Nyquist Frequency Calculator
Frequency (Hz)
22050
How it works
The Nyquist frequency is half the sampling rate: f_Nyquist = f_sample / 2. It represents the highest frequency that can be unambiguously represented in a digital signal without aliasing. Any signal component above the Nyquist frequency aliases to a lower frequency.
**Aliasing explained** Aliasing occurs when an analog signal contains frequencies above the Nyquist limit. These high-frequency components are indistinguishable from lower-frequency components after sampling. A 9 kHz signal sampled at 10 kHz (Nyquist = 5 kHz) aliases to 1 kHz (|9 - 10| = 1 kHz). The alias appears as a spurious signal that cannot be separated from the real signal.
**Baseband vs. bandpass sampling** Baseband sampling: the standard case — signal from DC to f_max requires f_s > 2 × f_max. Bandpass (undersampling): if a signal occupies a narrow band centered at a high frequency, it can be sampled at a lower rate (f_s > 2 × bandwidth) and the signal aliases to baseband. This is intentional aliasing used in software-defined radio to downconvert signals without a hardware mixer.
**Nyquist in control systems** The Nyquist stability criterion analyzes open-loop frequency response to determine closed-loop stability. Separately, control systems are sampled systems — the sampling rate must be much higher than the process bandwidth (typically 10–20× the bandwidth, not just 2×) to avoid stability problems from sampling delay.
**Two-sided spectrum and negative frequencies** The full spectrum from -f_s/2 to +f_s/2 has total width f_s. Negative frequencies are mathematical artifacts of complex exponential representation; for real signals, the spectrum is symmetric about zero. The Nyquist interval is therefore f_s in bandwidth, but usable for real signals only from 0 to f_Nyquist.
Frequently Asked Questions
- Sample rate must be at least 2× the highest frequency of interest. For machine vibration monitoring: rolling element bearing defect frequencies (BPFI, BPFO) are typically 3–10× shaft frequency. At 3600 RPM (60 Hz), bearing frequencies may reach 600 Hz. To capture harmonics up to 10th: 6,000 Hz × 2 = 12 kHz sample rate minimum. For rotating machinery: common practice is sample at 2.56× the maximum analysis frequency (analysis frequency = Nyquist × 0.8 to allow for anti-aliasing filter roll-off). At 12 kHz sample rate: f_Nyquist = 6 kHz, maximum reliable analysis frequency ≈ 4,800 Hz.
- Aliasing sources: insufficient anti-aliasing filter before ADC (most common), non-bandlimited signal source, undersampled CCD or CMOS sensors in cameras (moiré patterns on fine textures), and intentional undersampling without proper filtering. Prevention: always implement a hardware anti-aliasing filter with cutoff at least at f_Nyquist, with sufficient rolloff by f_Nyquist. Practical filter order: for a 5× oversampled ADC (anti-aliasing at 5× Nyquist), a 4th-order filter achieves >80 dB attenuation at the image frequency. Digital cameras use optical low-pass filters (birefringent crystal) before the sensor to prevent spatial aliasing.
- The Nyquist stability criterion (different from sampling Nyquist) analyzes stability of a closed-loop feedback system by examining the open-loop frequency response. Plot the open-loop transfer function G(jω)H(jω) on a complex plane for ω from -∞ to +∞. Count encirclements of the -1+j0 point. Stability criterion: N = Z - P, where N is clockwise encirclements, Z is closed-loop RHP (unstable) poles, P is open-loop RHP poles. If P = 0, no encirclements of -1 → stable. Gain margin (how much gain can increase before instability) and phase margin (how much phase lag before instability) are read from the Nyquist plot.
- Shannon's capacity theorem extends Nyquist: C = B × log₂(1 + SNR), where C is channel capacity (bits/s), B is bandwidth (Hz), and SNR is signal-to-noise ratio. For a 20 MHz Wi-Fi channel at SNR = 30 dB (1000:1): C = 20×10⁶ × log₂(1001) ≈ 200 Mbps theoretical maximum. Practical Wi-Fi achieves 60–80% of this. Nyquist's theorem says maximum symbol rate = 2B symbols/second (Nyquist rate). With 1024-QAM (10 bits/symbol): data rate = 2 × 20 MHz × 10 bits = 400 Mbps/spatial stream. MIMO adds multiple streams: Wi-Fi 6 with 8×8 MIMO achieves up to 9.6 Gbps theoretical maximum.