7.6 Summary

• Sampling can be viewed as multiplying a signal by an impulse train. In the frequency domain, this replicates the signal’s spectrum at every integer multiple of \(f_s\).

• Perfect reconstruction is possible whenever we can isolate the original baseband spectrum from its copies. The Nyquist-Shannon sampling theorem guarantees this when \(f_s > 2 f_{\max}\). The frequency \(f_s/2\) is the Nyquist frequency.

• When \(f_s \le 2 f_{\max}\), the spectral copies overlap and high frequencies alias to lower ones. The apparent frequency is \(f_{\text{alias}} = \min(f \bmod f_s,\, f_s - (f \bmod f_s))\). Aliasing is audible, and it appears throughout nature (the wagon-wheel effect, strobe lights).

• For audio, human hearing tops out near 20 kHz, so \(f_s > 40\) kHz suffices. Standard rates of 44.1 and 48 kHz add headroom for a real anti-aliasing filter, which must remove content above the Nyquist frequency before sampling.

• Quantization, unlike sampling, is lossy: it adds quantization noise. The decibel, \(20\log_{10}(a/a_0)\), is the logarithmic unit for amplitude. Each additional bit halves the noise, worth about 6 dB of dynamic range, so 16 bits (\(\approx\) 96 dB) covers the roughly 100 dB range of human hearing.

• Resampling reads a signal at interpolated positions to change its sample rate (\(M = N f_s^2/f_s^1\)). Downsampling requires anti-alias filtering first.