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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.

For an experimental-noise interpretation of Nyquist's theorem, a resistor $R$ in thermal equilibrium at temperature $T$ has one-sided open-circuit voltage-noise spectral densityin the classical range $hf\ll k_{B}T$, over an approximately rectangular bandwidth $\Delta f$. Across a matched load, half the open-circuit voltage appears, giving available noise power $k_{B}T\Delta f$. This is thermal agitation, present without a mean current, and is distinct from shot noise. A real readout weights the spectrum with its transfer function: $⟨v_{\mathrm{out}}^2⟩={\int }_0^{\infty }{S}_V(f)|H(f){|}^{2}df$. Cooling, reduced bandwidth and suitable impedance/noise matching reduce measured noise; the resistor's noise does not simply disappear in an ideal amplifier model.^[ref]

The bare name also has a sampling interpretation. The Nyquist–Shannon sampling theorem permits exact reconstruction of a bandlimited signal from sufficiently fast uniform samples. The practical condition is $f_s>2f_{\max }$, with an anti-aliasing filter before sampling; otherwise distinct continuous frequencies produce identical samples. Ideal reconstruction uses shifted sinc functions. Critical-rate equality needs an appropriate signal class and can fail for unrestricted edge-frequency sinusoids with unknown phase. Sampling fast enough prevents aliasing, but does not remove thermal noise. This distinction covers the source's unspecified use of “Nyquist's theorem”.^[ref]

Uniform signal sampling at rate $f_s$ multiplies a continuous signal by a time-domain Dirac comb, giving frequency-domain replicas spaced by $f_s$:If a baseband signal is restricted to $|f|\le B$ and $f_s>2B$, these replicas can be separated by an ideal reconstruction filter. Frequencies differing by integer multiples of $f_s$ have the same sampled phase progression; for real signals they fold into the interval from zero to $f_s/2$. This ambiguity is aliasing.

For example, $0.8f_s$ and $0.2f_s$ cosines produce identical sample values. Apply an analogue Anti-aliasing filter before digitization and choose enough sample-rate margin for its finite transition band. A digital filter applied afterward cannot distinguish components already folded onto one another. Out-of-band noise also aliases unless filtered. The sample rate sets temporal/spectral coverage; quantization resolution sets amplitude precision and is a separate limitation. Bandpass Signal sampling has additional conditions beyond the simple baseband rule.^[ref]

The fastest clearly resolved late oscillations occupy roughly $0.004$–$0.005\mathrm{s}$ per cycle when compared with the $0.05\mathrm{s}$ major-tick spacing. Thus $f_{\max }$ is of order $200$–$250\mathrm{Hz}$ and the Nyquist–Shannon sampling theorem requiresA sampling rate of order $500\mathrm{samples}{\mathrm{s}}^{-1}$ is the estimated lower boundary for the plotted signal bandwidth; practical acquisition needs anti-alias filtering and margin. This is not a determination of LIGO's actual acquisition rate.^[ref]

The bit-depth requirements for waveform digitization need a specified precision. The numerical bit depth is underdetermined by the figure. If the digitizer covers a full-scale range $V_{\mathrm{FS}}$ and the required quantization step is at most $\Delta$, thenFor a uniform quantizer its ideal quantization-noise rms is approximately $\Delta /\sqrt{12}$. No permitted quantization error, noise criterion or calibrated digitizer range is specified. The plotted reconstruction band does not itself define one. Therefore no unique finite bit count, or exact lossless digitization of an arbitrary real-valued waveform, follows from the supplied data.

Walt Kester is an electrical engineer whose technical writing concerns data converters and sampled-data electronics. He edited The Data Conversion Handbook edited by Walt Kester, which brings together converter architectures, testing, interfacing and practical circuit design.^[ref]

A useful distinction in this subject is between signal sampling, which selects values at particular times, and quantization, which selects a representable numerical level. An analog-to-digital converter combines both. Kester's explanation of the Nyquist–Shannon sampling theorem connects the sampling rate with the frequencies present at the converter input, making filtering part of the measurement design.^[ref]

Technical publications: www.analog.com/en/resources/technical-books/data-conversion-handbook.html