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E. if the sequence x [n] is squaresummable. A signal with this property is sometimes referred to as a finiteenergy signal. For a simple example of the converse, note that a periodic signal which is not identically zero is not square-summable. (3) We will see later that a more “correct” approximation to differentiation is given by a filter H(e j ω ) = j ω. For most applications, however, the first-order difference will suffice. Prandoni and M. e. their energy dilutes to zero over an infinite time duration).

23) this is called the periodic extension of the finite length signal x [n]. This type of extension is the “natural” one in many contexts, for reasons which will be apparent later when we study the frequency-domain representation of discrete-time signals. Note that now an arbitrary shift of the periodic sequence corresponds to the periodization of a circular shift of the original finite-length signal. 6. Finally, the energy of a periodic extension becomes infinite, while its power is simply the energy of the finite-length original signal scaled by 1/N .

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.