Why Digitize — Data Conversion

Along the way

1
Time first, amplitude later. Ideal sampling is just instants: \(x[n]=x_c(nT_s)\). Quantization — rounding each voltage to a code — is a separate knob (Part 3). Real ADCs do both; we split them so each demo has one meaning.
2
Why digitize at all. Bits copy and compute without analog drift. Filters, FFTs, and control laws become software. The front end still ages — you just moved the analog problem to the pin.
3
Samples per cycle. \(N_{\mathrm{cyc}}=f_s/f_{\mathrm{in}}\). Two per period is the Nyquist edge for a sine. In practice you want headroom for filters, jitter, and anything that is not a sine.
4
ZOH is playback, not what the ADC saw. Amber stairs are a common DAC reconstruction. The converter stored points. Jagged hold ≠ “the ADC failed” — reconstruction quality is a different question (Part 2).
5
Watch the Nyquist readout. When \(f_{\mathrm{in}}\) crosses \(f_s/2\) the dots can still look plausible. Part 2 explains why that case is dangerous.

No aperture jitter, track-and-hold bandwidth, quantization, or anti-alias filter on this page.

Next: Part 2 — the same dots can come from two different continuous tones (aliasing).

Continuous tone → sample train

Gray = analog \(x_c(t)\) · blue stems = samples · amber = zero-order hold (what a DAC often plays back)

Analog \(x_c(t)\) Samples \(x[n]\) ZOH playback

\(f_{\mathrm{in}}\) (Hz) \(f_s\) (Hz) Show ZOH

Sample period T_s —

Samples / cycle —

Nyquist f_s/2 —

f_in vs Nyquist —

\[ T_s = 1/f_s \]

= —

\[ x[n] = x_c(n\,T_s) \]

→ sequence of voltages

\[ N_{\mathrm{cyc}} = f_s / f_{\mathrm{in}} \]

= —

Ideal impulse sampling (no aperture window, no quantization yet). Part 2 treats aliases; Part 3 adds amplitude stairs.

Try it: drop \(f_s\) toward \(2f_{\mathrm{in}}\) — the blue dots still land on the gray curve, but the amber hold looks jagged. The sequence still “knows” the tone; reconstruction quality is a different question (Shannon / Part 2).

Analog Intuition

Why digitize — sampling in time

Continuous tone → sample train