Paste samples, set rate, and choose processing options. See Hilbert output with clear numerical summaries. Download results, compare cases, and validate spectra easily now.
| i | t (s) | x |
|---|---|---|
| 0 | 0.0000 | 0.000000 |
| 1 | 0.0100 | 0.309017 |
| 2 | 0.0200 | 0.587785 |
| 3 | 0.0300 | 0.809017 |
| 4 | 0.0400 | 0.951057 |
| 5 | 0.0500 | 1.000000 |
| 6 | 0.0600 | 0.951057 |
| 7 | 0.0700 | 0.809017 |
| 8 | 0.0800 | 0.587785 |
| 9 | 0.0900 | 0.309017 |
The continuous Hilbert transform of a real signal \(x(t)\) is the principal-value integral:
For discrete samples, this calculator forms the analytic signal using an FFT-based frequency filter:
The inverse FFT gives \(z[n]\) where the imaginary part is the Hilbert transform: \(z[n] = x[n] + j\,H\{x\}[n]\). The envelope is \(|z[n]|\), and phase is \(\angle z[n]\). If enabled, instantaneous frequency uses the unwrapped phase derivative: \(f[n] = \frac{\Delta \phi[n]}{2\pi}\,f_s\).
For a real input sequence x[n], the tool builds the analytic signal z[n] = x[n] + j y[n], where y[n] is the discrete Hilbert transform. From z[n] it reports envelope |z[n]|, phase ∠z[n], and optional instantaneous frequency derived from phase change and the sampling rate.
The discrete Hilbert transform can be implemented by suppressing negative-frequency content and doubling positive-frequency content in the spectrum. After multiplying the FFT bins by the mask M[k], the inverse FFT produces a complex-valued signal whose imaginary part is the quadrature component.
Accurate envelope and phase require a sampling rate comfortably above the highest frequency present. As a rule of thumb, aim for at least 5–10 samples per cycle of the dominant oscillation. For example, a 5 Hz waveform is well-resolved at 100 Hz (20 samples per cycle).
Mean removal prevents DC offsets from dominating the analytic signal. Linear detrending reduces slow drift that can distort phase. Windowing (Hann, Hamming, Blackman) lowers spectral leakage when the record does not contain an integer number of cycles, improving envelope stability for narrowband signals.
Hilbert transforms are non-local, so endpoints are sensitive to truncation. Expect the first and last few percent of samples to show larger phase and envelope bias, especially with sharp transients. Increasing duration, applying a smooth window, or zero-padding (larger NFFT) can reduce visible ringing.
The envelope |z[n]| tracks amplitude modulation for narrowband signals. In vibration and acoustics it highlights slow amplitude changes; in communications it approximates AM demodulation. For broadband impulses, the envelope is still useful but may blend multiple components into a single magnitude trace.
Raw phase is wrapped to (−π, π]. Unwrapping removes 2π jumps, enabling a smooth derivative. Instantaneous frequency is computed as Δφ/(2π)·fs and can be noisy; the smoothing window averages the derivative over multiple samples to emphasize trends rather than sample-to-sample jitter.
A quick check is a pure sinusoid: the Hilbert transform should be a 90°-shifted cosine/sine pair, the envelope should be approximately constant, and the instantaneous frequency should cluster near the set tone. Add controlled noise to evaluate robustness and tune smoothing length.
It creates a quadrature companion signal, enabling analytic-signal methods such as envelope detection, instantaneous phase, and instantaneous frequency. It is common in vibration analysis, radar/sonar, biomedical time series, and communications.
Endpoint samples suffer from truncation because the transform is non-local. Use longer records, apply a smooth window, and interpret the first/last few percent cautiously. Zero-padding can also reduce visible ringing.
Yes when offsets or drift exist. Mean removal prevents DC bias from inflating the envelope. Detrending helps when slow ramps are present, improving phase behavior and making instantaneous frequency more meaningful.
Use a window when your data segment does not align with whole cycles or when leakage contaminates nearby frequencies. Hann and Hamming are good general choices; Blackman offers stronger sidelobe suppression with more mainlobe widening.
It increases the FFT grid density and can smooth the reconstructed analytic signal, but it does not add new information. It may reduce visual artifacts and help with interpolation, especially when analyzing narrowband signals.
It depends on a phase derivative, which amplifies noise and sharp transitions. Enable phase unwrapping and use a smoothing window. For highly broadband or impulsive signals, interpret instantaneous frequency as a qualitative indicator.
Try a clean sinusoid: the envelope should be nearly constant and the frequency estimate should match the tone. Then add small noise to see how much smoothing is needed for stable trends.
This calculator constructs the analytic signal z[n] = x[n] + j* y[n], where y[n] is the discrete Hilbert transform. From z[n] it reports the envelope |z[n]| and the phase arg(z[n]). These outputs are commonly used to estimate amplitude modulation, track phase, and compute instantaneous frequency.
The discrete transform is computed with an FFT: the spectrum X[k] is multiplied by a mask that doubles positive-frequency bins, keeps DC (and Nyquist for even lengths), and zeros negative frequencies. After the inverse FFT, the imaginary part becomes y[n].
Enter a correct sampling rate f_s to interpret time and frequency. The output time step is dt = 1/f_s and the instantaneous frequency option uses f[n] = (f_s/2pi) * dphi/dn. For example, a 5 Hz tone sampled at 100 Hz should produce a near-constant 5 Hz estimate away from edges.
FFT processing assumes the record repeats, so sharp start or end points can leak into the envelope and phase. Windowing reduces this leakage, while a larger NFFT (zero-padding) refines the frequency grid and can make the quadrature estimate smoother. Padding does not add new information, but it can improve interpolation of the analytic signal.
The discrete transform is computed with an FFT: the spectrum X[k] is multiplied by a mask that doubles positive-frequency bins, keeps DC (and Nyquist for even lengths), and zeros negative frequencies. After the inverse FFT, the imaginary part becomes y[n].
Set the sampling rate carefully because time step is dt = 1/fs. Instantaneous frequency is reported in hertz using the phase difference f[n] = (dphi/(2*pi))*fs. For example, a 5 Hz tone sampled at 100 Hz should yield a stable 5 Hz estimate away from edges.
FFT processing assumes the record repeats. If your signal is not periodic within the window, discontinuities at the ends can leak energy and distort the envelope and phase near the boundaries. Zero padding increases frequency sampling and can smooth plots, but it does not remove boundary bias.
Mean removal suppresses a large DC component that can dominate the analytic magnitude. Linear detrending reduces slow ramps that inflate low-frequency bins. Windowing (Hann, Hamming, or Blackman) reduces spectral leakage, which is useful when the record length is short or contains sharp transitions.
The envelope |z[n]| tracks slowly varying amplitude. A common test is an AM signal such as x(t) = (1 + 0.3*cos(2*pi*2*t))*cos(2*pi*20*t). With a sampling rate above 40 Hz, the envelope should follow 1 + 0.3*cos(2*pi*2*t) after transients.
The phase of z[n] can jump by plus or minus pi when it crosses the branch cut. Unwrapping removes these discontinuities so the phase derivative is meaningful. Instantaneous frequency is sensitive to noise, so the optional moving-average smoothing can reduce jitter.
Validate using a pure sinusoid: the Hilbert transform should be the same sinusoid shifted by 90 degrees, and the envelope should be constant. In practice, analytic envelopes are used in vibration diagnostics, seismology, radar, and biomedical time series to highlight bursts and estimate local oscillation rates.
FFT processing assumes the record repeats. End discontinuities create leakage and bias the Hilbert pair. Try windowing, detrending, longer records, or ignore a short edge region.
Padding increases FFT length and interpolates the frequency grid. It can make curves smoother and reduce wraparound artifacts, but it does not add new information or fix boundary discontinuities.
Use a sampling rate comfortably above the highest frequency of interest. As a rule of thumb, 5 to 10 samples per cycle improves envelope and phase stability and reduces instantaneous-frequency noise.
Enable it when you need a smooth phase trace or instantaneous frequency. Without unwrapping, phase jumps by plus or minus pi can create large spikes in the derivative.
A strong DC offset and slow ramps concentrate power at very low frequencies and can dominate the analytic magnitude. Removing them often yields a cleaner envelope and more interpretable phase.
Yes, but interpret results locally. Sudden transients and wideband content can make instantaneous frequency unstable. Smoothing and appropriate windowing help, and longer records reduce variance.
Test with a clean sinusoid. The Hilbert output should be the same sinusoid shifted by 90 degrees, the envelope should be constant, and the instantaneous frequency should match the tone value away from edges.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.