Page iv CONTENTS
3.1.9 Derivatives of f(x)=a
x
............... 32
3.1.10 Back to e ....................... 33
3.1.11 Sidebar on Mathematica . . . . . . . . . ...... 34
3.1.12 Back to e
jθ
...................... 34
3.2 Informal Derivation of Taylor Series . . . . . . ...... 36
3.3 Taylor Series with Remainder ................ 37
3.4 Formal Statement of Taylor’s Theorem . . . . . ...... 39
3.5 Weierstrass Approximation Theorem . . . . . . ...... 40
3.6 Differentiability of Audio Signals . . . . . . . . ...... 40
4 Logarithms, Decibels, and Number Systems 41
4.1 Logarithms . . . ....................... 41
4.1.1 Changing the Base . . ................ 43
4.1.2 Logarithms of Negative and Imaginary Numbers . 43
4.2 Decibels . . . . . ....................... 44
4.2.1 Properties of DB Scales . . . . . . . . . ...... 45
4.2.2 Specific DB Scales . . ................ 46
4.2.3 Dynamic Range . . . . ................ 52
4.3 Linear Number Systems for Digital Audio . . . ...... 53
4.3.1 Pulse Code Modulation (PCM) . . . . . ...... 53
4.3.2 Binary Integer Fixed-Point Numbers . . ...... 53
4.3.3 Fractional Binary Fixed-Point Numbers ...... 58
4.3.4 How Many Bits are Enough for Digital Audio? . . 58
4.3.5 When Do We Have to Swap Bytes? . . . ...... 59
4.4 Logarithmic Number Systems for Audio . . . . ...... 61
4.4.1 Floating-Point Numbers . . . . . . . . . ...... 61
4.4.2 Logarithmic Fixed-Point Numbers . . . ...... 63
4.4.3 Mu-Law Companding ................ 64
4.5 Appendix A: Round-Off Error Variance . . . . ...... 65
4.6 Appendix B: Electrical Engineering 101 . . . . ...... 66
5 Sinusoids and Exponentials 69
5.1 Sinusoids . . . . ....................... 69
5.1.1 Example Sinusoids . . ................ 70
5.1.2 Why Sinusoids are Important . . . . . . ...... 71
5.1.3 In-Phase and Quadrature Sinusoidal Components . 72
5.1.4 Sinusoids at the Same Frequency . . . . ...... 73
5.1.5 Constructive and Destructive Interference . . . . . 74
5.2 Exponentials . . ....................... 76
DRAFT of “Mathematics of the Discrete Fourier Transform (DFT),” by J.O.
Smith, CCRMA, Stanford, Winter 2002. The latest draft and linked HTML
version are available on-line at http://www-ccrma.stanford.edu/~jos/mdft/.
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