Detailed derivation of the Discrete Fourier Transform (DFT) and its associated mathematics, including elementary audio signal processing applications and matlab programming examples.

common dB scale in audio recording is the dBm scale in which the reference power is taken to be a milliwatt (1 mW) dissipated by a 600 Ohm resistor. (See Appendix 4.6 for a primer on resistors, voltage, current, and power.) DBV Scale Another dB scale is the dBV scale which sets 0 dBV to 1 volt. Thus, a 100-volt signal is 100V = 40 dBV 20 log10 1V DRAFT of “Mathematics of the Discrete Fourier Transform (DFT),” by J.O. Smith, CCRMA, Stanford, Winter 2002. The latest draft and linked HTML version

4.6. APPENDIX B: ELECTRICAL ENGINEERING 101 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/. Chapter 5 Sinusoids and Exponentials This chapter provides an introduction to sinusoids, exponentials, complex sinusoids, t60 , in-phase and quadrature sinusoidal components, the analytic signal, positive and negative frequencies,

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/. Page 96 5.5. ACKNOWLEDGEMENT 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/. Chapter 6 Geometric Signal

some scalar c. We can quickly show this for real vectors x ∈ RN , y ∈ RN , as follows: If either x or y is zero, the inequality holds (as equality). Assuming both are nonzero, let’s scale them to unit-length by deﬁning the normalized ∆ ∆ vectors x ˜ = x/ x , y˜ = y/ y , which are unit-length vectors lying on the 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

the cross-correlation of its input and output signals x(n) and y = h ∗ x, respectively. To see this, note that, by the correlation theorem, x y ↔ X · Y = X · (H · X) = H · |X|2 Therefore, the frequency response is given by the input-output crossspectrum divided by the input power spectrum: H= X·Y |X|2 In terms of the cross-spectral density and the input power-spectral density (which can be estimated by averaging X · Y and |X|2 , respectively), this relation can be written as H(ω) = Rxy (ω) Rxx