Analog to Digital and Digital to Analog Converters

Why it matters

ADCs sample and quantize analog signals into numbers. DACs map numbers into analog levels; a reconstruction filter can smooth the resulting steps. Resolution alone does not guarantee accuracy.

Key ideas

An N-bit converter has 2^N codes. Sampling rate is measured in samples per second. Quantization maps a range of inputs to a discrete representation; offset, gain error, nonlinearity and noise add further errors. DAC settling time is the time to enter a specified error band after a code change. DNL measures step-size deviation and INL departure from a reference straight line.

Equations and conventions

For an ideal uniform ADC covering input span V_span, bin width q = V_span/2^N. With midpoint reconstruction or rounding and no overload, quantization error magnitude is at most q/2. Its RMS value is q/√12 only under the usual uniform-error statistical model.

For a common unipolar ideal DAC, V_out = D V_ref/2^N, where D ranges from 0 to 2^N−1. Its maximum output is V_ref(1−2^−N). If a problem instead defines both endpoint codes to match endpoint voltages exactly, its step is (V_max−V_min)/(2^N−1); do not mix that convention with ADC bin width.

Worked example

An ideal 8-bit ADC spans 0 to 3 V, using 256 uniform bins and midpoint reconstruction.

q = 3/256 = 0.01171875 V = 11.71875 mV. Maximum quantization-error magnitude is q/2 = 5.859375 mV, away from overload. RMS error under the uniform-error model is about 3.383 mV.

For V_in = 1 V and floor-coded bins, D = floor(1/q) = 85. That code covers [0.99609375, 1.0078125) V. Its midpoint is 1.001953125 V, giving reconstructed-minus-input error +1.953125 mV, within half an LSB.

A separate 8-bit DAC with V_ref = 3 V and the D/256 convention gives 1.5 V at code 128 and 2.98828125 V at code 255.

Common mistakes

Calling one LSB the maximum rounding error; mixing 255 and 256 conventions; equating more bits with guaranteed accuracy; and ignoring aliasing. A band-limited baseband signal needs a suitable anti-alias filter and sampling above twice its highest frequency, with practical transition margin.

Quick check

1. How many codes does an 8-bit ADC have?
2. What is the ideal midpoint quantization bound?
3. Does doubling the sample rate add one bit of resolution automatically?

Answers: 1. 256. 2. ±q/2 without overload. 3. No.