# Distortion Introduced by Sampling

July 17, 2019

Back to: Sampling & Reconstruction

In general, sampling introduces three types of distortion due to signal bandwidth, quantization, and digital to analog converter (DAC) interpolation.

### Distortion Due to Signal Bandwidth

In Lesson 2, we learned that perfect reconstruction without distortion is theoretically possible for band-limited signals. However, in the real world, no signal is truly band-limited. If nothing else, there is always thermal noise.

If the sample rate đť–Ą_{s} is not at least the Nyquist rate, then aliasing occurs (see Figure 1.10.) Typically, aliasing is prevented by making the signal band-limited with a low-pass filter, often called an anti-aliasing filter. However, any kind of filteringâ€”no matter useful it isâ€”changes the signal and introduces distortion.

### Distortion Due to Quantization

Quantization is the mapping from a continuous waveform to a discrete quantity. Quantization forces continuous signals with an infinite number of possible values to be represented as discrete-valued sequences with a very finite number of possible values.

For example, a signal in the range of Â±1 volt has an infinite number of possible values; however, when sampled by an 8 bit ADC, it is forced to be one of 28 = 256 possible values (see Figure 1.11.) The difference between the ideal value and the discrete value is, by definition, distortion.

### Distortion due to Digital to Analog Converter (DAC) Interpolation

The DAC must convert a sequence of discrete values back to a continuous voltage or current waveform. From a theoretical standpoint, this waveform is ideally a sequence of weighted â€śinfinitely high and infinitely narrowâ€ť pulses called â€śDirac delta functions.â€ť However, real-world systems can’t support infinite voltage levels so we must settle for more mundane solutions.

In practice, the simple â€śsample-and-holdâ€ť circuitry is often implemented, or more generally by Lagrange interpolation (see Figure 1.12.)

Sample-and-hold is simple because it can be implemented with D-flip-flops. However, the sample-and-hold operation is equivalent to convolution with a rectangle function. As such, it introduces distortion in both the time and frequency domains.