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WEIGHTED DIGRAPHS AND GRAPHS A weighted digraph is a pair (V, w) where V is a finite set of vertices and w is a function that assigns to each pair (x, y) of vertices either a positive integer or (infinity). The function w is called the weight function, and its value w(x, y) can be interpreted as the cost (or time or distance) for moving directly from x to y. The value w(x, y) = indicates that there is no edge from x to y. A weighted graph is a weighted digraph (V,w) whose weight function w is symmetric, that is, w(y,x) w(x,y) for all x,y V. Just as every digraph has an embedded graph, every weighted digraph has an embedded weighted graph (V, w) and an embedded (unweighted) digraph. The weight function for the embedded weighted graph can be defined as w (x, y) = min{w(x,y), w(y,x)}, where w is the weight function of the weighted digraph. The vertex set for the embedded digraph can be defined as E = {(x,y) : w(x,y) < }. The properties described above for digraphs and graphs apply to weighted digraphs and weighted graphs. In addition there are some extended properties that depend upon the underlying weight function in the obvious manner. For example, the weighted path length is the sum of the weights of the edges along the path. And the shortest distance from x to y would be the minimum weighted path length among all the paths from x to y. EXAMPLE 15.21 A Weighted Digraph and Its Embedded Structures

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where Once the coefficients a ( k ) have been determined, the coefficients h ( k ) are found using the PadC approach of' forcing h ( n ) = h d ( n )for n = 0, 1. . . . , 9 :

int[][] a;

The coefficients a ( k ) that minimize E may be found by setting the partial derivatives of E equal to zero,

Figure 15.20 shows a weighted digraph together with its embedded weighted graph, its embedded digraph, and its embedded graph. The weights are shown on the edges.

where is the correlation of hd(n).

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An incidence matrix for a digraph (V, E) is a two-dimensional integer array obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and the edges E = {e0 , e1, . . . , em 1} and then assigning 1 to a[i][j] and 1 to a[j][i] if there exists an edge emanating from vertex vi and terminating at vertex vj, and assigning 0 everywhere else. EXAMPLE 15.17 An Incidence Matrix for a Digraph

The inverse of a linear shift-invariant system with unit sample response g(n) and system function G ( z ) is the system that has a unit sample response, h(n),such that

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In graph G1 the weighted path length of the path cabd is |cabd| = 2 + 3 + 2 = 7, and the shortest distance from c to d is 6 (along the path cad). But in graph G2 that shortest distance is 1 (along the path cd). Note that graph G3 is the same as that in Example 15.13 on page 292, and graph G4 is the same as that in Example 15.1 on page 285. Figure 15.21 shows the adjacency matrix, the incidence matrix, and the adjacency list for graph G1.

solution. One of the reasons is that, In most applications, the system function H ( z ) = I/G(z) is not a v~able unless G ( z )is minimum phase, I / G ( z )cannot be both causal and stable. Another consideration comes from the fact that, in some applications, it may be necessary to constrain H ( 2 ) to be an FIR filter. Because l / G ( z )will be infinite in length unless G ( z )is an all-pole filter, constraining h ( n )to be FIR would only be an approximation to the inverse filter. In the FIR least-squares inverse filter design problem. the goal is to find the FIR filter h ( n )of length N such that h ( n )* ~ ( n ) S(n) The filter that minimizes the squared error

Figure 15.17 shows an incidence matrix for the digraph in Figure 15.13 on page 292. The first row indicates that two edges emanate from vertex a and one edge terminates there. The last 1 is in the row for vertex d and the last column. The only other nonzero entry in that column is the -1 in the row for vertex c, meaning that this edge emanates from vertex d and terminates at vertex c.

where In many cases, constraining the least-squares inverse filter to minimize the difference between h ( n )* ~ ( n ) and S(n) is overly restrictive. For example. if a delay may be tolerated, we may consider finding the filter h ( n ) so that h ( n )* ~ ( n ) 6(n - no) for some delay no. In most cases, a nonzero delay will produce a better approximate inverse filter and, in many cases, the improvement will be substantial. The least-squares inverse filter with delay is found by solving the linear equations

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