Analog to Digital Conversion

1. Introduction to Analog-to-Digital Conversion (ADC)

Analog-to-Digital Conversion is the process of converting a continuous analog signal, like the sound of your voice, into a discrete digital signal that a computer can understand and process. This is fundamental to all modern digital audio. The process involves three main steps: Sampling, Quantization, and Encoding.

2. Sampling

Sampling is the first step, where we take snapshots of the analog signal at regular, discrete intervals of time. The rate at which we take these snapshots is called the Sampling Rate (f_s). In this simulation, you control the 'Number of Samples' taken over a fixed duration.

• A higher sampling rate captures the original signal more accurately, preserving higher frequencies.
• The time interval between samples is Ts = 1/fs.
• The sampled signal is represented as xa(nTs), which means the value of the analog signal xa at the time of the n-th sample.

3. Quantization

After sampling, each sample still has a continuous amplitude. Quantization approximates this continuous amplitude to the nearest value from a finite set of discrete levels. The precision of this step is determined by the number of Quantization Bits (N).

• The number of available levels (L) is calculated by: L = 2N.
• For example, with N=4 bits, we have 24 = 16 distinct levels.
• The distance between these levels is the Step Size (Δ), calculated as: Δ = Vrange / L, where V_range is the total possible voltage or amplitude range of the signal.
• Quantization Error: This is the unavoidable difference between the true analog sample value and its rounded, quantized value. More bits (a larger N) lead to a smaller step size and less error, resulting in a higher quality digital signal.

4. Encoding

In the final step, each quantized level is assigned a unique binary code. With N bits, we can represent L = 2^N levels. The simulation shows this binary string for each sample in the 3D view and the digital stream output.

Summary Formula

The entire process can be summarized by the formula you see on the control panel:

xd[n] = Q(xa(nTs))

This means the n-th sample of the final digital signal (xd[n]) is the result of applying the Quantization function (Q) to the sampled analog signal (xa(nTs)).

Bonus: Speed of Sound

The simulation also calculates the speed of sound based on the air temperature you set. The approximate formula used is:

v ≈ 331.3 + 0.606 × T

Where 'v' is the speed in meters per second (m/s) and 'T' is the temperature in degrees Celsius (°C).