090112HomeWork1

# 090112HomeWork1 - ECE 700 Digital Signal Processing Winter...

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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 )? 2. Show that the upsampler and downsampler operators, 2 and commute. 3 3. Suppose x c ( t ) is bandlimited to 1.25/ T , and x 0 [ n ] = x c ( nT ), x 1 [ n ] = x c ( nT - T /3) and x 2 [ n ] = x c ( nT - 2 T /3). Design the black box below and show the corresponding details of the sinc reconstruction so that y c ( t ) = x c ( t ) i.e., x c ( t ) is perfectly recovered. 1

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4. For each of the following H ( z ), derive closed-form expressions for the R = 2 polyphase components, H 0 ( z ) and H 1 ( z ).
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090112HomeWork1 - ECE 700 Digital Signal Processing Winter...

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