Sound Is Frequencies

• Most sounds have high periodicity

• Fourier's Theorem (FOO-ree-YAY or thereabouts) says that an infinitely repeating sound can be represented as a sum of sinusoids

• The ear hears/decomposes a sum of sinusoids

• Yet PCM is a sequence of samples over time: frequency is not represented

• The Nyquist Limit is hard to think of as a signal change over time thing

Time and Frequency

• We have: a continuous waveform, a function $$f(t)$$ representing sound pressure

• We want: a continuous spectrum, showing the amplitude and phase of sine waves at every frequency $$\hat{f}(\omega)$$

• Wait, amplitude and phase from a single function? Yes, representing a frequency as a complex number with the usual geometric interpretation

$$f(\omega) = a + b i$$

$$|f(\omega)| = \sqrt{a^2 - b^2}$$

$$\theta(f(\omega)) = tan^{-1}(a, b)$$

• Note: you will see both i and j for $$\sqrt{-1}$$ in different contexts

• Note: we freely mix between angular frequency $$\omega$$ and "normal" frequency f (dammit — we'll be using f as a symbol for both frequency and a generic function) via

$$\omega = 2 \pi f$$

because once around the circle is one cycle

The Euler Formula

• Euler's Formula says complex exponential is a sinusoid:

$$e^{i (\omega t + \theta)} = cos(\omega t + \theta) + i~sin(\omega t + \theta)$$

$$= e^{i \omega t} e^{i \theta}$$

• Starting point for "phasor analysis"

• Now our sum of sinusoids can be represented as a sum of exponentials, making things easier (?)

The Time Domain and the Frequency Domain

• Reminder: Fourier claims that every $$f(t)$$ can be represented as some $$\hat{f}(\omega)$$ (more or less)

• We think of the first kind of thing as "in the time domain", the second as "in the frequency domain"

• Converting from a single frequency to its time domain representation is "easy":

$$f(t) = e^{-i \omega t}$$

• Even for a single sinusoid, converting the other way isn't immediately obvious

Continuous Fourier Decomposition: The Fourier Transform

• Let's just get the Fourier Transform out there:

$$\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt$$

$$f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i \omega t} d\omega$$