Traditional multichannel signal analysis treats data samples independently. However, recent multidimensional signal analysis techniques, such as multiple signal classification (MUSIC), principal component analysis (PCA), and independent component analysis (ICA), demonstrate the advantages of treating multichannel data samples jointly. This article briefly introduces the fundamentals of MUSIC, PCA, and ICA, and describes their applications in multichannel signal analysis. You can find all the functions that this article uses in the LabVIEW Advanced Signal Processing Toolkit.
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2. Comparing the MUSIC and FFT-Based Methods
The following example compares MUSIC and traditional Fast Fourier Transform (FFT) analysis methods by analyzing the same set of multichannel signals with the two methods. First, this example simulates vibration signals from a steel-reinforced concrete beam. This example then applies the MUSIC and FFT-based methods to compute the resonant frequencies of the vibration signals. Figure 1 shows the block diagram of this example.
Figure 1. Analyzing Vibration Signals with the MUSIC and FFT-Based Methods
As the block diagram shows, after acquiring the vibration signals, this example uses the TSA MUSIC VI and the TSA Periodogram VI to analyze those signals. The TSA MUSIC VI computes the frequency components of the signals using the MUSIC method. The TSA Periodogram VI computes the frequency components of the signals using the FFT-based method. Figure 2 shows the original vibration signals and the resulting frequency components using each method.
Figure 2. Analysis Results of the MUSIC and FFT-Based Methods
By comparing the resulting plots in Figure 2, you can see that the MUSIC method provides a more comprehensive view of the frequency components than the FFT-based method: the resulting MUSIC plot displays four frequency peaks, whereas the resulting FFT-based plots contain no more than three frequency peaks. The reason is that the MUSIC method considers the correlations between all signal channels and processes these signals jointly, whereas the FFT-based method processes these channels separately.
3. Comparing the PCA and ICA Methods
The MUSIC method maps a set of multichannel signals into a one-dimensional frequency plot. Unlike the MUSIC method, the PCA and ICA methods convert the set of multichannel signals into another set of time waveforms and make the attributes of the signals more explicit. Mathematically, you can formulate both the PCA and ICA methods as a linear system with the following equation:
Xm×L = Am×nSn×L
where X is a set of measured multichannel data samples, and X is also called the observation matrix.
m is the number of signal channels.
L is the length of the multichannel data samples.
A denotes a time-invariant system matrix.
n is the number of signal sources.
S is a matrix for decomposed components.
Both the PCA and ICA methods try to determine S using the measured X. Generally, you cannot solve this equation unless you apply some constraints to S or A. Different constraints lead to different methods and solutions.
For the PCA method, S is a matrix that has the same dimensions as X, whose row vectors are orthogonal to each other. The principal component implies the largest combination of the row vectors in X. For example, consider signals you acquire using triaxial accelerometers. Unlike classical accelerometers, which provide only one vibration signal, triaxial accelerometers provide vibration signals from three directions, x, y, and z, simultaneously. Figure 3 shows a triaxial accelerometer.
Figure 3. Triaxial Accelerometer
If the vibration signals have only one vibration source and the system noise, such as the noise from sensors, is negligible, then the principal component represents the sum of the vibrations from all three directions. In this case, the three vibration signals are projections of the vibration on the x, y, and z axes, as shown in Figure 4.
Figure 4. Projections of the Vibration Signals
The result of the PCA method represents the joint effects of multichannel data samples. This method does not return signals that physically exist. To recover the true signals, you must apply the ICA method.
The ICA method assumes that:
- The source signals are statistically independent.
- The observation X is a superposition of the source signal S.
Figure 5 shows the result of using the ICA method to analyze magneto encephalogram (MEG) signals.
Figure 5. ICA for MEG Signals Analysis
In this example, 148 sensors located in a helmet system acquire the MEG signals from a scalp. MEG signals are the magnetic signals from electric dipoles around a human brain. All cognitive activities in the human brain generate magnetic signals. Heartbeats, eye blinking, and breathing also generate magnetic signals. These signals are superimposed on the measured brain signal. As the Heartbeats and eye blinking signals plot shows, the ICA method identifies the eye blinking and heartbeat-related components from the MEG signals. After you remove those components, the remaining components form an enhanced brain signal that can help you better understand brain activities.
Many multichannel medical signals can be approximated by linear superposition. Therefore, the ICA method often produces dramatic differences compared to other methods in medical signal analysis. If multichannel signals cannot be approximated by linear superposition, the ICA method might not be an appropriate analysis method. For example, you cannot use the ICA method to analyze voices recorded from a "cocktail party" environment, because the noise in that environment is not the simple summation of multiple individual speakers.
Use the TSA Principal Component Analysis VI and Independent Component Analysis VI to perform the PCA and ICA methods, respectively. Refer to the LabVIEW Help for more information about these VIs.
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