DSP Calculator

The Nyquist frequency is exactly half of the sampling rate. It represents the absolute maximum frequency that a digital system can represent without distortion. For example, CD audio sampled at 44,100 Hz has a Nyquist frequency of 22,050 Hz, which covers the full human hearing range of roughly 20 Hz to 20,000 Hz. Any frequency above the Nyquist frequency will fold back into the spectrum as aliasing artifacts.

The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a continuous signal from its samples, the sampling rate must be at least twice the highest frequency present in the signal. This minimum rate (2 x fmax) is called the Nyquist rate. Sampling below this rate causes aliasing, where high-frequency components masquerade as lower frequencies in the digital representation.

Oversampling means sampling at a rate higher than the Nyquist rate minimum. It provides three key benefits: it relaxes anti-aliasing filter requirements (allowing a gentler rolloff), it spreads quantization noise across a wider bandwidth (improving effective SNR), and it provides a guard band between the signal bandwidth and the Nyquist frequency. Common oversampling ratios are 2x, 4x, and 8x. Professional audio often uses 96 kHz or 192 kHz sampling for a 20 kHz signal bandwidth.

Each additional bit of depth adds approximately 6.02 dB of dynamic range. A 16-bit system provides about 98 dB of dynamic range (6.02 x 16 + 1.76), which is sufficient for most consumer audio. A 24-bit system provides roughly 146 dB, exceeding the range of human hearing (about 120 dB). The 1.76 dB constant accounts for the quantization noise floor of a uniform quantizer.

Frequency resolution equals the sampling rate divided by the FFT size (fs / N). A larger FFT provides finer frequency resolution but requires more samples and processing time. For example, at 44,100 Hz sampling with a 1,024-point FFT, the resolution is 43.07 Hz. Doubling the FFT to 2,048 points halves the resolution to 21.53 Hz. This creates a fundamental tradeoff between frequency precision and time resolution in spectral analysis.

Aliasing occurs when a signal contains frequency components above the Nyquist frequency (half the sampling rate). These components fold back into the spectrum as false lower frequencies that cannot be distinguished from genuine signals. Prevention requires two steps: first, apply an analog anti-aliasing low-pass filter before the ADC to attenuate frequencies above the Nyquist frequency; second, ensure the sampling rate is at least twice the highest frequency of interest. Oversampling provides additional protection.

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