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Note that if we were to use a Hamming or a Blackman window instead of a Hanning window, the stopband and passband ripple requirements would have been exceeded, and the required filter order would have been larger. With a Blackman window, for example, the filter order required to meet the transition band requirement is

public class Network { Vertex start; private class Vertex { Object object; Edge edges; Vertex nextVertex; boolean done; int dist; Vertex back; } private class Edge { Vertex to; int weight; Edge nextEdge; }

W e would like t o filter an analog signal x,(t) with a n analog low-pass filter that has a cutoff frequency f,. = 2 kHz, a transition width Af = 500 Hz, a n d a stopband attenuation o f 50 dB. This filter is t o b e implemented digitally, a s illustrated in the following figure:

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Dijkstra s algorithm finds the shortest path from one vertex v0 to each other vertex in a digraph. When it has finished, the length of the shortest distance from v0 to v is stored in the vertex v, and the shortest path from v0 to v is recorded in the back pointers of v and the other vertices along that path. (See Example 15.24.) The algorithm uses a priority queue, initializing it with all the vertices and then dequeueing one vertex on each iteration. Algorithm 15.1 Dijkstra s Shortest Paths Algorithm (Precondition: G = (V,w) is a weighted graph with initial vertex v0.) (Postcondition: Each vertex v in V stores the shortest distance from v0 to v and a back reference to the preceding vertex along that shortest path.) 1. Initialize the distance field to 0 for v0 and to for each of the other vertices. 2. Enqueue all the vertices into a priority queue Q with highest priority being the lowest distance field value. 3. Repeat steps 4 10 until Q is empty. 4. (Invariant: The distance and back reference fields of every vertex that is not in Q are correct.) 5. Dequeue the highest priority vertex into x. 6. Do steps 7 10 for each vertex y that is adjacent to x and in the priority queue. 7. Let s be the sum of the x s distance field plus the weight of the edge from x to y. 8. If s is less than y s distance field, do steps 9 10; otherwise go back to Step 3. 9. Assign s to y s distance field. 10. Assign x to y s back reference field. EXAMPLE 15.24 Tracing Dijkstra s Algorithm

4 1 )

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

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This is a trace of Algorithm 15.1 on a graph with eight vertices. On each iteration, the vertices that are still in the priority queue are shaded, and vertex x is labeled. The distance fields for each vertex are shown adjacent to the vertex, and the back pointers are drawn as arrows.

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