By Lothar Papula
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Visible pondering - visible mind's eye or conception of diagrams and image arrays, and psychological operations on them - is omnipresent in arithmetic. is that this visible pondering simply a mental reduction, facilitating snatch of what's collected by means of different capability? Or does it even have epistemological services, as a method of discovery, realizing, or even evidence?
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Extra info for Mathematik fuer Ingenieure und Naturwissenschaftler 3: Vektoranalysis, Wahrscheinlichkeitsrechnung (2008)
Speciﬁcally, x[n] can be decomposed as the sum of the three sequences x[−2]δ[n + 2], xδ[n], and xδ[n − 3] representing the three nonzero values in the sequence x[n]. The sequences x[−2]h[n + 2], xh[n], and xh[n − 3] are the system responses to x[−2]δ[n + 2], xδ[n], and xδ[n − 3], respectively. The response to x[n] is then the sum of these three individual responses. Although the convolution-sum expression is analogous to the convolution integral of continuous-time linear system theory, the convolution sum should not be thought of as an approximation to the convolution integral.
Discrete-time signals are deﬁned at discrete times, and thus, the independent variable has discrete values; that is, discrete-time signals are represented as sequences of numbers. Signals such as speech or images may have either a continuous- or a discrete-variable representation, and if certain conditions hold, these representations are entirely equivalent. Besides the independent variables being either continuous or discrete, the signal amplitude may be either continuous or discrete. Digital signals are those for which both time and amplitude are discrete.
One of the important aspects of the impulse sequence is that an arbitrary sequence can be represented as a sum of scaled, delayed impulses. 4 can be expressed as p[n] = a−3 δ[n + 3] + a1 δ[n − 1] + a2 δ[n − 2] + a7 δ[n − 7]. 4) More generally, any sequence can be expressed as ∞ x[k]δ[n − k]. 5) k=−∞ We will make speciﬁc use of Eq. 5) in discussing the representation of discrete-time linear systems. 3b) is deﬁned as 1, n ≥ 0, 0, n < 0. 7) k=−∞ that is, the value of the unit step sequence at (time) index n is equal to the accumulated sum of the value at index n and all previous values of the impulse sequence.
Mathematik fuer Ingenieure und Naturwissenschaftler 3: Vektoranalysis, Wahrscheinlichkeitsrechnung (2008) by Lothar Papula