With \(\mathrm{V}_{\text {ref }}=2.56 \mathrm{~V}\), find the \(\mathrm{V}_{\text {in }}\) for the following outputs: (a) D9-D0 = 1111111111 (b) D9-D0 \(=1000000001\) (c) D9-D0 \(=1100110000\)

In the realm of electronics, an Analog to Digital Converter (ADC) stands as a bridge between analog and digital worlds. It translates continuously varying analog signals into discrete digital numbers that computers can process. A real-world example might be the conversion of a microphone's analog audio signal into a digital format that your phone or computer can understand and process.

As demonstrated in the textbook exercise, an ADC with a 10-bit resolution can differentiate between \(2^{10}\) or 1024 levels. This allows it to represent an analog voltage with 1024 possible digital values. In simpler terms, the higher the resolution of the ADC, the more detailed the digital representation of the analog signal will be, much like how a higher pixel count leads to a clearer digital image.

Understanding the quantization step is pivotal in comprehending how ADCs work. The quantization step, also known as the Least Significant Bit (LSB) value, is the smallest change in the analog input signal that leads to an alteration in the digital output. Think of it as the smallest 'step' that the ADC can measure.

In practical terms, once the reference voltage is set (in the exercise it's \(2.56 V\)), this voltage is subdivided into equal 'steps' by the ADC's resolution. With our 10-bit ADC, the quantization step is calculated by dividing the reference voltage by 1024 levels, yielding a step of 0.0025 V or 2.5 mV. This value indicates how sensitive the ADC is to changes in the input signal—a smaller quantization step would mean a more sensitive converter, which is desirable in high-precision applications.

Binary to decimal conversion is a fundamental concept in digital electronics, as it allows humans to interpret the binary outputs from digital circuits such as ADCs. Binary numbers are composed of only 0s and 1s, with each '1' representing a power of 2, depending on its position in the sequence. To convert from binary to decimal, you sum the products of each '1' in the binary number by \(2^{position}\), counting the position from right to left starting with zero.

For instance, in the exercise output (b) \(1000000001\), the far-right '1' represents \(2^{0}\) or 1, and the '1' near the left end represents \(2^{9}\) or 512. Summing these gives the decimal equivalent of 513. This process is integral for interpreting the digital output of an ADC into a meaningful analog voltage value.