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Chapter 4 Continuous -Time Fourier Transform

ELG 3120 Signals and Systems Chapter 4 1/4 Yao Chapter 4 Continuous -Time Fourier Transform Introduction A periodic signal can be represented as linear combination of complex exponentials which are harmonically related. An aperiodic signal can be represented as linear combination of complex exponentials, which are infinitesimally close in frequency. So the representation take the form of an integral rather than a sum In the Fourier series representation, as the period increases the fundamental frequency decreases and the harmonically related components become closer in frequency. As the period becomes infinite, the frequency components form a continuum and the Fourier series becomes an integral. Representation of Aperiodic Signals: The Continuous -Time Fourier Transform Development of the Fourier Transform Representation of an Aperiodic Signal Starting from the Fourier series representation for the Continuous -Time periodic square wave: <<<=2/,0,1)(11 TtTTttx, ( ) 1T1T 2TT2T T T2T2 )(tx The Fourier coefficients ka for this square wave are TkTkak010)sin(2 =.

Starting from the Fourier series representation for the continuous-time periodic square wave: < < < = 0, /2 1, ( ) 1 1 T t T t T x t, (4.1) −T 1 T 1 2 T 2 − − 2T − T 2T x(t) The Fourier coefficients a k for this square wave are k T k T a k 0 2sin( 0 1) w w = . (4.2) or alternatively

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Transcription of Chapter 4 Continuous -Time Fourier Transform

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