In "Frequency Analysis from the Basics (25) - Basics of Vibration Measurement Part 4," when discussing damping ratios, I explained the logarithmic damping method. There, I introduced a method for calculating the logarithmic damping ratio from the envelope of a damped free vibration waveform using the Hilbert transform.
This time, I'll talk about the Hilbert transform.
In general, the time signals we can observe are real signals. For simplicity, let's consider a sinusoidal function x(t) here.
x(t)=Acos(ωt)=Acos(θ)
The value of the real signal x(t) in equation (1) is determined solely by the time variable, provided that the amplitude A and angular frequency ω are time-invariant, and can be easily obtained using the Fourier spectrum.
Next, consider the case where both change with time (equation (2) below).
x(t)=A(t)cos(ω(t)t)=A(t)cos(θ(t))
Now, A(t) and θ(t) (or ω(t)) are values that change with time, and as they are, we cannot tell which one changed to cause the change in the real signal x(t). In order to simultaneously determine both the instantaneous amplitude A(t) and the instantaneous phase θ(t), the time signal x(t) needs to be a complex signal (having two values) rather than a real signal (having only one value).
As shown in Figure 1, a sine wave function can be considered as the projection of a point P(x,y) rotating on the complex plane onto the x-axis (real axis) as a cosine wave, and its projection onto the y-axis (imaginary axis) as a sine wave. Therefore, if we can create a sine wave from a cosine wave, we can simultaneously determine the amplitude A and phase θ.
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Figure 1. Rotating vector OP and sine wave function
If we express point P using the complex function z(t), then:
It will be.
Since the sine wave function can be obtained by delaying the phase of the cosine wave function by 90°, the complex function z(t) can be obtained by delaying the real function x(t) by 90°. One way to shift any time function by 90° is the Hilbert transform.
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Figure 2: How to convert a real function to a complex function.
The Hilbert transform of a real function x(t) ∧ x(t) is defined as the convolution of x(t) with 1/πt.
In other words;
Here, * represents convolution, and the integral is Cauchy's principal value integral.
Taking the Fourier transform of both sides of equation (7) gives:
Therefore, taking the inverse Fourier transform of both sides of equation (8) gives:
The flow of these processes is illustrated in Figure 3.
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Figure 3 Hilbert transform using FFT
Also, the imaginary unit
Multiplying by this is equivalent to advancing the phase by 90 degrees (= π/2), so from equations (8) and (9), the phase characteristics of the Hilbert transform are as shown in Figure 4.
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Figure 4 Phase characteristics of Hilbert transform
As shown in Figure 2, the complex function z(t) created using the real function x(t) and its Hilbert transform is called the analytic signal of x(t).
If Z(f) is the Fourier transform of the analyzed signal z(t), then Z(f) has the following properties (Figure 5).
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Figure 5 Frequency characteristics of Z(f)
In other words, Z(f) has the property of not having negative frequency components. Conversely, by utilizing this property, it is possible to calculate the analysis signal z(t) relatively easily using FFT.
The procedure for obtaining the analytical signal z(t) from the real signal x(t) is as follows:
- Perform an FFT on x(t) to find X(f).
- We find Z(f) from equation (12).
- Perform an IFFT on Z(f) to find z(t).
To summarize, by creating an analytical signal z(t) using the Hilbert transform, we can then determine the instantaneous amplitude and instantaneous phase of the real signal x(t).
Instantaneous amplitude (envelope)
Instantaneous phase
Here are some specific data examples.
Figure 6 shows an example of obtaining the envelope of an amplitude-modulated (AM) sine wave.
- Carrier frequency: 5kHz
- Modulation frequency: 100Hz
- Modulation level: 20%
Both the demodulated envelope waveform and the spectrum indicate a variability of 20%.
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Figure 6. Envelope of AM modulated wave (Left: Original carrier wave, Center: Envelope, Right: Carrier wave spectrum)
As shown again in Figure 7, an example is shown in which the analysis signal is obtained from the decay time waveform, the instantaneous amplitude (envelope) is calculated from that, the vertical axis is displayed logarithmically, and the decay ratio is determined from the slope using the logarithmic decay method.
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Figure 7: Example of determining the logarithmic decay rate and decay ratio from decay time waveform and envelope data.
If the instantaneous phase is determined, the instantaneous frequency can also be found by differentiating it with respect to time.
The instantaneous frequency f(t) is:
It will be.
Figure 8 shows an example of analyzing a chirp sine wave swept rapidly from 125 Hz to 100 kHz using an FFT time window of 8 ms. The X axis represents time and the Y axis represents frequency, and it can be seen that the frequency changes linearly up to 100 kHz.
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Figure 8. Instantaneous frequency of a chirp sine signal (Top: Original time waveform, Middle: Instantaneous phase, Bottom: Instantaneous frequency)
By using instantaneous frequency, the rotational fluctuation component can be determined.
Another application of the Hilbert transform is its property that if the system is linear, the result of the Hilbert transform of the system transfer function is equal to the original transfer function. This property is used to check for nonlinearity in a system.
Finally, here's a summary.
- Real-world functions cannot simultaneously determine instantaneous amplitude and instantaneous phase.
- By using the Hilbert transform to convert a real signal into a complex-time signal, we can simultaneously determine the instantaneous amplitude and instantaneous phase.
- A complex function whose real and imaginary parts are derived from a real function and its Hilbert transform is called an analytic signal.
- The analytical signal is a function that does not have negative frequency components. This property allows us to relatively easily determine the analytical signal z(t) from a real function x(t).
- If the instantaneous phase is determined, the instantaneous frequency can be found by differentiating it with respect to time, and from that, rotational fluctuation information can also be obtained.
- Another application of the Hilbert transform is its use in checking the linearity of a system.
【keyword】
Real signal, amplitude, phase, instantaneous amplitude, instantaneous phase, cosine wave, sine wave, complex signal, Hilbert transform, convolution, phase characteristics, analytic signal, envelope, instantaneous frequency
【reference】
"Digital Fourier Analysis (II) - Advanced Level -" edited by Kenichi Kido, Corona Publishing Co., Ltd. (2007)
(Excerpt from the email newsletter issued on September 26, 2016)
