PROGRAMMING IN MATHEMATICA, A PROBLEM ... - Duke …
In class one cannot go on forever showing students just how commands in Mathematica work; on the other hand it would be very di cult to follow the codes if one writes a program having more than ve lines in class (especially as Mathematica’s style of programming provides a condensed code). Thus this note. This note
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Table 3: Properties of the z-Transform - Duke University
pfister.ee.duke.eduTable 3: Properties of the z-Transform Property Sequence Transform ROC x[n] X(z) R x1[n] X1(z) R1 x2[n] X2(z) R2 Linearity ax1[n]+bx2[n] aX1(z)+bX2(z) At least the intersection of R1 and R2 Time shifting x[n −n0] z−n0X(z) R except for the possible addition or deletion of the origin
Chapter 5 Linear Transformations and Operators
pfister.ee.duke.eduible. For vector spaces, the relevant structure is given by vector addition and scalar multiplication. Since a linear transformation preserves both of these operation, it is also a vector space homomorphism. Likewise, an invertible linear transformation is a vector space isomorphism. 5.2 Linear Functionals on Vector Spaces Definition 5.2.1.
Table of Discrete-Time Fourier Transform Pairs
pfister.ee.duke.eduTable of Discrete-Time Fourier Transform Pairs: Discrete-Time Fourier Transform : X() = X1 n=1 x[n]e j n Inverse Discrete-Time Fourier Transform : x[n] = 1 2ˇ Z 2ˇ X()ej td: x[n] X() condition anu[n] 1 1 ae j jaj<1 (n+ 1)anu[n] 1 (1 ae j)2 jaj<1 (n+ r 1)! n!(r 1)! anu[n] 1 (1 ae j)r jaj<1 [n] 1 [n n 0] e j n 0 x[n] = 1 2ˇ X1 k=1 (2ˇk) u[n ...
Chapter 1 Logic and Set Theory - Duke University
pfister.ee.duke.edu4 CHAPTER 1. LOGIC AND SET THEORY strange at first glance, this truth table is universally accepted by mathematicians. To motivate this definition, one can think of P !Qas a promise that Qis true whenever Pis true. When Pis false, the promise is kept by default. This defini-tion allows one to combine many statements together and detect ...
Chapter 7 Canonical Forms - Duke University
pfister.ee.duke.edu2 6= 1. Then, we have 1 2w Hv= wHA Av= wA2v= 2 1 w Hv: We also assume, without loss of generality, that 1 6= 0 . Therefore, if 2 6= 1, then wHv= 0 and the eigenvectors are orthogonal. 7.2 Applications of Eigenvalues 7.2.1 Differential Equations It is well known that the solution of the 1st-order linear differential equation d dt x(t) = ax(t) is ...
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