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How to Calculate Sampling Rate
Understanding how to calculate sampling rate is fundamental to digital signal processing (DSP) and audio engineering. The sampling rate determines how frequently an analog signal is measured to convert it into a digital format. This calculator helps you determine the minimum requirements based on the Nyquist-Shannon sampling theorem and estimates the resulting file size for uncompressed audio.
The Nyquist-Shannon Sampling Theorem
The core principle behind calculating the minimum necessary sampling rate is the Nyquist-Shannon sampling theorem. It states that to perfectly reconstruct a continuous signal from its samples, the sampling rate ($f_s$) must be strictly greater than twice the highest frequency component ($f_{max}$) present in the signal.
Minimum Sampling Rate > 2 × fmax
This threshold is known as the Nyquist Rate. If you sample at a rate lower than this, a phenomenon called aliasing occurs, where high-frequency signals become indistinguishable from lower-frequency signals, causing distortion that cannot be removed later.
Example Calculation
The limit of human hearing is generally considered to be 20,000 Hz (20 kHz). To capture the full range of human hearing without aliasing, we calculate:
- Max Frequency ($f_{max}$): 20,000 Hz
- Nyquist Calculation: 20,000 Hz × 2 = 40,000 Hz
This is why the standard Audio CD sampling rate is 44,100 Hz. It provides a small safety margin (guard band) above the theoretical minimum of 40,000 Hz to allow for practical anti-aliasing filters.
Calculating Data Rate and File Size
Once you have established your sampling rate, you can calculate the data rate (bit rate) and the total file size. This is crucial for determining storage requirements for recording equipment or bandwidth for streaming.
The Formula
To find the uncompressed data rate in bits per second (bps):
Bit Rate = Sampling Rate × Bit Depth × Channels
To find the total file size in bytes:
Total Size (Bytes) = (Bit Rate × Duration in Seconds) / 8
Variables Defined
- Sampling Rate ($f_s$): The number of samples per second (e.g., 44,100 Hz).
- Bit Depth: The number of bits of information in each sample (e.g., 16-bit, 24-bit). Higher bit depth increases dynamic range (signal-to-noise ratio).
- Channels: The number of audio tracks (e.g., 1 for Mono, 2 for Stereo).
Why Oversampling is Used
While the calculation $2 \times f_{max}$ gives the theoretical minimum, engineers often use oversampling. Sampling significantly above the Nyquist rate relaxes the design requirements for the analog anti-aliasing filters and can improve the resolution of the signal through digital filtering techniques.
Common Sampling Rates
- 44.1 kHz: Standard CD Audio.
- 48 kHz: Standard for Video (DVD, Blu-ray, TV).
- 88.2 kHz / 96 kHz: High-Resolution Audio formats.
- 192 kHz: Professional mastering and archival quality.
function calculateSampling() {
// 1. Get Input Values
var maxFreqInput = document.getElementById('maxFrequency').value;
var sampleRateInput = document.getElementById('samplingRate').value;
var bitDepthInput = document.getElementById('bitDepth').value;
var channelsInput = document.getElementById('channels').value;
var durationInput = document.getElementById('duration').value;
// 2. Parse Values to Numbers
var fMax = parseFloat(maxFreqInput);
var fS = parseFloat(sampleRateInput);
var bitDepth = parseInt(bitDepthInput);
var channels = parseInt(channelsInput);
var durationMinutes = parseFloat(durationInput);
// 3. Flag for displaying results container
var showResults = false;
// 4. Calculate Nyquist Rate (if Max Freq is provided)
if (!isNaN(fMax) && fMax > 0) {
var nyquistRate = fMax * 2;
document.getElementById('resNyquist').innerHTML = nyquistRate.toLocaleString() + " Hz";
// If user hasn't typed a sampling rate yet, suggest one?
// For this logic, we just calculate Nyquist.
showResults = true;
} else {
document.getElementById('resNyquist').innerHTML = "Enter Max Frequency";
}
// 5. Calculate File Size & Bit Rate (if Sampling Rate & Duration are provided)
if (!isNaN(fS) && !isNaN(bitDepth) && !isNaN(channels) && !isNaN(durationMinutes)) {
// Bit Rate Calculation (bits per second)
var bitRateBps = fS * bitDepth * channels;
var bitRateKbps = bitRateBps / 1000;
// Total Duration in Seconds
var durationSeconds = durationMinutes * 60;
// Total Bits
var totalBits = bitRateBps * durationSeconds;
// Total Bytes
var totalBytes = totalBits / 8;
// Convert to MB (Megabytes) – utilizing 1 MB = 1024 * 1024 Bytes (Binary)
// Or 1 MB = 1,000,000 Bytes (Decimal). Storage usually decimal, Memory binary.
// We will use Binary MB (MiB) as it's standard for file systems usually.
var sizeMB = totalBytes / (1024 * 1024);
// Total Samples
var totalSamples = fS * durationSeconds * channels;
// Update DOM
document.getElementById('resBitRate').innerHTML = Math.round(bitRateKbps).toLocaleString() + " kbps";
document.getElementById('resFileSize').innerHTML = sizeMB.toFixed(2) + " MB";
document.getElementById('resTotalSamples').innerHTML = Math.round(totalSamples).toLocaleString();
showResults = true;
} else {
document.getElementById('resBitRate').innerHTML = "-";
document.getElementById('resFileSize').innerHTML = "-";
document.getElementById('resTotalSamples').innerHTML = "-";
}
// 6. Show the box if any calculation occurred
if (showResults) {
document.getElementById('results').style.display = 'block';
}
}