Analog, Discrete & Digital Signals

“Analog,” “sampled,” and “digital” describe different properties. A continuously varying voltage can encode a continuous quantity; selecting its value at discrete times creates a sequence; mapping each selected value into one of finitely many codes makes that sequence digital. Keeping those operations separate makes their error mechanisms obvious.

Three representations #

A waveform can also be continuous in time but discrete in allowed level, as at a digital logic pin. Conversely, a software floating-point sequence is discrete in time and has a very large but still finite representable alphabet.

flowchart TB
  PHY["physical quantity"] --> ANA["analog signal x(t)"]
  ANA -->|"sample at nT"| DISC["discrete-time x[n]"]
  DISC -->|"quantize + encode"| DIG["digital code c[n]"]
  DIG -->|"decode + hold"| STAIRS["piecewise analog output"]
  STAIRS -->|"reconstruction filter"| REC["reconstructed waveform"]

Sampling changes the time description #

Sampling records values at $t=nT$, with sample rate $f_s=1/T$. It does not inherently round amplitude. The sampling theorem and aliasing concern whether spectral replicas overlap; aliasing is a time-discretization failure that no later amplitude precision can repair.

Quantization changes the value description #

A quantizer partitions an input range into decision intervals and assigns one output code to each. Values inside one interval become indistinguishable. For a uniform $N$-bit converter with range $V_{FS}$, the nominal step is $\Delta=V_{FS}/2^N$. Increasing sample rate does not shrink $\Delta$; increasing bit depth does not raise the Nyquist frequency.

Encoding assigns bit patterns to quantized levels. Binary, two's-complement, offset-binary, thermometer, Gray, and serialized line codes can represent the same underlying values with different transition and interface properties.

A complete signal chain #

Real acquisition also includes a sensor transfer function, analog bandwidth, anti-alias filter, sample-and-hold settling, reference, clock, noise, distortion, common-mode limits, and digital framing. Digital output is not automatically accurate or noise-free; it is simply represented by codes robust enough to regenerate logic levels within an electrical contract.

On reconstruction, a DAC emits levels or pulses corresponding to codes. A zero-order hold produces spectral images and sinc-shaped droop; an analog reconstruction filter removes out-of-band images. It cannot recreate information lost to clipping, aliasing, or quantization.

Choosing a representation #

Specify the signal's bandwidth, dynamic range, acceptable total error, latency, time alignment, and physical interface. Then allocate error separately to sampling, clock, quantization, analog front end, conversion, numerical processing, and reconstruction. Using “digital” as a blanket synonym for exact hides every one of those budgets.