Frequency Domain and the Fourier Transform

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:

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

  • The minus sign in the exponent is easy to lose

  • The infinite integral is alarming

  • Still, we now can math up what we wanted

• What about going the other way? Turns out that

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

• So the transform is almost self-inverse