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Interactive Math: Definition of Laplace Transform
this introduction to Laplace Transform. Reminder: Unit, Ramp and Impulse Functions Unit step function: f(t) = u(t): Ramp function: f(t) = (t): Impulse function `f(t)=δ(t)`: `f(t)=δ(t)` represents an impulse at t = 0 and has value 0 otherwise. We do not...
Interactive Math: Laplace Transform
metric system. The Laplace Transform is widely used in engineering applications (mechanical and electronic), especially where the driving force is discontinuous. It is also used in process control. What Does the Laplace Transform Do? The main idea behind the Laplace Transformati...
Stony Brook: What is a System of Linear Equations?
CSE 373 Course page NIST Dictory of Algorithms and Data Structures World of Mathematics Programming Challenges Graduate Study Opportuinties 1.2.1 Solving Linear Equations INPUT OUTPUT Input Description: An m x n matrix A, and an m x 1 vector b, representing m linear equations w...
Orthogonal Matrices
The proof of the equivalence is nearly identical to the proof of Theorem 2 from the previous section and so we’ll leave it to you to fill in the details. Since it is much easier to verify that the columns/rows of a matrix are orthonormal than it is to...
Stanford Encyclopedia of Philosophy: quantum entanglement and information
Suppose that Alice has an additional photon in an unknown state of polarization |u>, where the notation ‘| >’ denotes a quantum state. It is possible for Alice to perform an operation on the two photons in her possession that will transform...
 Detecting Meter in Recorded Music
serves two functions: to strip the signal of pitch, while still preserving the rough relationships between high and low sounds, and to reduce the amount of data, thus making the DFT and PT computations shorter. Consider an arbitrary signal, s, sampled at 44.1 kHz. To begi...
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serves two functions: to strip the signal of pitch, while still preserving the rough relationships between high and low sounds, and to reduce the amount of data, thus making the DFT and PT computations shorter. Consider an arbitrary signal, s, sampled at 44.1 kHz. To begin, s is stripped to a mono signal, that is, a vector of length n). The algorithm moves along s from beginning to end taking windows—vectors consisting of some £xed number of consecutive entries from s. We overlap our windows so that prominent
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http://www.sju.edu/~rhall/Rhythms/banff.pdf#page=1
www.sju.edu/~rhall/Rhythms/banff.pdf#page=1
Detecting Meter in Recorded Music Joseph E. Flannick, Rachel W. Hall, and Robert Kelly Dept. of Math and C. S. Saint Joseph’s University 5600 City Avenue Philadelphia, PA 19131, USA E-mail: rhall@sju.edu Abstract We use the Discrete Fourier <span class="highlight">Transform</span> and the Periodicity <span class="highlight">Transform</span> (Sethares and Staley, 1999) <span class="highlight">to</span> identify the primary rhythmic content of pieces of popular music. The meter of such pieces is deter- mined by a repeating pattern of accents. In order <span class="highlight">to</span> use the Fourier and Periodicity Transforms
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http://www.sju.edu/~rhall/Rhythms/banff.pdf#page=2
www.sju.edu/~rhall/Rhythms/banff.pdf#page=2
serves two functions: <span class="highlight">to</span> strip the signal of pitch, while still preserving the rough relationships between high and low sounds, and <span class="highlight">to</span> reduce the amount of data, thus making the DFT and PT computations shorter. Consider <span class="highlight">an</span> arbitrary signal, s, sampled at 44.1 kHz. <span class="highlight">To</span> begin, s is stripped <span class="highlight">to</span> a mono signal, that is, a vector of length n). The <span class="highlight">algorithm</span> moves along s <span class="highlight">from</span> beginning <span class="highlight">to</span> end taking windows—vectors consisting of some £xed number of consecutive entries <span class="highlight">from</span> s. We overlap our windows so that prominent
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http://www.sju.edu/~rhall/Rhythms/banff.pdf#page=3
www.sju.edu/~rhall/Rhythms/banff.pdf#page=3
∗, the closest periodic vector <span class="highlight">to</span> x with respect <span class="highlight">to</span> this metric. By subtracting x∗ <span class="highlight">from</span> x, we get a residual vector r. We then search for the closest periodic vector <span class="highlight">to</span> r, subtract that vector <span class="highlight">from</span> r, and the process is repeated. Finally, we have a decomposition of x = x∗ + r∗1 + r∗2 + . . . into periodic vectors. Like the basis elements in the DFT, these periodic vectors give us <span class="highlight">an</span> idea of the relative strengths of periodicities within x. 4.1. The space of p-periodic vectors. Recall that x[k], k ∈ Z is p
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http://www.sju.edu/~rhall/Rhythms/banff.pdf#page=5
www.sju.edu/~rhall/Rhythms/banff.pdf#page=5
the periodic subspaces Pp are not <span class="highlight">orthogonal</span> <span class="highlight">to</span> each other. Therefore, the representation of <span class="highlight">an</span> arbitrary signal s as a linear combination of the basis elements is not unique. Furthermore, there is not a unique order <span class="highlight">to</span> choose projection onto periodic subspaces, since different orders may yield different results. 4.4. Algorithms. At the heart of the PT is its ability <span class="highlight">to</span> choose among these subspaces and determine the most relevant order in which <span class="highlight">to</span> project. Sethares and Staley have proposed the Small-<span class="highlight">to</span>-Large
Interactive Math: Solving Integro-Differential Equ
» Friday math movie - math problem solving with mind maps This week's movie gives you some good tips on how to solve math problems.... more » 9. Solving Integro-Differential Equations An "integro-differential equation" is an equation that involves...
Orthonormal Basis
and yes it does require an inner product space to construct. However, before we do that we’re going to need to get some preliminary topics out of the way first. We’ll first need to get a set of definitions out of way. Definition 1 Suppose that S is...
Convolution Integral
take the inverse transforms of a product of transforms. Fact Let’s work a quick example to see how this can be used. Example 1 Use a convolution integral to find the inverse transform of the following transform. Solution First note that we could use #11 fro...
Cornelius Lanczos
teachers who were to make a great impression on Lanczos. His physics teacher was Eötvös who first interested Lanczos in relativity. His mathematics teacher was Fejér and Lax writes in [2]:- Lanczos was much influenced by Fejér; he learnt from him about Fourier...
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