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A.1 A.2 A.3 A.4 A.5 A.6 A function f() is called idempotent if f(f(x)) = f(x) for all x in the domain of f(). Explain why the floor and ceiling functions are idempotent. What is a logarithm What is the difference between weak induction and strong induction How can you decide when to use strong induction What is Euler s constant What makes Stirling s formula useful

(6) Determine whether or not the filter is the same as that which would be designed using the impulse

n! =

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Therefore, because the Laplace transform of the step response is related to the system function H,(s) as follows,

A.1 A.2 A.3 Prove Theorem A.1 on page 319. Prove Theorem A.2 on page 320. True or false: a. f = o(g) g = (f) b. f = O(g) g = (f) g = (f) c. f = (g) d. f = O(g) f = (g) f = (g) e. f = (g) f. f = (h) g = (h) g. f = (h) g = (h) h. n 2 O(n lg n) i. n 2 (n lg n) j. n 2 (n lg n) k. lg n (n) l. lg n o(n)

To design a digital filter using step invariance, we first perform a partial fraction expansion of S,(s),

k = 1 2 3 4

Therefore. Sampling s,,(t),

f + g = (h) f g = (h)

APPENDIX]

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s ( n ) = s,(nT,)

Note that although H ( z ) has the same poles as the filter designed using step invariance, the system functions are not the same. Therefore, the two designs are not equivalent.

Prove Theorem A.5 on page 323. Prove Theorem A.6 on page 323. Prove Theorem A.7 on page 323. Run a program that tests De Moivre s formula on page 325 by comparing the values obtained from it with those obtained from the recursive definition of the Fibonacci numbers.

Unlike the harmonic sequence, the factorial sequence grows exponentially. This is reflected by Stirling s formula:

Suppose that we would like to design a discrete-time low-pass filter by applying the impulse invariance method to a continuous-time Butterworth tilter that has a magnitude-squared function

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Show that the design is not affected by the value of the sampling period that is used in the impulse invariance technique. In the absence of aliasing, the impulse invariance method is a linear mapping from H,,(jQ) to H(eJL" for Iwl 5 n. This mapping is H W " ) = H,,(;w "! = ,, IwI 5 n Let us assume that there is no aliasing (this will be approximately true if the filter order is large enough). The required filter order is then log d N Z log k where d, the discrimination factor, is (I

A.1 A.2 A.3 The floor and ceiling functions are idempotent because they return integer values, and according to Theorem A.1 on page 319, the floor or ceiling of an integer is itself. A logarithm is an exponent. It is the exponent on the given base that produces the given value. The First Principle of Mathematical Induction ( weak induction) allows the inductive hypothesis that assumes that the proposition P(n) is true for some single value of n. The Second Principle of Mathematical Induction ( strong induction) allows the inductive hypothesis that assumes that all the propositions P(k) are true for all k less than or equal to some value of n. Use weak induction (the first principle) when the proposition P(n) can be directly related to its predecessor P(n 1). Use strong induction (the second principle) when the proposition P(n) depends upon P(k) for k < n 1. Euler s constant is the limit of the difference (1 + 1/2 + 1/3 + . . . + 1/n) lnn. Its value is approximately 0.5772. Stirling s formula is a useful method for approximating n! for large n (e.g., n > 20).

12n ,

d = [

A.1 Proof of Theorem A.1 on page 319: a. The relationships x = max{m Z | m x}, and restatements of the definitions of x and x .

- 6,J2 -

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