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 the analysis window.

• 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.

5. What your real hardware is doing

Everything you see here is driven by genuine samples from your machine's audio codec — the ADC chip on your sound card, laptop board or USB microphone. The panel on the left reports what that chip negotiated with your operating system:

• Codec rate: the hardware sampling frequency, typically 44100 Hz or 48000 Hz. Your browser cannot change it; it is set by the driver.
• Nyquist limit: half the codec rate. Frequencies above this cannot be represented and are removed by an anti-alias filter before the ADC.
• Driver DSP: whether the operating system applied echo cancellation, noise suppression or automatic gain control before handing samples to the browser. This simulation requests all three to be switched off so you see the rawest signal your hardware will give.
• Buffer latency: the block size the audio thread works with, expressed in milliseconds. Real ADCs deliver samples in blocks, not one at a time.
• Capture path: AudioWorklet runs the capture on the real-time audio thread, which is how professional audio software does it.

The browser hands those samples over as 32-bit floating point numbers between -1.0 and +1.0, already converted from the codec's native integer PCM. The 'Quantization Bits' slider then re-quantizes that real signal down to N bits so you can hear and see exactly what precision loss sounds like.

6. The converter chain running in this page

Nothing on this page is a stand-in for conversion. The full signal path is:

1. Transducer & codec ADC: your microphone's diaphragm and the converter chip on your sound card produce integer PCM at the hardware rate. The browser hands those words to the page as float32; the panel shows the original signed 16-bit words in hex.
2. AudioWorklet capture: the samples are collected on the real-time audio thread, in blocks, exactly as professional audio software does it. No synthesis, no interpolation, no stand-in waveform.
3. Anti-alias low-pass: before the rate is reduced, the signal passes a windowed-sinc FIR with a Blackman window, cut off at 90 % of the new Nyquist frequency. This is the filter that sits in front of every real converter. Switch it off and you will hear real aliasing — high frequencies folding down into audible tones.
4. Decimation: the filter is evaluated only at the output instants, which is precisely how a decimating filter is built in silicon — the discarded samples are never computed.
5. N-bit quantizer: each sample is mapped to a real integer code on a mid-rise ladder of 2^N steps. Those integers are the binary strings streaming across the 3D scene and the ones written to the exported files.
6. DAC: to play the result back, the code words are handed to your sound card at their own sample rate, so the hardware's own interpolator performs the reconstruction, the same job a physical DAC's output filter does. Turn the reconstruction filter off and the page feeds a zero-order-hold staircase at the codec rate instead, so you hear the image harmonics a DAC normally removes.

The Export .wav button writes a genuine RIFF/WAVE file at the decimated rate, carrying values snapped to your N-bit grid — open it in Audacity and the quantization steps are physically in the file.

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: Signal to Quantization Noise Ratio

For a full-scale signal the theoretical best case is:

SQNR ≈ 6.02 N + 1.76 dB

The panel shows this theoretical figure alongside the SQNR actually measured from your recorded voice, so you can see how close real speech gets to the ideal.

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).