ADC - DAC — resolution, sampling rate, Nyquist: Question bank

Before we start, every symbol this page reuses, defined in plain words:

resolution = how finely the amplitude (voltage height) is sliced → measured in bits ( $N$ ).
sampling rate $f_{s}$ = how often in time we grab a value → measured in Hz.
$f_{ma x}$ = the highest frequency component present in the signal — the fastest wiggle hiding inside whatever you feed the ADC. Nyquist is stated against this number.
$V_{r e f}$ = the converter's full-scale reference voltage — the top of the range it can read (e.g. 3.3 V). Any input from 0 up to $V_{r e f}$ maps to a code.
LSB (Least-Significant-Bit step) = the smallest voltage step the converter can distinguish, one rung of the ladder: $LSB = V_{r e f} / (2^{N} - 1) \approx V_{r e f} / 2^{N}$ .
$q$ = the quantization step size, which is just one LSB (the width of one rounding "bin") — used in the noise formula $σ_{q}^{2} = q^{2} /12$ .
aliasing = a fast wave, under-sampled, disguising itself as a slow one.

True or false — justify

Sampling at exactly

f_{s} = 2 f_{ma x}

always lets you reconstruct the wave.

False. At exactly twice the frequency you can keep landing on the zero crossings and read

0, 0, 0, \dots

, so amplitude and phase are lost. The criterion is strict:

f_{s} > 2 f_{ma x}

A 16-bit ADC will always give a cleaner recording than a 12-bit one.

False. More bits only improve amplitude precision. If your

f_{s}

under-samples the signal, the extra bits faithfully record the wrong (aliased) waveform — cleaner garbage.

Doubling the number of bits doubles the SNR in dB.

False. The point is per-bit: adding one bit (which ×2 the number of levels) adds ~6 dB, from

6.02 N + 1.76

. So SNR grows linearly with the bit count

N

. If you literally double

N

from 8 to 16 you add eight bits →

\approx 48

dB, not a factor of two. See Quantization Noise & SNR for the full derivation.

An anti-aliasing filter goes after the ADC.

False. It must sit before the ADC (in the analog domain). Once a frequency has aliased during sampling it is already tangled with real low-frequency content and cannot be untangled digitally — a low-pass filter blocks the high frequencies while they are still analog. See Anti-Aliasing Filters.

Nyquist guarantees reconstruction for any signal if

f_{s}

is high enough.

False. It requires the signal be band-limited — no energy above

f_{ma x}

. A signal with unbounded frequency content (e.g. a perfect square wave) has no finite

f_{ma x}

and can never be sampled without aliasing.

The Nyquist frequency

f_{s} /2

is a property of the signal.

False. It is a property of the sampler — it depends only on

f_{s}

. The signal has an

f_{ma x}

; the sampler has an

f_{s} /2

. Aliasing happens when the signal's

f_{ma x}

exceeds the sampler's

f_{s} /2

Quantization error is truly random noise.

Mostly false — it's a useful model. The error is deterministic (it depends on the input value), but for busy, large signals it behaves like uniform noise of width

q

(one LSB), which is why

σ_{q}^{2} = q^{2} /12

works. For a slow DC input it is a fixed offset, not noise.

A DAC reproduces the exact voltage that entered the original ADC.

False. It reproduces

code \times LSB

, which is the quantized value — rounded to the nearest ladder rung. Everything finer than one LSB was thrown away and cannot return.

Using

LSB = V_{r e f} / 2^{N}

instead of

V_{r e f} / (2^{N} - 1)

is a mistake.

Half-true. It is an approximation: there are

2^{N} - 1

gaps between

2^{N}

codes, so the exact step is

V_{r e f} / (2^{N} - 1)

. The

/ 2^{N}

form is common and its error is negligible for large

N

— acceptable, not "exactly correct".

Spot the error

"To capture a 20 kHz tone I sample at 20 kHz."

Error: you need

f_{s} > 2 \times 20 = 40

kHz (here

f_{ma x} = 20

kHz). Sampling at the signal frequency gives one point per cycle — a constant reading, no oscillation seen at all.

"My signal only has energy up to 5 kHz, so I don't need an anti-alias filter at

f_{s} = 12

kHz."

Error: noise and interference (RF pickup, EMI, thermal) live above 5 kHz too. Without a filter, that out-of-band junk aliases down into your 0–6 kHz band. Band-limiting must be enforced, not assumed.

"CD audio uses 44.1 kHz because humans hear up to 44.1 kHz."

Error: humans hear to ~20 kHz, so

f_{ma x} \approx 20

kHz, needing

f_{s} > 40

kHz. The extra 4.1 kHz is filter margin — room for the real anti-alias filter's gradual roll-off, since no filter is a perfect brick wall.

"The reconstruction filter's job is to add back the bits we lost."

Error: it removes the staircase's high-frequency images (from the zero-order hold — the flat steps in the figure above), smoothing steps into a curve. It cannot restore quantization detail — that information is gone at the ADC. See Zero-Order Hold & Reconstruction.

"Increasing

V_{r e f}

improves resolution."

Error: raising

V_{r e f}

widens the full-scale span while keeping

2^{N}

codes, so each LSB step

V_{r e f} / (2^{N} - 1)

gets bigger — resolution gets worse. Lowering

V_{r e f}

(to match your signal's true range) shrinks the step.

"Aliasing adds a new high frequency to my recording."

Error: aliasing folds high frequencies down to a lower apparent frequency below

f_{s} /2

. A 1800 Hz tone at

f_{s} = 1000

Hz appears as 200 Hz — a fake low tone, not a new high one.

Why questions

Why exactly two samples per period, not three or four?

Two points (one per hump) are the theoretical minimum to fix a sinusoid's frequency and amplitude. Fewer leaves an oscillation ambiguous; two is the mathematical floor Nyquist proves is sufficient (in the strict

>

sense).

Why does sampling make frequencies

f

and

f + k f_{s}

indistinguishable?

Sampled values are

cos (2 π f n / f_{s})

; adding

k f_{s}

inside gives

cos (2 π f n / f_{s} + 2 π k n)

, and cosine is

2 π

-periodic so the extra term vanishes. Identical samples ⇒ the sampler literally cannot tell them apart. See Fourier Transform & Frequency Domain.

Why is the quantization variance

q^{2} /12

and not, say,

q^{2} /4

Because

12

is the variance denominator of a uniform distribution: for a flat spread over width

q

(one LSB), variance

= (b - a)^{2} /12

. It reflects the spread of the rounding error, not its maximum (

\pm q /2

Why does each extra bit add ~6 dB of SNR, not some other number?

An extra bit halves the step

q

, so noise power

q^{2} /12

drops to one quarter. In decibels a quarter-power is

10 lo g_{10} (4) \approx 6.02

dB — the constant in

6.02 N + 1.76

Why can't a purely digital algorithm remove aliasing after sampling?

Once folded, the aliased frequency occupies the same bin as legitimate low-frequency signal — they are algebraically summed and indistinguishable. No digital operation can separate a sum back into unknown parts.

Why does the DAC produce a staircase rather than a smooth curve on its own?

A basic DAC holds each code's voltage constant until the next sample arrives — zero-order hold (the amber staircase in the figure). Holding a level for

T_{s}

makes flat steps, whose sharp edges are exactly the high-frequency images the smoothing filter must erase.

Why does a filter with only "cut-off ≈

f_{s} /2

" leave margin for design?

Real filters roll off gradually, not instantly. Placing the highest signal well below

f_{s} /2

(as CD does) gives the transition band physical room, so unwanted energy is genuinely attenuated by the Nyquist point. See Anti-Aliasing Filters.

Edge cases

What happens if a frequency sits exactly at

f_{s} /2

You get two samples per cycle, but if they land on the zero crossings the reconstructed amplitude reads zero — the boundary is unreliable, which is why the criterion excludes equality.

What does the ADC output for an input below 0 V or above

V_{r e f}

It saturates (clips) to the lowest code (0) or the highest code (

2^{N} - 1

). Everything out of range collapses to the rail — a hard, non-recoverable distortion, unlike quantization.

What is the alias of a DC signal (0 Hz)?

DC never aliases — it is already below every

f_{s} /2 > 0

, and

f = 0

trivially satisfies

f_{s} > 2 f

. It samples to a constant, exactly as intended.

For a perfectly noiseless, constant DC input, what is the "quantization noise"?

It's a fixed offset, not noise — the input rounds to one nearest code every time. The

q^{2} /12

noise model breaks down because there's no variation to average out.

What if

f_{ma x} = 0

(a signal with no time variation)?

Any

f_{s} > 0

satisfies Nyquist since

f_{s} > 2 \cdot 0

always holds. A truly static signal can be sampled arbitrarily slowly with no aliasing.

f_{s}

is extremely high (massive over-sampling), is resolution automatically better?

No — resolution is set by bits, independent of

f_{s}

. But over-sampling can be traded for effective bits by averaging (the principle behind sigma-delta converters). See Successive Approximation vs Sigma-Delta ADC.

Connections

ADC - DAC — resolution, sampling rate, Nyquist (index 5.5.4)
Quantization Noise & SNR
Anti-Aliasing Filters
Zero-Order Hold & Reconstruction
Fourier Transform & Frequency Domain
Successive Approximation vs Sigma-Delta ADC