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Problem_Set_2

Course Number: EE 648, Spring 2001

College/University: Purdue

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EE 648 Problem Set 2, due Monday, 3/10/2003. For on-campus students, due in class. For o-campus students, must be postmarked 3/10. Problem 1. Prove the Pythagorean theorem: if z and w are vectors in an inner product space, and z w, then z 2 + w 2 = z + w 2 . Here, v = v, v . Problem 2. Prove that, if we dene v = v, v for any vector v in an inner product space V, then the following inequality will hold for any two...

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648 EE Problem Set 2, due Monday, 3/10/2003. For on-campus students, due in class. For o-campus students, must be postmarked 3/10. Problem 1. Prove the Pythagorean theorem: if z and w are vectors in an inner product space, and z w, then z 2 + w 2 = z + w 2 . Here, v = v, v . Problem 2. Prove that, if we dene v = v, v for any vector v in an inner product space V, then the following inequality will hold for any two vectors x and y in V: x+y x + y . (You may use the Cauchy-Schwartz inequality.) Problem 3. Problem 3.6 from the textbook. Let g[n] = (1)n h[n]. Relate g () to h(). If h is a low-pass lter, can g be a low-pass lter? Problem 4. Problem 3.7 from the textbook. Prove the convolution Theorem 3.3: If f l1 (Z) and h l1 (Z), then g = f h l1 (Z) and g () = ()h(). f Problem 5. Problem 3.9 from the textbook. Let a and b be two integers with many digits. Relate the product ab to a convolution. Explain how to use FFT to compute this product. Problem 6. (a) Use the identity sin 2 = 2 sin cos to show that cos Use this to prove that 2 cos . . . cos N 4 2 = sin 2N 1 1 sin sin 2 ... . 2N sin sin sin 2 N 2 4 cos p=1 2p = 1 sin . (b) Using Part (a), show that the innite product formula (7.37) of Theorem 7.2 holds for the Haar multiresolution. In other words, use Part (a) to calculate the innite product (7.37) the for Haar lter 1 h[n] = 2 ([n] + [n 1]), and show that it is equal to the Fourier transform of the Haar scaling function. 1 (c) Suppose h() = 22 (1 + ei )2 . Use Part (b) to nd the corresponding innite product given by (7.37). What is the corresponding time domain function? Is it an orthogonal scaling function? If it is, prove it; if it is not, why is Theorem 7.2 not applicable in this case? Problem 7. Problem 3.3 from the textbook. An interpolation function f (t) satises f (n) = [n]. (a) Prove that k= f ( + 2k) = 1 if and only if f is an interpolation function. (Hint. You can use Proposition 3.1.) 1 (b) Suppose that f (t) = relate f () to (). n= h[n](t n) with L2 (R). Find such h() that f (n) = [n], and Problem 8. Consider the wavelet series expansion of a continuous-time signal f : f= j= n= f, j,n j,n . Assume that is the Haar wavelet, and let f = 1[0,1] (i.e., f is the Haar scaling function). (a) Give the expansion coecients in the above expression. (b) Verify that j= n= | f, j,n |2 = 1 (Parsevals identity). (c) Consider the signal f (t) = f (t 2k ), where k is a positive integer. (Here, the symbol does not denote a derivative!) Give the range of scales over which the expansion coecients of the wavelet expansion of f are dierent from zero. (d) Repeat Part (c) for f (t) = f (t 1/ 2). 2