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Hilbert Transform / Analytic Signal

Overview & Motivation

Every real-valued signal contains both positive and negative frequency components that carry redundant information. The Hilbert transform discards the negative half and pairs the original signal with a version shifted by exactly −90° at every frequency, producing a complex analytic signal whose magnitude tracks instantaneous amplitude and whose angle tracks instantaneous phase.

This is the standard technique for AM/DSB demodulation, vibration envelope detection, bearing-fault analysis, single-sideband modulation, and instantaneous-frequency measurement — all tasks where the underlying amplitude or phase varies slowly relative to a carrier and must be tracked in real time.

Mathematical Theory

Hilbert Transform

For a real signal $x(t)$ the Hilbert transform is the convolution

$$\mathcal{H}{x}(t) = \frac{1}{\pi} , \text{p.v.} \int_{-\infty}^{\infty} \frac{x(\tau)}{t - \tau} , d\tau$$

which is equivalently a multiplication of each frequency component by $-j \operatorname{sgn}(f)$, i.e. a $-90°$ phase rotation for positive frequencies and $+90°$ for negative.

Analytic Signal

The analytic signal is

$$z(t) = x(t) + j,\mathcal{H}{x}(t)$$

Its one-sided spectrum satisfies $Z(f) = 0$ for $f < 0$, $Z(0) = X(0)$, and $Z(f) = 2X(f)$ for $f > 0$.

Instantaneous Attributes

Quantity Formula
Amplitude (envelope) $A(t) =
Phase $\phi(t) = \arg z(t) = \operatorname{atan2}(\mathcal{H}{x}, x)$
Frequency $f_i(t) = \frac{1}{2\pi}\frac{d\phi}{dt}$ (phase unwrapped before differencing)

FFT Method (Marple)

Given the $N$-point DFT $X[k]$ of a real sequence, the analytic signal is recovered by:

$$H[k] = \begin{cases} X[0] & k = 0 \ 2X[k] & 1 \le k < N/2 \ X[N/2] & k = N/2 \ 0 & N/2 < k < N \end{cases}$$

followed by the inverse DFT of $H$.

FIR Approximation

A Type III/IV antisymmetric FIR filter with impulse response

$$h[k] = \frac{2}{\pi k} w[k], \quad k \text{ odd}; \quad h[k] = 0, \quad k \text{ even}$$

(with a Hamming window $w[k]$) approximates the ideal $-90°$ phase shift across its passband. The delayed original signal (center-tap copy) is paired with the filtered output to form the approximate analytic signal.

Complexity Analysis

Method Time per call Space Notes
FFT (block) $O(N \log N)$ $2N$ complex words Exact, latency $N$
FIR (streaming) $O(P)$ per sample $P$ words state Approx., latency $(P-1)/2$
Feature extraction $O(1)$ None atan2, sqrt, subtract

$P$ = number of FIR taps.

Step-by-Step Walkthrough

FFT method on $N = 8$, input $x = \cos(2\pi n / 8)$:

  1. Forward DFT yields $X[1] = 4$, $X[7] = 4$, all others zero.
  2. Apply one-sided weighting: $H[1] = 8$, $H[7] = 0$.
  3. Inverse DFT of $H$ produces the imaginary part $\sin(2\pi n / 8)$.
  4. Analytic signal: $z[n] = \cos(2\pi n/8) + j\sin(2\pi n/8)$, amplitude $\equiv 1$.

Pitfalls & Edge Cases

  • DC and Nyquist bins — their imaginary parts must remain zero; the one-sided formula preserves this.
  • Phase unwrapping — before computing instantaneous frequency, the phase difference must be mapped to $(-\pi, \pi]$ to suppress $2\pi$ jumps.
  • Block-edge artefacts — the FFT method treats the block as periodic; trim the first and last few samples when asserting accuracy.
  • FIR group delay — the FIR output is delayed by $(P-1)/2$ samples relative to the input; align before comparing to the FFT method.
  • FIR bandwidth — the FIR Hilbert approximation degrades near DC and Nyquist; use only over the filter's flat passband.

Variants & Generalizations

  • Block FFT method — exact, no ripple, requires a complete block; suitable for offline or buffered processing.
  • FIR streaming method — causal, constant memory, approximate; length and window choice trade accuracy against latency.
  • Quadrature-oscillator method — for narrowband signals, a simple IIR resonator can approximate the 90° shift with even lower cost.

Applications

  • AM/DSB envelope demodulation and AGC.
  • Vibration and bearing-fault analysis (envelope of resonance band).
  • Single-sideband (SSB) radio modulation/demodulation.
  • Instantaneous frequency tracking in FM receivers and Doppler radar.
  • Phase and frequency estimation in PLLs.

Connections to Other Algorithms

  • Uses FastFourierTransform / FastFourierTransformRadix2Impl as the block back-end.
  • RealFastFourierTransform is an alternative back-end for purely real inputs.
  • ConvolutionCorrelation provides the FIR convolution primitive used by the streaming path.
  • SignalDetectors (RMS envelope, peak hold) offers cheaper non-coherent envelope estimation.

References & Further Reading

  • S. L. Marple, "Computing the Discrete-Time Analytic Signal via FFT," IEEE Transactions on Signal Processing, 47(9), 2600–2603, 1999.
  • A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed., Prentice Hall, Ch. 12.
  • S. W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing, Ch. 9 (available free online).