Problem Set 5.1 Spring 2009
Arrays for Signal Processing

A.  Objectives

To gain familiarity with the formal notation commonly associated with arrays and matrices.  This problem set provides students with experience manipulating arrays in setting up and carrying out computations that will be of help in completing Project 5 on voice quality analysis.  Some problems in this problem set require the use of Excel.  You will need to submit a printout of your work.

1. Compute the following summations based on the array values given in the matrix below. Following standard conventions, the index i refers to rows and the index j refers to columns.

   4	3	7	1	9
   1	4	6	3	8
   2	3	8	2	9
   3	5	7	4	8

   1. 

   2. 
      
      
   3. 
      
      
   4. Explain in words each of the following summations (you need not compute them):

                                                      

   5. In summation notation, the mean (or average) of the elements in the second column is expressed as:
      
      
      
      

      where n is the number of values being averaged (in this example, 4). Compute the mean of the elements in the second column;. In part (f) below, we will refer to this value as  .

   6. Compute the sum of the differences between the values in the second column and their mean:
      
      
      
      

      Why did you get the result that you did?

       

   7. Compute the sum of the squares of the differences you found in (e):
      
      
      
      

      Summing the squares of differences between values and their mean avoids the result you obtained in (e); that result wasn't very helpful as a way to measure the cumulative effect of the individual differences.

       

2. Enter the matrix below into an Excel spreadsheet.  Calculate (a) the row sums, (b) the column sums, (c) the row averages, (d) the column averages, and (e) the sum of the squares of the first column values (the "harmonic power" = H), (f) the sums of squares of the differences between the individual column values and the respective column means for the first column, (the "noise power" = N), (g) the harmonic to noise ratio for the first column in dB, HNR = 10*log(H/N) for log base10. Enter the necessary formulas into the cells near the lower or right borders of the matrix as appropriate.

   9	12	20	17	2	2	7	9	10
   11	10	22	16	3	4	6	8	10
   9	10	21	18	4	3	7	8	9
   10	10	20	16	5	4	7	8	11
   8	9	23	18	1	3	5	9	10