Exercise 2.1. Compute the arithmetic code for the two sequences given the shared spreadsheet. The probability mass function is also provided in the spreadsheet.
Exercise 2.2. Compute the LZ78 parsing for the three sequences given in the shared spreadsheet.
Exercise 2.3 (1-bit quantization). In this exercise, you will implement a scalar quantizer that minimizes the mean squared error distortion. For any x P R, consider a 1-bit/2-level quantizer which does the following:
#a if x ą t
Qpxq “ b if x ď t
where a ą b are the two levels, and t is the threshold. We assume that a ą t ą b.
For any real-valued random variable X, the mean squared error of the quantizer for a fixed a,b,t is
MSE “ EpX ´ QpXqq2
Assume that X is uniformly distributed over the interval rα,βs. Set up the optimization problem and find the a,b,t which minimizes the mean squared error. These would be functions of α,β.
Now use the files mentioned in the shared spreadsheet. Each file contains a set of 100 points uniformly distributed over some rα,βs. Estimate α,β from the set given to you. Use this, and the optimal 1-bit quantizer that you have designed above. Find Qpxq for all x in the file, and compute the mean squared error
ř .
2-1
