1.5 Summary

• Physical sound is a traveling pattern of air-pressure variation. Analog sound is a continuous signal \(x(t) : \mathbb{R} \to \mathbb{R}\) describing the time-varying pressure measured at a single point.

• Amplitude is, by convention, a unitless quantity in \([-1, 1]\), proportional to the underlying pressure: \(x(t) = p(t) / p_{\max}\).

• Analog-to-digital conversion (ADC) discretizes time and amplitude: sample at rate \(f_s\), then quantize each amplitude to a \(b\)-bit signed integer in \(\mathbb{Z}_b\) via \(\hat{x}[n] = \lfloor (2^{b-1} - 1) \cdot x[n] \rfloor\).

• The bitrate \(f_s \cdot b\) tells you how much disk space uncompressed audio takes (CD-quality mono is about \(88 \frac{\text{kilobytes}}{\text{seconds}}\)).

• The discrete representation \(x[n] = x(n / f_s)\) is what computers manipulate; we use parentheses for continuous time, square brackets for sample indices. In memory we use floats for arithmetic convenience; quantization shows up at the storage boundary.

• A DAC reconstructs an analog signal by smoothing the discrete samples back into a continuous voltage; under conditions we will study later, this reconstruction can be made perceptually indistinguishable from the original.

• Be wary of values outside \([-1, 1]\), which will clip. Keep headphones off until your output is bounded.