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Digital signal processing transforms or analyzes signals represented by numerical sequences. A physical input commonly passes through signal sampling and quantization before an algorithm filters or transforms it. Sampling chooses times; quantization chooses a finite set of representable values. A discrete-time signal is a sequence indexed by integers, whereas a digital implementation also has finite precision.^[ref]

An elementary digital filter averages adjacent samples:For a constant sequence it preserves the value. For an alternating sequence $x[n]=(-1{)}^n$ it gives zero. If the input noise samples are independent with variance ${\sigma }^2$, the output noise variance is ${\sigma }^2/2$, but neighboring output samples are now correlated because they share an input sample. Smoothing changes both the signal and its noise statistics.

Sampling can also erase distinctions before any algorithm runs. At sampling frequency $1000\mathrm{Hz}$,These distinct analog frequencies produce the same samples. An anti-aliasing filter limits unwanted input frequencies before conversion; later computation cannot uniquely recover which of these unrestricted inputs produced the sequence.^[ref]

Fourier analysis and the discrete Fourier transform help reveal frequency content, while finite-word arithmetic introduces an additional error budget. The sampling rate, numerical representation and intended physical bandwidth are therefore part of the processing design.

Julius O. Smith III works on applications of digital signal processing to music and audio. Stanford identifies him as a professor emeritus of music and, by courtesy, electrical engineering, based at its Center for Computer Research in Music and Acoustics. His doctorate in electrical engineering was completed at Stanford in 1983.^[ref]

His Spectral Audio Signal Processing by Julius O. Smith III explains how to extract a spectral envelope rather than treating every narrow harmonic peak as part of the slowly varying sound shape. One method uses a cepstrum: transform a logarithmic magnitude spectrum, retain an appropriate low-quefrency region, and transform back. This separates broad spectral shape from more rapidly oscillating harmonic detail under suitable assumptions.^[ref]

For example, two played notes can have different harmonic spacings while sharing a similar broad envelope associated with the instrument's resonances. Distinguishing spacing from envelope helps explain why changing a note's pitch and changing its timbre are different signal-processing tasks. The separation is approximate: window choice and closely spaced spectral features affect what the method preserves.