Module1_Discretetime Fourier series and transforms

Module1_Discretetime Fourier series and transforms - Module...

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Module 1 Discrete Discrete-Time Fourier Series and Time Fourier Series and Fourier Transforms 2010 R Chen 9/2/ L Module 1 : Discrete : Discrete-Time Fourier Series and Fourier Transforms Time Fourier Series and Fourier Transforms 1 Content • Motivation for signal representation by means of sinusoidal • Motivation for signal representation by means of sinusoidal components • Discrete-time Fourier Series; Fourier Series of periodic signals; synthesis and analysis equations; properties • Discrete-time Fourier Transforms; origin; synthesis and analysis equations; properties Module 1 : Discrete : Discrete-Time Fourier Series and Fourier Transforms Time Fourier Series and Fourier Transforms 2
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Learning Objectives • Be able to determine the DT Fourier series of DT signals • Understand the properties of the DT Fourier series • Be able to determine the DT Fourier Transform of DT signals and vice-versa • Understand the properties of the DT Fourier Transform 2010 • Be able to use MATLAB to analyze DT signals using DT Fourier Series or DT Fourier Transforms R Chen 9/2/ L Module 1 : Discrete : Discrete-Time Fourier Series and Fourier Transforms Time Fourier Series and Fourier Transforms 3 Bibliography • “Fourier Series Representation of Periodic Signals” (Chapter 3) in Signals and Systems , 2nd edition by A. V. Oppenheim, A. S. Willsky, and S. H. Nawab. • “The Discrete-Time Fourier Transform” (Chapter 5) in Signals and Systems , 2nd edition by A. V. Oppenheim, A. S. Willsky, and S. H. Nawab. Module 1 : Discrete : Discrete-Time Fourier Series and Fourier Transforms Time Fourier Series and Fourier Transforms 4
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Motivation for signal representation by means of sinusoids • The use of signal representation by means of sinusoidal components stems from their utility in analyzing linear-time invariant (LTI) systems. • In particular, the response of an LTI system to a sinusoidal input signal is also a sinusoidal signal but with a different amplitude and phase . Continuous-time (CT) signals ¾ the output of an LTI system can be expressed in terms of a convolution integral between the input x ) and the system 2010 convolution integral between the input ( t ) and the system transfer function h ( t ): = ) ( ) ( ) ( τ t x h d y sinusoid R Chen 9/2/ ¾ if we put x ( t ) = e j ω t , then we get where ) ( ) ( ω j H e = ωτ L Module 1 : Discrete : Discrete-Time Fourier Series and Fourier Transforms Time Fourier Series and Fourier Transforms 5 = ) ( ) ( Motivation for signal representation by means of sinusoids Continuous-time (CT) signals ¾ note that H ( j ω ) is a complex number that will alter the amplitude and phase of the sinusoidal signal. Discrete-time (DT) signals ¾ the output of a DT-LTI system can be written as n −∞ = = k n ] [ ] [ ] [ if we put x [ n ] = z , then ) ( ] [ z = where = ] [ ) ( Module 1 : Discrete : Discrete-Time Fourier Series and Fourier Transforms Time Fourier Series and Fourier Transforms 6 −∞ =
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Motivation for signal representation by means of sinusoids Discrete-time (DT) signals ¾ if further we put z = e j
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Module1_Discretetime Fourier series and transforms - Module...

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