By Julius O. Smith III

Targeted derivation of the Discrete Fourier rework (DFT) and its linked arithmetic, together with straight forward audio sign processing functions and matlab programming examples.

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Extra resources for Mathematics of the Discrete Fourier Transform (DFT): with Audio Applications (2nd Edition)

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It can easily be computed in Matlab as e=exp(1). O. Smith, CCRMA, Stanford, Winter 2002. edu/~jos/mdft/. CHAPTER 2. O. Smith, CCRMA, Stanford, Winter 2002. edu/~jos/mdft/. 6. 1000i Another thing to note about Matlab is that the transpose operator ’ (for vectors and matrices) conjugates as well as transposes. O. Smith, CCRMA, Stanford, Winter 2002. edu/~jos/mdft/. 0000i CHAPTER 2. 7 Numerical Tools in Mathematica In Mathematica, we can ﬁnd the roots of simple polynomials in closed form, while larger polynomials can be factored numerically in either Matlab or Mathematica.

Log(x) = log10 (x). This is the common logarithm. 1. LOGARITHMS (The use of lg() for base 2 logarithms is common in computer science. ) By far the most common bases are 10, e, and 2. Logs base e are called natural logarithms. ) The inverse of the natural logarithm y = ln(x) is of course the exponential function x = ey , and ey is its own derivative. In general, a logarithm y has an integer part and a fractional part. The integer part is called the characteristic of the logarithm, and the fractional part is called the mantissa.

The kth sinusoid is given by ∆ WNkn = ej2πkn/N = ejωk tn = cos(ωk tn )+j sin(ωk tn ), ∆ n = 0, 1, 2, . . , N −1 ∆ where ωk = 2πk/N T , tn = nT , and T is the sampling interval in seconds. 6 Real Exponents The closest we can actually get to most real numbers is to compute a rational number that is as close as we need. It can be shown that rational numbers are dense in the real numbers; that is, between every two real numbers there is a rational number, and between every two rational numbers is a real number.