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and let X ( z ) be its z-transform. If we sample X ( z ) at zk = e x p ( j % k ) fork = 0, 1 , 2 , 3 , we obtain a set of DFT coefficients X ( k ) . Find the sequence, y ( n ) , that has a four-point DFT equal to these samples.

Sampling X(z) at four equally spaced points around the unit circle produces an aliased version of x(n):

Prove Corollary 101 on page 188 Prove Corollary 102 on page 188 Derive the formula for the path length of a full tree of order d and height h The St Petersburg Paradox is a betting strategy that seems to guarantee a win It can be applied to any binomial game in which a win or lose are equally likely on each trial and in which the amount bet on each trial may vary For example, in a coin-flipping game, bettors may bet any number of dollars on each flip, and they will win what they bet if a head comes up, and they will lose what they bet if a tail comes up The St Petersburg strategy is to continue playing until a head comes up, and to double your bet each time it doesn t.

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Which traversals always visit: a. the root first b. the left-most node first c. the root last d. the right-most node last The level order traversal follows the pattern as reading a page of English text: left-to-right, row-by-row. Which traversal algorithm follows the pattern of reading vertical columns from left to right Which traversal algorithm is used in the call tree for the solution to Problem 9.32 on page 184

Using the tabular method to evaluate this sum, noting that x(n) and x(n +4) are the only sequences that have nonzero values for 0 5 n 5 3, we have

Consider a finite-length sequence x ( n ) that is zero outside the interval 10. N - 11. Suppose that we form a new sequence i ( n ) as follows:

For example, the sequence of tosses is {T, T, T, H}, then the bettor will have bet $1 and lost, then $2 and lost, then $4 and lost, then $8 and won, ending up with a net win of $1 + $2 + $4 + $8 = $1 Since a head has to come up eventually, the bettor is guaranteed to win $1, no matter how many coin flips it takes Draw the transition diagram for this strategy showing the bettor s winnings at each stage of play Then explain the flaw in this strategy Some people play the game of craps allowing 3 to be a possible point In this version, player Y wins on the first toss only if it comes up 2 or 12 Use a transition diagram to analyze this version of the game and compute the probability that X wins.

Derive the formula for the path length of a full tree of order d and height h The St Petersburg Paradox is a betting strategy that seems to guarantee a win It can be applied to any binomial game in which a win or lose are equally likely on each trial and in which the amount bet on each trial may vary For example, in a coin-flipping game, bettors may bet any number of dollars on each flip, and they will win what they bet if a head comes up, and they will lose what they bet if a tail comes up The St Petersburg strategy is to continue playing until a head comes up, and to double your bet each time it doesn t.

where M < N. Find the M-point DFT of the sequence i(n) O l n c M otherwise expressing the answer in terms of the D T F T of x(n). This problem is most easily solved if we take advantage of what we know about the DFT. Recall that the M-point , DFT corresponds to M samples of the DTFT, X ( e J U )at wi = 2 n k l M for k = 0, 1, . . ., M - 1. In addition, if these samples are used for the DFT coefficients of a sequence y ( n ) of length M, y ( n ) is related t o x ( n ) as follows:

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