Spectral Analysis

Summary

• Plugin name : Spectral Analysis
• Version : 1.2
• Author : Laurent Bonnet / Quentin Barthelemy
• Company : Mensia Technologies
• Short description : Performs a Spectral Analysis using FFT.
• Documentation template generation date : Jan 24 2024

Description

Performs a Spectral Analysis using FFT.

The Spectral Analysis box performs spectrum computations on incoming signals and possible outputs include the spectrum amplitude (the power of the signal in a number of frequency bands), as well as its phase, real part and imaginary part.
Output computations may be enabled/disabled from the settings dialogue box. The analysis is performed using a Fast Fourier Transform .
Do not forget to apply a Windowing step before spectral analysis.

Considering an input signal \( X \in \mathbb{R}^{C \times N} \), composed of \( C \) channels and \( N \) temporal samples, this plugin computes the spectrum of this signal \( \Phi \in \mathbb{C}^{C \times F} \), composed of \( C \) channels and \( F \) frequencies.
Input signal being real, the spectrum exhibits conjugate symmetry: consequently, only half of the spectrum is returned with \( F = \left\lfloor N/2 \right\rfloor + 1 \).
For the \( c^{ \text{th} } \) channel and the \( f^{ \text{th} } \) frequency, the spectrum is defined as:

\[ \Phi(c,f) = \Phi_r(c,f) + \mathsf{i} \times \Phi_i(c,f) = \left| \Phi(c,f) \right| \times e^{\mathsf{i} \arg(\Phi(c,f))} \]

with \( \mathsf{i} \) being the imaginary unit.

Using these notations, for the \( c^{ \text{th} } \) channel, the Parseval's Theorem gives:

\[ \sum_{n=0}^{N-1} \left| X(c,n) \right|^2 = \frac{1}{N} \sum_{f=0}^{F-1} \left| \Phi(c,f) \right|^2 \]

with \( \left| \Phi(c,f) \right|^2 = \Phi_r(c,f)^2 + \Phi_i(c,f)^2 \).

Inputs

1. Input signal

An input multichannel signal \( X \in \mathbb{R}^{C \times N} \), composed of \( C \) channels and \( N \) temporal samples.

• Type identifier : Signal (0x5ba36127, 0x195feae1)

Outputs

1. Amplitude

An output spectral amplitude (absolute value) \( \left| \Phi \right| \in \mathbb{R}^{C \times F} \).

• Type identifier : Spectrum (0x1f261c0a, 0x593bf6bd)

2. Phase

An output spectral phase \( \arg(\Phi) \in \mathbb{R}^{C \times F} \), in radians.

• Type identifier : Spectrum (0x1f261c0a, 0x593bf6bd)

3. Real Part

An output real part of the spectrum \( \Phi_r \in \mathbb{R}^{C \times F} \).

• Type identifier : Spectrum (0x1f261c0a, 0x593bf6bd)

4. Imaginary Part

An output imaginary part of the spectrum \( \Phi_i \in \mathbb{R}^{C \times F} \).

• Type identifier : Spectrum (0x1f261c0a, 0x593bf6bd)

Settings

1. Amplitude

Activate or not the Amplitude output.

• Type identifier : Boolean (0x2cdb2f0b, 0x12f231ea)
• Default value : [ true ]

2. Phase

Activate or not the Phase output.

• Type identifier : Boolean (0x2cdb2f0b, 0x12f231ea)
• Default value : [ false ]

3. Real Part

Activate or not the Real Part output.

• Type identifier : Boolean (0x2cdb2f0b, 0x12f231ea)
• Default value : [ false ]

4. Imaginary Part

Activate or not the Imaginary Part output.

• Type identifier : Boolean (0x2cdb2f0b, 0x12f231ea)
• Default value : [ false ]

Examples

Practical example : visualising the power spectrum of a signal.

Let's use a Signal Oscillator box to generator sinusoidal signals on one channel. Next we add a Spectral Analysis box and connect boxes together. We make sure the 'Amplitude' of the signal is computed by checking the appropriate setting in the settings dialog box (see image below). Finally, we connect the 'Amplitude' output connector of the Spectral Analysis box to the input connector of a Power Spectrum Display box. The player may now be launched to visualize the power spectrum of the signal.

Visualising the power spectrum of sinusoidal signals.

Miscellaneous

To verify the Parseval's Theorem, in version 1.1, spectra have been multiplied by \( \sqrt{2} \) with respect the previous version 1.0.
DC bin and Nyquist bin (when \( N \) is even) are not concerned by this correction.