## Mathematics of the Discrete Fourier Transform (DFT): With Audio Applications"The DFT can be understood as a numerical approximation to the Fourier transform. However, the DFT has its own exact Fourier theory, and that is the focus of this book. The DFT is normally encountered as the Fast Fourier Transform (FFT)--a high-speed algorithm for computing the DFT. The FFT is used extensively in a wide range of digital signal processing applications, including spectrum analysis, high-speed convolution (linear filtering), filter banks, signal detection and estimation, system identification, audio compression (such as MPEG-II AAC), spectral modeling sound synthesis, and many others. In this book, certain topics in digital audio signal processing are introduced as example applications of the DFT"--Back cover. |

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first 5 chapters are great

### Contents

Introduction to the DFT | 1 |

Complex Numbers | 7 |

Proof of Eulers Identity | 19 |

Sinusoids and Exponentials | 31 |

Geometric Signal Theory | 69 |

The DFT Derived | 99 |

Fourier Theorems for the DFT | 115 |

DFT Applications | 165 |

Continuous Fourier Theorems | 213 |

Sampling Theory | 219 |

E Taylor Series Expansions | 231 |

F Logarithms and Decibels | 239 |

G Digital Audio Number Systems | 251 |

H Matrices | 265 |

MatlabOctave Examples | 271 |

291 | |

### Common terms and phrases

algebraic aliasing amplitude Appendix audio axis bandlimited binary bits bytes called coeﬃcients complex numbers complex plane complex sinusoid components compute convolution coordinates corresponds cross-correlation cycles per sample dB scale decimal deﬁned deﬁnition denotes derivation DFT sinusoids diﬀerent diﬀerentiable discrete-time DTFT Euler’s identity example exponential exponents factor FFT algorithm ﬁeld Figure ﬁnd ﬁnite ﬁrst ﬁxed-point Fourier theorems Fourier transform Frequency cycles frequency domain Hann window imaginary impulse response inﬁnitely inner product input signal integer L2 norm length linear combination linear phase linearly logarithmic mathematical Matlab matrix modulation Mth root multiplying negative nonzero norm Normalized Frequency notation Note Octave oﬀ operator orthogonal output periodic plot polynomial projection Proof radians real numbers real signal roots of unity sampled sinusoids sampling rate scalar signal processing signal x(t signiﬁcand spectral magnitude spectrogram Taylor series term unit circle vector space zero zero-padding