EE 5345/7345

Medical Signal Analysis

 

Instructor: Carlos Davila

Dept. of Electrical Engineering, Southern Methodist University

 

Module 1: Sampling Theory and A/D Converters

 

Homework Problems:

  1. Suppose a signal x(t) is band-limited to 100 rad/sec. What is the minimum sampling frequency in samples/sec which will enable the signal to be recovered from its samples, x[n]? Sketch a block diagram showing how x(t) would be recovered from x[n].
  2. Consier a 4-bit successive approximation A/D converter. The full scale voltage is Vref = 8V and Vin = 5.5. Find the contents of the successive approximation register at each step of the algorithm.
  3. Sketch the quantization characteristic for the A/D converter having 3 bits.
  4. Explain why a flash A/D converter is faster than a successive approximation A/D converter?
  5. What is quantization error? How is it modeled?
  6. What is the power spectral density? What is the relationship between signal power and power spectral density?
  7. Explain why an oversampled PCM A/D converter has lower quantization noise power in the signal band than Nyquist-rate sampled PCM A/D converter?
  8. Derive equation (11) in Aziz, Sorensen, and Van Der Spiegel and explain why the Sigma-Delta converter has even lower quantization error power in the signal band than oversampled PCM A/D converters.

 

Project:

 

In this project, we will investigate the characterstics of pulse code modulation (PCM) and Sigma-Delta A/D converters. You can do this using a programming language of your choice, though Matlab or Simulink is recommended. If you have not worked with Simulink, you may want to visit The Mathworks web page (www.mathworks.com) where you can do several tutorials which will help you get started.

 

  1. Implement a PCM A/D converter. Use a Gaussian white noise signal for the quantizer input, x[n], having zero mean and a variance of 1. Compute the quantization error, e[n] and estimate the probability density function (pdf) of e[n] for 8, 16, 32, 64, and 128 quantization levels using a histogram. Comment on how the pdf changes as the number of quantization levels increases.
  2. Using the results from part 1, experimentally determine the SNR for each of the quantizers that were tested and comment on the extent to which the SNR approaches the theoretical value discussed in lecture. Assume x[n] has been sampled at the Nyquist rate.
  3. For a 128 level PCM A/D converter, experimentally find the SNR when the input signal, x(t), is sampled at four times the Nyquist rate. Here you will have to lowpass filter your Gaussian white noise samples in order to model sampling at four times the Nyquist rate. This can be done by designing a lowpass filter using the Matlab filter design and analysis tool (fdatool). Compare the resulting SNR with that which results from Nyquist rate sampling in part 2. Also, compare your results with theory.
  4. Experimentally find the SNR for a first-order Sigma-Delta Modulation A/D converter assuming x(t) is sampled at four times the Nyquist rate. Compare your results with theory.