Homework 2

Data attributes and digital convolution

You can either write your answers to theoretical questions on paper or edit them in the file hw2/paper.tex. Please show all the mathematical derivations that you perform.

1. The varimax attribute is defined as
   

   $$\phi[\mathbf{a}] = \frac{\displaystyle N\,\sum\limits_{n=1... ...n^4}{\displaystyle \left(\sum\limits_{n=1}^{N} a_n^2\right)^2}$$ (1)

   
   

   Suppose that the data vector $\mathbf{a}$ consists of random noise: the data values $a_n$ are independent and identically distributed with a zero-mean Gaussian distribution: $E[a_n]=0$, $E[a_n^2]=\sigma^2$, $E[a_n^4]=3\,\sigma^4$. Find the mathematical expectation of $\phi[\mathbf{a}]$.

2. The matrix in equation (2) represents a convolution operator with zero boundary conditions.
   

   $$\mathbf{F} = \left[\begin{array}{llllll} f_1 & f_0 & 0 & 0 &... ... f_0 \\ 0 & 0 & 0 & f_3 & f_2 & f_1 \\ \end{array}\right]\;.$$ (2)

   
   

   The operator is implemented in the C function hw2/conv.c.

   void conv_lop (bool adj, bool add, 
   		int nx, int ny, float* xx, float* yy) 
   /*< linear operator >*/
   {
       int f, x, y, x0, x1;
   
       assert (ny == nx);
       sf_adjnull (adj, add, nx, ny, xx, yy);
       
       for (f=0; f < nf; f++) {
   	x0 = SF_MAX(0,1-f);
   	x1 = SF_MIN(nx,nx+1-f);
   	for (x = x0; x < x1; x++) {
   	    if( adj) {
   		/* add code */
   	    } else {
   		yy[x+f-1] += xx[x] * ff[f];
   	    }
   	}
       }
   }
   

   1. Modify the matrix and the program to implement periodic boundary conditions.
   2. Add the code for the adjoint (matrix transpose) operator.

3. The C code in hw2/filter.c implements a recursive filtering operator.

   void filter_lop (bool adj, bool add, 
   		int nx, int ny, float* xx, float* yy) 
   /*< linear operator >*/
   {
       int i;
       float t;
   
       assert (ny == nx);
       sf_adjnull (adj, add, nx, ny, xx, yy);
       
       if (adj) {
   	/* add code */
       } else {
   	t = a*xx[0];
   	yy[0] += t;
   	for (i = 1; i < nx; i++) {
   	    t = a*xx[i] + b*xx[i-1] + c*t;
   	    yy[i] += t;
   	}
       }
   }
   

   1. Express this filter in the $Z$-transform notation as a ratio of two polynomials.
   2. Add code for the adjoint operator.

Homework 2