Intuition The one core ideaA computer can only store numbers taken now and then, rounded to a fixed set of allowed values — but the real world is a smooth, continuous wiggle. This whole topic is about the two rulers we use to copy that wiggle into numbers (how finely in height , how often in time ) and the one law (Nyquist) that says how often is often enough to not be fooled.
Before you can read the parent note ADC/DAC topic , you need every squiggle it throws at you to already mean something. Below, each symbol is built from nothing, given a picture, and justified.
A quantity is continuous if it can take any value with no gaps — between 1.0 V and 1.1 V there are infinitely many voltages (1.05, 1.0500001, …). Time is continuous; a real voltage is continuous.
A quantity is discrete if it can only take values from a separated list — like the whole numbers 0 , 1 , 2 , 3
WHY the topic needs this. The entire job of an ADC is to cross the bridge from the left picture (smooth line) to the right picture (dots on a grid). Every other symbol below is a tool for measuring where on the grid a smooth value lands.
V Voltage is the "height" of an electrical signal — how strongly it pushes, measured in volts (V) . Think of it as the up/down position of the wiggling line in the picture above.
Definition Reference voltage,V r e f V r e f biggest voltage the converter is allowed to see — the top of its measuring cup. A common value is 3.3 V 0 V r e f
V r e f height of a ruler stood up next to the signal. The signal is a mark somewhere on that ruler; the ADC's job is to read off which tick the mark is nearest.
WHY the topic needs it. You cannot say a voltage is "half" without knowing "half of what ". V r e f
A bit is one yes/no switch — it holds 0 1
N N how many of those switches the converter uses to describe one reading. A "12-bit ADC" uses N = 12
Worked example Feel the doubling
N = 8 ⇒ 2 8 = 256 N = 10 ⇒ 2 10 = 1024 N = 12 ⇒ 2 12 = 4096 doubles the number of height-ticks the ruler has.
WHY the exponent, not multiplication? Because switches combine by choices multiplying , not adding. This is exactly why one more bit doubles quality — and later, why one more bit adds a fixed ~6 dB of SNR.
A code is the actual integer the ADC outputs — one of the 2 N 0 2 N − 1 "the signal was closest to tick number [code]" .
Definition LSB — Least Significant Bit (step size)The LSB is the voltage gap between two neighbouring ticks — the smallest change the ADC can notice. It is the "1
Common mistake Off-by-one: gaps vs ticksWhy it feels right: "2 N 2 N The fix: 2 N dots leave only 2 N − 1 spaces between them . Dividing by 2 N 2 N − 1 ≈ 2 N N
WHY the topic needs it. LSB is the ruler's finest tick . It sets how truthfully we copy the signal's height, and it is the width q
Rounding means snapping a value to the nearest allowed tick. If a voltage sits between two ticks, it jumps to whichever is closer.
Intuition Why the error is at most half a stepIf you always jump to the nearest tick, the worst you can be off is when you sit exactly halfway — that is 2 1 − 2 1 LSB + 2 1 LSB ∣ ⋯ ∣
WHY the topic needs it. This unavoidable "snap" is the source of quantization error — the built-in tiny lie every ADC tells, which the parent turns into the 6.02 N + 1.76
t T s t seconds (s) . The sampling period T s gap in time between two measurements — "grab a value, wait T s
Definition Sampling rate,f s f Frequency f how many full up-and-down cycles happen per second , in hertz (Hz) . The sampling rate f s how many samples we grab per second , also in Hz.
WHY the topic needs it. Resolution slices height ; sampling slices time . f s f
Definition Sine / cosine waveA sine (or cosine) is the smoothest possible repeating wiggle — the shape of a pure tone, a swinging pendulum, an AC voltage. We write it cos ( 2 π f t )
2 π One full cycle of the wave corresponds to the angle 2 π cos ( 2 π f t ) f f
repeat identically every 2 π 2 π 2 π nothing you can measure. Hold this thought: it is the entire reason a fast wave can disguise itself as a slow one — the heart of aliasing.
WHY the topic needs it. Nyquist and aliasing are statements about sine waves. "Two samples per period" and "f f + k f s 2 π
Definition round(x) and |x|
round ( x ) nearest whole number to x round ( 1.8 ) = 2 ∣ x ∣ absolute value — the size of x ∣ − 200∣ = 200
Intuition Why the parent's alias formula reads the way it doesf a l ia s = f − f s ⋅ round ( f / f s ) "take your true frequency, subtract off the nearest whole multiple of the sampling rate, and report the size of what's left." That leftover is the fake low frequency you actually hear. You now have every piece: subtraction, multiplication, round, and absolute value.
A decibel is a ratio squished onto a friendlier scale used for "how much bigger is signal than noise". You don't need its full definition here — just know bigger dB = cleaner signal , and the parent's rule "≈ 6 dB per bit" means each extra switch makes the copy noticeably cleaner .
SNR = Signal-to-Noise Ratio = how tall the real signal is compared with the background fuzz . High SNR = crisp; low SNR = hissy.
WHY the topic needs it. It turns "more bits" into a concrete, measurable improvement (6.02 N + 1.76
Quantization noise and SNR
Can you say, in one sentence, the difference between a continuous and a discrete quantity? Continuous can be any value with no gaps; discrete only comes from a separated list of allowed values.
Do you know what V r e f It's the full-scale voltage (top of the ruler); every step is a fraction of it, so "half" only means something relative to V r e f
Can you explain why N 2 N Each bit is a doubling switch, so N 2 × 2 ⋯ = 2 N
Do you know why we divide by 2 N − 1 2 N 2 N 2 N − 1
Can you state the largest possible rounding error and why? ± 2 1
Do you know the relationship between f s T s They are reciprocals: f s = 1/ T s
Can you explain why a wave repeats identically every 2 π Cosine returns to the same value after 2 π 2 π
Do you know what round ( x ) ∣ x ∣ round ∣ x ∣
Can you read "6 dB per bit" in plain words? Each extra bit roughly halves the noise step, making the digital copy measurably cleaner.
