# Cours2 - Chapter 2 Signal Representation ELEC 2880 Modem...

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Chapter 2: Signal Representation ELEC 2880: Modem Design () Chapter 2: Signal Representation ELEC 2880: Modem Design 1 / 18

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Complex baseband signal Why is it useful? For bandpass signal, such as modulated signal Simple, low pass representation Used for performance computation Also used in actual signal processing Course structure Reminder of basic principles/equations Application to elements in communication chain Modulation Filters Random noise Typical example: Sinusoid in AWGN () Chapter 2: Signal Representation ELEC 2880: Modem Design 2 / 18
Analytical Signal Let x ( t ) be a real signal Bandpass: | X ( w ) | = 0 for | ω - ω 0 | > π B Analytical Signal x a ( t ): Keep positive frequencies ( ω 0) with factor 2 x ( t ) = { x a ( t ) } Add imaginary part to cancel negative frequencies () Chapter 2: Signal Representation ELEC 2880: Modem Design 3 / 18

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Complex baseband representation X(w) X a (w) E x (w) () Chapter 2: Signal Representation ELEC 2880: Modem Design 4 / 18
Complex baseband representation Complex baseband, also called complex envelope: E x ( ω ) = X a ( ω + ω 0 ) e x ( t ) = e - j ω 0 t x a ( t ) Depends on choice of ω 0 (not unique!) Equivalent representation, contains all information x ( t ) = { e x ( t ) e j ω 0 t } () Chapter 2: Signal Representation ELEC 2880: Modem Design 5 / 18

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Rice components Rice components defined as e x ( t ) = x 1 ( t ) + jx 2 ( t ) If ω 0 well chosen, they have same bandwidth as complex baseband (but not same shape) x 1 ( t ) = { e x ( t ) } = [ e x ( t ) + e * x ( t )] / 2 X 1 ( ω ) = [ E x ( ω ) + E * x ( - ω )] / 2 They represent I/Q components: (! sign) x ( t ) = x 1 ( t ) cos( ω 0 t ) - x 2 ( t ) sin( ω 0 t )
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• Spring '10
• wiener
• Signal Representation, Modem Design

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