Understanding Cross-Spectrum

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Portable balancer & Vibration analyzer Balanset-1A

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Vibration sensor

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Optical Sensor (Laser Tachometer)

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Balanset-4

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Magnetic Stand Insize-60-kgf

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Reflective tape

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Dynamic balancer “Balanset-1A” OEM

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Cross-spectrum — also called the 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. Where an auto-spectrum shows the frequency content of a single channel, the cross-spectrum reveals which frequencies are common to both signals and the phase relationship between them at every frequency.

This makes the cross-spectrum the mathematical foundation of advanced multi-channel analysis: transfer function estimation, coherence analysis, and Operating Deflection Shape (ODS) measurements all rest on it. In practical terms it lets an engineer see how vibration propagates through a structure and identify cause-and-effect relationships between measurement locations — something a single-channel spectrum simply cannot do.

1. Mathematical Definition

Computation

The defining relationship is compact:

G_xy(f) = X(f) × Y*(f)

• X(f) is the FFT of signal x(t).
• Y*(f) is the complex conjugate of the FFT of signal y(t).
• The result is complex-valued, carrying both magnitude and phase.

Components

• Magnitude — |G_xy(f)|: shows the strength of the frequency content the two signals share.
• Phase — ∠G_xy(f): shows the phase difference between the signals at each frequency.
• Real part: the in-phase, or co-spectral, component.
• Imaginary part: the quadrature, or 90°-out-of-phase, component.

2. Properties

Three properties set the cross-spectrum apart from the familiar auto-spectrum, and each one matters in interpretation.

It Is Complex-Valued

• Unlike the auto-spectrum, which is real only, the cross-spectrum is complex.
• It therefore carries both magnitude and phase.
• That phase information is the whole point — it is what reveals how the two signals relate in time.

It Is Not Symmetric

• In general G_xy(f) ≠ G_yx(f).
• Order matters — which signal you treat as the reference changes the result.
• Formally, G_yx(f) is the complex conjugate of G_xy(f), so the phase simply flips sign.

It Requires Averaging

• A single cross-spectrum is noisy and unreliable.
• Averaging many cross-spectra produces a stable estimate.
• Uncorrelated noise components average toward zero because their phase is random from block to block.
• Genuinely correlated components keep a consistent phase and reinforce — which is exactly why averaging cleans the estimate up.

3. Applications

Transfer Function Calculation

This is the single most important application:

H(f) = G_xy(f) / G_xx(f)

• Here x is the input and y is the output.
• The result shows how the system responds to excitation.
• Its magnitude shows amplification or attenuation at each frequency.
• Its phase shows time delay and resonance behaviour.
• It is the core measurement of modal analysis and structural dynamics, closely related to the frequency response function.

Coherence Calculation

• Coherence is defined as |G_xy|² / (G_xx × G_yy).
• It measures the correlation between the two signals at each frequency.
• It ranges from 0 to 1: a value of 1 means perfect correlation, 0 means none at all.
• It validates measurement quality and flags where the result is being corrupted by noise — indispensable during a bump test or modal survey.

Phase Relationship Determination

• The phase from the cross-spectrum reveals time delay or resonance directly.
• 0°: the signals are in phase, moving together.
• 180°: the signals are out of phase, moving in opposition.
• 90°: quadrature, indicating resonance or a pure time delay.
• This is the diagnostic basis for mode shapes and for tracing vibration transmission.

Common-Mode Rejection

• The cross-spectrum isolates the frequency components common to both channels.
• Uncorrelated noise cancels through averaging.
• The true, shared signal components emerge from the background.
• The practical payoff is a better signal-to-noise ratio.

4. Practical Measurement Scenarios

The abstract idea becomes concrete the moment two sensors go onto a real machine. Three everyday set-ups show the value.

Bearing Comparison

• Signal X: vibration at bearing 1. Signal Y: vibration at bearing 2.
• The cross-spectrum shows the frequencies that affect both bearings at once.
• That separates a shared, rotor-related issue from a problem local to one bearing.

Input–Output Analysis

• Signal X: force or vibration at the input — a coupling or the driver bearing.
• Signal Y: the response at the output — the driven-equipment bearing.
• The cross-spectrum reveals the transmission characteristics between them.
• The derived transfer function then quantifies exactly how vibration travels across a coupling.

Structural Transmission

• Signal X: bearing-housing vibration. Signal Y: foundation or frame vibration.
• The cross-spectrum shows which frequencies actually reach the structure.
• That guides decisions on isolation or stiffening, and connects directly to foundation stiffness and structural resonance problems.

5. Interpreting the Cross-Spectrum

High Magnitude at a Frequency

• Indicates strong correlation between the signals at that frequency.
• Points to a common source or strong coupling between the two locations.
• The component is genuinely present in both signals.

Low Magnitude at a Frequency

• Indicates little correlation — weak coupling, or no shared source.
• The component may exist in one signal but not the other.
• Or it may simply be uncorrelated noise from different sources.

Phase Information

• 0°: the signals move together — a rigid connection, or operation below resonance.
• 180°: the signals move oppositely — above resonance, or across a line of symmetry.
• 90°: quadrature — at resonance, or arising from a specific geometry.
• Frequency-dependent phase: the way phase changes with frequency exposes the dynamic behaviour of the structure.

6. Advanced Applications

Multiple Input / Output Analysis

• Several reference signals are paired with several response signals.
• The result is a full matrix of cross-spectra.
• It identifies multiple, simultaneous transmission paths.
• This is how genuinely complex systems are characterised.

Operating Deflection Shapes

• Cross-spectra are taken between many measurement points around a machine.
• Their phase relationships define the deflection pattern.
• The motion of the whole structure can then be visualised and animated.
• Resonant modes stand out clearly in the result.

7. Cross-Spectrum in Field Balancing

Although the cross-spectrum is most associated with modal and structural work, the same two-channel mathematics underpins everyday field balancing. A portable two-channel instrument such as the Balanset-1A records vibration at two bearing planes simultaneously and references both to the once-per-revolution tachometer pulse, so it can resolve the amplitude-and-phase of the 1× component at each plane and compute the cross-coupled influence coefficients that link a weight in one plane to the response in the other. That two-channel, phase-referenced relationship is conceptually a cross-spectrum focused on running speed — and it is exactly what makes correct two-plane dynamic balancing possible on an assembled machine.

In short, the cross-spectrum extends frequency analysis from one channel to many, exposing the relationships between signals that enable transfer-function calculation, coherence validation, and an understanding of how vibration travels through a machine and its supports. More demanding than the auto-spectrum, it is nonetheless essential for modal testing, structural dynamics, and any sophisticated diagnostic that relies on multi-point measurement.

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