Understanding Cross-Spectrum

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Definition: What is Cross-Spectrum?

Cross-spectrum (also called cross-power spectrum or cross-spectral density) is the frequency domain representation of the relationship between two simultaneously measured vibration signals. It is computed by multiplying the FFT of one signal by the complex conjugate of the FFT of the other signal. Unlike an auto-spectrum which shows frequency content of a single signal, cross-spectrum reveals which frequencies are common to both signals and the phase relationship between the signals at each frequency.

Cross-spectrum is fundamental to advanced multi-channel vibration analysis including transfer function estimation, coherence analysis, and Operating Deflection Shape (ODS) measurements. It enables understanding how vibration propagates through structures and identifying cause-effect relationships between measurement locations.

Mathematical Definition

Computation

• Gxy(f) = X(f) × Y*(f)
• Where X(f) = FFT of signal x(t)
• Y*(f) = complex conjugate of FFT of signal y(t)
• Result is complex-valued (has both magnitude and phase)

Components

• Magnitude: |Gxy(f)| shows common frequency content strength
• Phase: ∠Gxy(f) shows phase difference between signals at each frequency
• Real Part: In-phase (co-spectral) component
• Imaginary Part: Quadrature (90° out-of-phase) component

Properties

Complex-Valued

• Unlike auto-spectrum (real only), cross-spectrum is complex
• Contains both magnitude and phase information
• Phase crucial for understanding signal relationships

Not Symmetric

• Gxy(f) ≠ Gyx(f) generally
• Order matters (which signal is reference)
• Gyx(f) = complex conjugate of Gxy(f)

Averaging Required

• Single cross-spectrum noisy and unreliable
• Average multiple cross-spectra for stable estimate
• Noise components average toward zero (uncorrelated)
• Correlated components reinforce

Applications

1. Transfer Function Calculation

Most important application:

• H(f) = Gxy(f) / Gxx(f)
• Where x = input, y = output
• Shows how system responds to excitation
• Magnitude shows amplification/attenuation
• Phase shows time delay or resonance behavior
• Used in modal analysis, structural dynamics

2. Coherence Calculation

• Coherence = |Gxy|² / (Gxx × Gyy)
• Measures correlation between signals at each frequency
• Values 0-1: 1 = perfect correlation, 0 = no correlation
• Validates measurement quality and identifies noise

3. Phase Relationship Determination

• Phase from cross-spectrum shows time delay or resonance
• 0° phase: signals in-phase (moving together)
• 180° phase: signals out-of-phase (moving opposite)
• 90° phase: quadrature (resonance or time delay)
• Diagnostic for mode shapes, vibration transmission

4. Common Mode Rejection

• Cross-spectrum identifies frequency components common to both channels
• Uncorrelated noise cancels in averaging
• Reveals true signal components
• Improves signal-to-noise ratio

Practical Measurements

Typical Measurement Scenarios

Bearing Comparison

• Signal X: Vibration at bearing 1
• Signal Y: Vibration at bearing 2
• Cross-spectrum shows frequencies affecting both bearings
• Identifies rotor-related issues vs. individual bearing problems

Input-Output Analysis

• Signal X: Force or vibration at input (coupling, driver bearing)
• Signal Y: Response at output (driven equipment bearing)
• Cross-spectrum reveals transmission characteristics
• Transfer function shows how vibration transmits

Structural Transmission

• Signal X: Bearing housing vibration
• Signal Y: Foundation or frame vibration
• Cross-spectrum shows which frequencies transmit to structure
• Guides isolation or stiffening efforts

Interpretation

High Magnitude at Frequency

• Indicates strong correlation between signals at that frequency
• Common source or strong coupling
• Component present in both signals

Low Magnitude at Frequency

• Little correlation (uncorrelated or weak coupling)
• Component may be present in one signal but not other
• Or component uncorrelated (noise, different sources)

Phase Information

• 0°: Signals move together (rigid connection or below resonance)
• 180°: Signals move opposite (above resonance or symmetry)
• 90°: Quadrature (at resonance or specific geometry)
• Frequency-Dependent: Phase changes reveal dynamic behavior

Advanced Applications

Multiple Input/Output Analysis

• Multiple reference signals, multiple response signals
• Matrix of cross-spectra
• Identifies multiple transmission paths
• Complex system characterization

Operating Deflection Shapes

• Cross-spectra between many measurement points
• Phase relationships define deflection pattern
• Visualizes structural motion
• Identifies resonance modes

Cross-spectrum extends frequency analysis from single-channel to multi-channel, revealing relationships between signals that enable transfer function calculation, coherence validation, and understanding of vibration transmission paths. While more complex than auto-spectrum, cross-spectrum is essential for advanced vibration analysis including modal testing, structural dynamics, and sophisticated machinery diagnostics requiring multi-point measurements.

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