090207HomeWork1Solution

090207HomeWork1Solution - 1 ECE 700 Digital Signal...

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Unformatted text preview: 1 ECE 700 Digital Signal Processing Winter 2009 Home Work 1 Due 1/21/09 PART I 1. A continuous-time signal x a ( t ) is the sum of unit amplitude sine waves of frequencies 300 Hz, 500 Hz, 1.2 kHz, 2.15 kHz, and 3.5 kHz. x a ( t ) is sampled at 2.5 kHz, leading to the discrete time signal x [ n ]. x [ n ] is then converted to the continuous-time with an ideal low- pass filter with a cut-off frequency of 1.0 kHz leading to the signal y a ( t ). The arrangement is shown in the figure below. What are the frequencies (in Hz) and amplitudes of the components present in y a ( t )? Solution: From the sampling theorem, 1 1 1 2 ( ) ( ) j a k k X e X j T T =-- = Consider the components ( ) j X e will have between - < . Summation index k Frequency of component ( suppressed) 0 1200 500 300 300 500 1200 , , , , , 1250 1250 1250 1250 1250 1250--- 1 2150 350 3500 1000 2 ,2 1250 1250 1250 1250- =- = --1 2150 350 3500 1000 2 , 2 1250 1250 1250 1250- + = -- + = 2 So effectively,...
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090207HomeWork1Solution - 1 ECE 700 Digital Signal...

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