Fall '09
 AlbertK.Wong
 Fourier Series, Frequency, AKW Fall, Complex Sinusoids

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