By Paul C. Krause

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Prandoni and M. Vetterli, © 2008, EPFL Press This suggests another approach to embedding a finite-length signal x [n], n = 0, . . e. 24) where we have chosen M = 0 (but any other choice of M could be used). Note that, here, in contrast to the the periodic extension of x [n], we are actually adding arbitrary information in the form of the zero values outside of the support interval. This is not without consequences, as we will see in the following Chapters. In general, we will use the bar notation x¯ [n] for sequences defined as the finite support extension of a finite-length signal.

E. the norm of the vectors is not unitary).

L. Allen and D. W. Mills’ Signal Analysis (IEEE Press, 2004). 1: Review of complex numbers. ∞ (a) Let s [n] := 1 1 + j n . Compute s [n]. n 2 3 n =1 (b) Same question with s [n] := j 3 n . (c) Characterize the set of complex numbers satisfying z ∗ = z −1 . (d) Find 3 complex numbers {z 0 , z 1 , z 2 } which satisfy z i3 = 1, i = 1, 2, 3. ∞ (e) What is the following infinite product e j π/2 ? 2: Periodic signals. For each of the following discrete-time signals, state whether the signal is periodic and, if so, specify the period: n (a) x [n] = e j π (b) x [n] = cos(n) (c) x [n] = cos π n 7 +∞ (d) x [n] = y [n − 100 k ], with y [n] absolutely summable.