5650_2p_fa2010 - ECE 5650/4650 MATLAB Project 2 This...

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Introduction 1 ECE 5650/4650 MATLAB Project 2 This project is to be treated as a take-home exam, meaning each student is to due his/her own work. The project due date target is 5:00 PM Friday, November 19, 2010. To work the project you will need access to M ATLAB and at minimum the signal processing toolbox. Introduction In this second Matlab computer simulation project we will consider: Sampling theory Basic modeling to study aliasing and reconstruction Quantization effects Filter design Filter design basics Group delay Quantization effects in filters Applications Problems Sampling Theory In the first few problems we deal with basic sampling theory and modeling in M ATLAB . Consider the A/D- -D/A DSP system shown in Figure 1. In lowpass sampling the continuous-time or analog signal, , is first passed through an anti-aliasing filter to remove signals that lie above the Nyquist frequency . The A/D converter produces a sampled version of its input, in this case the sequence . A digital filter, or some discrete-time system then processes the signal to form the output . Assuming the signal needs to be returned to the continuous-time domain, a D/A converter converts the signal samples to an analog waveform, which needs further analog sig- nal processing to return it to a signal containing just the fundamental spectral translate. To model this system in Matlab, we will make some assumptions and simplifications. To Hz  x a t f s 2 xn  yn H ( z ) A/D Anti-aliasing Filter Reconstruction Filter D/A f s f s x a ( t ) x aa ( t ) y a ( t ) x [ n ] y [ n ] y ZOH ( t ) Figure 1: A basic A/D- -D/A DSP system.
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ECE 5650/4650 MATLAB Project 2 Introduction 2 begin with we will approximate as a highly over sampled discrete-time signal. The sam- pling rate used to approximate the analog input signal will be denoted . For all of the prob- lems in this section kHz. The analog anti-aliasing filter will be a third order Butterworh lowpass digital filter of the form >> [ba,aa] = butter(3,2*(fs/2)/96000); where fs is the sampling rate in Hz used in the A/D. Note the 3dB cutoff frequency of the filter is at the folding or Nyquist frequency, . Next we model the A/D as a downsampler followed by a bit quantizer. The quantizer is a nonlinearity represented as . Since all signals in the model are discrete, the input to the anti-aliasing filter is really and the output of the anti-aliasing filter is , so the A/D output is (1) The block diagram of the A/D model is shown in Figure 2. In M ATLAB the upsampling operation performed by the Signal Processing Toolbox (SPTB) function >> y = downsample(x,M); The sequence is the true discrete-time signal in the DSP system. It nominally enters a signal processing function block, e.g., , and returns the sequence . In the sampling theory problems we will assume that the DSP function block is simply a through connection, i.e., (2) The output signal (in this case also ) next enters the D/A model. This model is com- posed of an upsampler followed by a filter having a rectangular impulse response. A digital filter
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5650_2p_fa2010 - ECE 5650/4650 MATLAB Project 2 This...

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