Ve216LectureNotesChapter3Part1 - Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 3 Fourier Representations of Signals and LTI Systems 3.1

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Ve216 Lecture Notes Dianguang Ma Spring 2009
Chapter 3 Fourier Representations of Signals and LTI Systems
3.1 Introduction In this chapter, we represent a signal as a weighted superposition of complex sinusoids. If such a signal is applied to an LTI system, then the system output is a weighted superposition of the system response to each complex sinusoid. The study of signals and systems using sinusoidal representations is termed Fourier analysis .
3.2 Complex Sinusoids and Frequency Response of LTI Systems Consider the output of a discrete-time LTI system with impulse response h[n] and unit amplitude complex sinusoidal input x[n]. ( ) [ ] [ ] [ ]* [ ] [ ] [ ] [ ] [ ] ( ) ( ): the frequency response of the system j n k j n k j n j k k k j n j j x n e y n h n x n h k x n k h k e e h k e e H e H e =-∞ - - Ω =-∞ =-∞ = = = - = = =
3.2 Complex Sinusoids and Frequency Response of LTI Systems The output of an LTI system is a complex sinusoid of the same frequency as the input, multiplied by the frequency response of the system.
3.2 Complex Sinusoids and Frequency Response of LTI Systems We say that the complex sinusoid e jΩn is an eigenfunction of the LTI system H associated with the eigenvalue H(e ). ( ) j n j j n e H e e
3.2 Complex Sinusoids and Frequency Response of LTI Systems We write the complex-valued frequency response in polar form. { } { } arg ( ) ( ) [ ] ( ) ( ) ( ) : the magnitude response of the system arg ( ) : the phase response of the system j j j k k j H e j j j j H e h k e H e H e e H e H e - Ω =-∞ = =
3.2 Complex Sinusoids and Frequency Response of LTI Systems The system thus modifies the amplitude of the input by the magnitude response |H(e )| and and the phase by the phase response arg{H(e )}. ( arg{ ( )} [ ] [ ] ( ) j j n j j n H e x n e y n H e e Ω + = =
3.2 Complex Sinusoids and Frequency Response of LTI Systems Similar results are obtained for continuous-time LTI systems.