Question

Find the number of discrete voltages provided by the \(n\)-bit DAC for the follow- \(\begin{array}{lll}\text { (a) } n=8 & \text { (b) } n=10 & \text { (c) } n=12\end{array}\)

Short answer

The number of discrete voltages are: (a) 256, (b) 1024, (c) 4096.

Step-by-step solution

Understand the concept of a digital-to-analog converter (DAC)

An n-bit DAC (Digital-to-Analog Converter) can produce distinct discrete output voltages. The number of discrete voltages provided is determined by the number of bits. For every bit, there are 2 possible levels. Hence, an n-bit DAC can produce 2^n distinct output voltages.

Calculate the number of discrete voltages for an 8-bit DAC

For an 8-bit DAC, the number of discrete voltages provided is 2^8 which is equal to 256.

Calculate the number of discrete voltages for a 10-bit DAC

For a 10-bit DAC, the number of discrete voltages is 2^10, which equals 1024.

Calculate the number of discrete voltages for a 12-bit DAC

For a 12-bit DAC, calculate the number of distinct voltages as 2^12, which equals 4096.

Key concepts

n-bit DAC

The heart of digital-to-analog conversion lies within a device known as an n-bit Digital-to-Analog Converter (DAC). This piece of technology is essential in translating digital signals, which are represented by binary code, into analog voltages that can be used by electronic devices. The 'n' in n-bit DAC specifies the resolution of the converter. That is, it determines the number of different voltage levels that the DAC can produce. The higher the number n, the greater the precision of the analog output.

For instance, in an audio system, an n-bit DAC is responsible for converting the binary data from a digital audio file into the continuous electrical signal needed to drive a loudspeaker and create sound. The resolution of the DAC affects the subtlety and richness of the sound produced. A higher bit depth means the sound can be reproduced with more detail and less chance of introducing quantization error, which can manifest as distortion in the audio output.

Discrete Output Voltages

The concept of discrete output voltages from a DAC revolves around the ability of the converter to produce a set number of distinct voltage levels. These output voltages are spaced equally apart within the range that the DAC operates. This is where the digital precision influences analog quality. Each level corresponds to a possible digital value combination from the input binary code.

To visualize this, imagine a staircase where each step represents a possible output voltage from the DAC. An 8-bit DAC would have 256 steps, each step being a distinct voltage level, while a 12-bit DAC would have a considerably more detailed staircase of 4096 steps, allowing for smoother transitions and finer gradations in the output signal. In any given system, whether it's audio, video or sensor data, these discrete voltages enable the digital information to be reconstructed into a form that closely replicates the original analog waveforms.

Binary Levels

At the core of digital systems are binary levels, which are the two states, often referred to as 0 and 1, that represent the most basic form of data in digital electronics. When it comes to DACs, each binary level corresponds to a distinct voltage output. As the number of bits increases, the number of possible binary level combinations exponentially grows.

Specifically, an n-bit system can represent 2 to the power of n different combinations of binary levels. In other words, for a 1-bit system, there are 2 possible levels (0 or 1), for a 2-bit system, there are 4 possible level combinations (00, 01, 10, or 11), and it continues to double with each additional bit. Hence, with each increase in bit depth, a DAC becomes capable of providing a higher number of discrete voltage steps, leading to a more accurate and refined analog output signal.