Noise in a Nyquist Zone

Along the way

1
Height is \(e_n\). Width is \(B_n\). The blue slab’s height is the density knob. Area \(\propto e_n^2 B_n\) is noise power. RMS is \(e_n\sqrt{B_n}\) — louder wiggles below, even at the same \(B_n\).
2
Zones are \(k\cdot f_s/2\). Zone 1 is \([0,f_s/2]\). Zone 2 is \([f_s/2,f_s]\), and so on. After sampling they all land on zone 1.
3
Who sets \(B_n\)? Amplifier BW, an explicit AA filter (Part 10), or a digital sinc (Part 12). Precision DC often uses a wide quiet analog front end and lets the digital filter define \(B_n\).
4
Fold stacks power into zone 1. Rose is the part of \(B_n\) past \(f_s/2\). After sampling it lands on \([0,f_s/2]\) — taller baseband density \(e_n\sqrt{B_n/(f_s/2)}\). Time plots: left is analog noise that was already in zone 1; right is the sampled sequence (folds included). Same Y (µV). If it folds, the right trace gets louder. The dashed band is \(\pm 3\sigma\) for a long record; one millisecond of noise often never reaches it.
5
Wideband is the opposite trade. If you must keep the wide BW alive, you spend analog filter order. You cannot both keep zone 2 and pretend it will not fold.

1/f (not flat), shaped Σ-Δ noise (Part 13), and clock-correlated interferers. White density is still the first spreadsheet tool.

Next: Part 6 — nonlinearity adds tones, not a noise floor. SFDR vs the ideal SNR you just derived.

Density height, bandwidth width, then fold

Top: analog NSD · blue height = \(e_n\) · width = \(B_n\) · rose = folds · violet = after sampling · Bottom: time-domain RMS

Noise density \(e_n\) (nV/√Hz) Noise bandwidth \(B_n\) (kHz) Sample rate \(f_s\) (kHz)

Analog \(e_n\) (height) \(B_n\) (width) Folds into zone 1 After sampling Zone edge \(k\cdot f_s/2\)

Analog noise density vs frequency · height in nV/√Hz, axis fixed 0–100 (same as the \(e_n\) knob)

Time · same ±100 µV · dashed = ±3σ long-record envelope (\(V_{\mathrm{pp}}\approx 6\,V_{\mathrm{rms}}\)). A 1 ms slice often stays inside that band. Left = analog in zone 1 · right = after sampling

Nyquist \(f_s/2\)—

Analog RMS—

Folded NSD—

Zones spanned—

Fold risk—

\[ v_{\mathrm{rms}} = e_n\sqrt{B_n} \]

= —

\[ f_{\mathrm{Nyq}} = f_s/2 \]

= —

\[ e_{\mathrm{fold}} = e_n\sqrt{B_n / f_{\mathrm{Nyq}}} \quad (B_n > f_{\mathrm{Nyq}}) \]

= —

\[ N_{\mathrm{zones}} = \lceil B_n / f_{\mathrm{Nyq}} \rceil \]

= —

Every zone beyond the first — \([f_s/2,f_s]\), \([f_s,3f_s/2]\), … — maps onto \([0,f_s/2]\) after sampling. A wideband front end with no anti-alias filter (Part 10) lets all of that noise fold in.

Try it: drag \(e_n\) — both time plots get louder (same Y). Then “Wide / folds”: left (analog already in zone 1) stays modest; right (sampled) jumps up — folds packed into \(f_s/2\). “Narrow digital BW” keeps both traces quiet and matched.

Analog Intuition

Density height, bandwidth width, then fold