ADC / DAC Sampling & Quantization (2D)

Analog signal

Signal frequency f 3.0 Hz

ADC

Sample rate fs 20 Hz

Bit depth 3 bit (8 levels)

Readout

Nyquist rate (2f)

6.0 Hz

Sample rate fs

20.0 Hz

Quantization step

0.250

Ideal SNR (6.02N+1.76)

19.8 dB

RMS quant. error

0.00

Aliasing

None

Analog input x(t) Samples (ADC) DAC output (zero-order hold) Aliased reconstruction

How sampling & quantization work

An ADC does two separate, lossy things to a continuous signal x(t):

sample:    x[k] = x(k / fs)                 — picks discrete instants
quantize:  q[k] = Δ · round(x[k] / Δ)       — Δ = full_scale / 2^N levels
DAC out:   y(t) = q[k]  for  k/fs ≤ t < (k+1)/fs   (zero-order hold)

Quantization (bit depth) rounds each sample's voltage to the nearest of 2^N levels — the gap left over is quantization noise. Every extra bit halves the step and adds ≈6.02 dB of SNR.
Sampling (rate) only works if fs > 2f — the Nyquist criterion. Sample slower than that and the reconstruction can't tell the true frequency from a false, lower one: aliasing.
The dashed red curve is the actual lowest-frequency sinusoid that passes through the very same samples the ADC captured — it is what a DAC + reconstruction filter would have to output, because from the samples alone the two signals are indistinguishable.
Try "force alias": at fs = 1.3×f the reconstructed wave visibly crawls at a much lower frequency than the real input, even though every sample was taken correctly.

Analog x(t) Sample stems DAC (ZOH)

Sampling → quantization → DAC reconstruction

Nyquist OK

Quantization error per sample