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2.4 Show that any signal can be decomposed into an even and an odd component. Is the decomposition unique? Illustrate your arguments using the signal x(n) = {2, 3, 4, 5, 6}

2.4 Show that any signal can be decomposed into an even and an odd component. Is the decomposition unique? Illustrate your arguments using the signal x(n) = {2, 3, 4, 5, 6}

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2.4 Show that any signal can be decomposed into an even and an odd component. Is the decomposition unique? Illustrate your arguments using the signal x(n) = {2, 3, 4, 5, 6}

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An integer n is even if, and only if, n equals twice some integer. An integer n is odd if, and only if, n equals twice some integer plus 1. Symbolically, for any integer, n n is even ⇔ n = 2k for some integer k n is odd ⇔ n = 2k + 1 for some integer k Prove the following statement. The difference of any two odd integers is even. Proof: Let m and n be any odd integers. By definition of odd, there are integers r and s so that m can be expressed in terms of r and n can be expressed in terms of s as follows: m = n = Write m - n in terms of r and s and factor out a 2 to obtain m - n = Now is an integer because --- Select-- v of integers are integers. Therefore, m - n = 2 ⋅ (an integer), and so m - n is … - Select-- by definition of … - Select--.
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