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2. Chapter-3-4-5-AKW

# 2. Chapter-3-4-5-AKW - 1 AKW Fall 2010 Chapter 3-4-5 ELEC...

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Unformatted text preview: 1 AKW Fall 2010 Chapter 3-4-5 ELEC 211 Signals and Systems Module 2 Fourier Analysis and Frequency Domain Representation of Signals 2 AKW Fall 2010 Chapter 3-4-5 • 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 3-4-5 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 3-4-5 Chapter 2 - Impulses as Basic Signal • In Chapter 2, we defined the impulse response and derive the input-output 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 ( t-τ ) t ) ( t CT system ) ( t h t ) ( t h ( ) ( ) ( ) x t x t d ∞ −∞ = − ∫ LTI-CT 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 3-4-5 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 3-4-5 • We observe also that we can characterize the properties of many natural as well as man-made 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 (10-6 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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2. Chapter-3-4-5-AKW - 1 AKW Fall 2010 Chapter 3-4-5 ELEC...

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