Any Continuous Cycle Signal Can Be Made From a Suitable Set of Sinusoidal Combinations

Sinusoidal Component

Each sinusoidal component of such a process has the strongly non-Gaussian pdf of the form given in eqn (9).

From: Encyclopedia of Vibration , 2001

Images, sampling, and frequency domain processing

Mark S. Nixon , Alberto S. Aguado , in Feature Extraction & Image Processing for Computer Vision (Third Edition), 2012

2.7.4 Other transforms

Decomposing a signal into sinusoidal components was actually one of the first approaches to transform calculus, and this is why the Fourier transform is so important. The sinusoidal functions are actually called basis functions, the implicit assumption is that the basis functions map well to the signal components. As such, the Haar wavelets are binary basis functions. There is (theoretically) an infinite range of basis functions. Discrete signals can map better into collections of binary components rather than sinusoidal ones. These collections (or sequences) of binary data are called sequency components and form the basis of the Walsh transform (Walsh and Closed, 1923), which is a global transform when compared with the Haar functions (like Fourier compared with Gabor). This has found wide application in the interpretation of digital signals, though it is less widely used in image processing (one disadvantage is the lack of shift invariance). The Karhunen–Loéve transform (Loéve, 1948; Karhunen, 1960) (also called the Hotelling transform from which it was derived, or more popularly Principal Components Analysis—see Chapter 12, Appendix 3) is a way of analyzing (statistical) data to reduce it to those data which are informative, discarding those which are not.

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Signal Analysis in the Frequency Domain

John Semmlow , in Circuits, Signals and Systems for Bioengineers (Third Edition), 2018

3.1 Goals of This Chapter

In this chapter we determine how to find the sinusoidal components of a general signal. But this is something we have already done in Example 2.15 when we used correlation between an electroencephalography (EEG) signal and sinusoids to search for oscillatory behavior. ¹ Here we develop a computationally more efficient approach to do the same thing. More importantly, we dig deeper into the greater significance of those sinusoidal components. We also address some issues we glossed over in Example 2.15. For example, in our search for oscillatory behavior, we compared the EEG signal with a group of sinusoids ranging in frequencies between 1 and 25   Hz in 0.5-Hz intervals. Could we have missed some oscillatory behavior with this arbitrary range of frequencies and frequency increments? Should we have used a broader range of frequencies and/or finer intervals, or perhaps used fewer sinusoids and larger intervals? Here we offer a definitive answer to these questions.

We also find that correlating a signal with a series of sinusoids gives us more than just a measure of the signal's oscillatory behavior. If we choose the right combination of sinusoids, and we use enough of them, the correlation coefficients become an equivalent, alternative representation of the signal. In other words, the correlation coefficients give a complete representation of the signal and it is possible to reconstruct the original signal from just these coefficients. This works only if we use enough sinusoids at the right frequencies, but the technique for finding the right correlation coefficients is straightforward.

The correlation coefficient signal representation can be very useful both in providing a new description of the signal and in certain operations applied to the signal. The correlation coefficient representation of a signal is called the "frequency domain" representation since it is based on the correlations with sinusoids having a range of frequencies. Converting a signal from its time domain representation to its frequency domain representation is an example of a class of operations known as "signal transformations" or just "transformations." Again, the two representations are completely interchangeable. You can go from the time to the frequency domain and back again with no constraints; you just have to follow the mathematical rules.

In Example 2.15 we used digital signals in the digital domain, but in this chapter we also determine sinusoidal correlations analytically from continuous signals in the continuous domain. In fact, time–frequency domain transformations were first developed in the continuous domain and laboriously worked out by hand before the advent of the digital computer. We do a few easy examples on continuous domain operations because as engineers we should know the inner workings of this important transformation. Fortunately, all real-world time–frequency transformations are done in the digital domain on a computer.

To summarize, in this chapter we will:

•

Show how to decompose any periodic waveform into a series of sinusoidal components and how to do the opposite, recombine the sinusoidal components into a waveform.

•

Demonstrate how sinusoidal decomposition leads to the frequency characteristics of the spectrum of a waveform.

•

Describe the Fourier transform using complex notation.

•

Use the Fourier transform to find the frequency domain representation of a periodic waveform.

•

Show one method for tracking the changes in waveform's spectrum over time.

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Gravity Waves

Nikolaos D. Katopodes , in Free-Surface Flow:, 2019

3.5.1 Fourier Series

The idea that a complex wave form consists of many sinusoidal components was introduced by Fourier (1822) in connection with the solution of the heat equation. Fourier studied the transient temperature distribution in a ring heated from below. He thought of the ring as a circle where every point is uniquely described by the central angle, θ, and concluded that the temperature must be a periodic function of θ with period 2π. Therefore, the fundamental frequency of a periodic function is ${\omega }_0=1/2\pi$ .

Fourier realized that the distribution of temperature could not be described by a simple harmonic function like the one given by Eq. (I-1.69) for a spring. As a simple alternative, he proposed to model the temperature distribution, $f(t)$ , by the sum of N sinusoids, as follows

(3.47) $f(t)=\sum _{n=1}^N{A}_n(t)\sin (n{\omega }_{0}t+{\varphi }_n)$

where the transient coefficients, $An(t)$ , are to be determined. Because each term in the sum has a frequency, $n{\omega }_0$ , which is an integer multiple of the fundamental frequency, ${\omega }_0$ , the sum is also periodic. Notice that the frequency of the entire series is still ${\omega }_0$ , as its periodicity is dominated by the longest period in the sum.

To simplify the analysis further, let us neglect phase shift, and include an additional trigonometric component, and a constant term for $n=0$ . This is permissible since the sum in Eq. (3.47) is arbitrary. Hence

(3.48) $f(t)=\frac{a_0}{2}+\sum _{n=1}^N[a_n\cos \left(n{\omega }_{0}t\right)+b_n\sin \left(n{\omega }_{0}t\right)]$

This is known as the trigonometric form of the Fourier series. For arbitrarily large N, it states that any periodic function can be expressed as a sum of sinusoids. Using Euler's formula, i.e. Eq. (I-1.71), the series can also be written in exponential form, as follows

(3.49) $f(t)=\sum _{n=-\infty }^{\infty }{c}_n{e}^{\dot{\mathfrak{I}}{\omega }_{n}t}$

where ${\omega }_n=n{\omega }_0$ . The compactness of this form justifies the inclusion of the second sinusoid and constant term. Notice that the coefficients, $c_n$ , are complex, and satisfy $c_{-n}={\overline{c}}_n$ , with the exception of $c_0$ , which is real. This conjugate symmetry guarantees that the overall sum is real as well.

Finally, the coefficients in the series can be identified by multiplying both sides by $e^{-\dot{\mathfrak{I}}m{\omega }_{0}t}$ , and integrating over the period T. All terms cancel out except for $m=n$ , yielding (Gray and Goodman, 1995)

(3.50) $c_m=\frac{1}{T}{\int }_0^T{e}^{-\dot{\mathfrak{I}}{\omega }_{n}t}f(t)dt$

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Methods for Open-Channel Flow

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

7.2.4.2 Stability Analysis

The stability analysis of the Preissmann scheme is performed by substituting the sinusoidal components given by Eqs. (7.46) and (7.47) into the discretized continuity and momentum equations, i.e. Eqs. (7.98) and (7.99). The result is a system of linear equations for the two amplification factors that can be written as follows

(7.110) $\mathbf{F}{\mathbf{\Xi }}^{n+1}=\mathbf{G}{\mathbf{\Xi }}^n$

where $\mathbf{\Xi }={\left(\xi \eta \right)}^T$ , and F, G are complex matrices containing the flow parameters. It is too tedious to write out the components of these two matrices, but the solution can be found numerically by first inverting F, i.e.

(7.111) ${\mathbf{\Xi }}^{n+1}={\mathbf{F}}^{-1}\mathbf{G}{\mathbf{\Xi }}^n$

Therefore, ${\mathbf{F}}^{-1}\mathbf{G}$ is the amplification matrix of the Preissmann scheme. Following some tedious calculations, the eigenvalues can be found, along with a stability condition similar to that of the Crank-Nicolson scheme, i.e.

(7.112) $\frac{1}{2}\le \theta \le 1$

This is no surprise since the explicit version of the scheme is clearly unstable. At $\theta =1/2$ , the method is second-order accurate with respect to the time step, and of course second-order accuracy is true in the approximation of the spatial derivatives under any conditions, due to the centered spatial weighting. At $\theta =1$ , the method is fully implicit, and thus loses its second-order accuracy completely.

The algorithmic portrait of the method is shown graphically in Fig. 7.13 and Fig. 7.14. For $\theta =1/2$ the Preissmann scheme is non-dissipative under all flow conditions, thus only the relative celerity is shown. For ${\mathbf{F}}_r=0$ , the phase error is substantial, but it improves for higher Froude numbers. In both cases, the phase error persists for relatively long waves, thus the method will affect the propagation of real waves in an open channel.

Figure 7.13. Phase error of Preissmann scheme; Regressive wave

Figure 7.14. Algorithmic dissipation of Preissmann scheme; Regressive wave

A small increase in θ over the Crank-Nicolson standard does introduce dissipation to the Preissmann scheme, as shown in Fig. 7.14. Thus, at the loss of some accuracy, it should be possible to suppress spurious oscillations that arise in the vicinity of discontinuities. The dissipation profile for ${\mathbf{F}}_r=0$ is not particularly selective, but it does improve as the Froude number increases.

In summary, the strength of the Preissmann scheme lies in its ability to use large time steps, and the efficiency of the double sweep solver that makes the method very robust in practical applications.

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The Ship Environments

Eric C. Tupper BSc, CEng, RCNC, FRINA, WhSch , in Introduction to Naval Architecture (Fifth Edition), 2013

The Sinusoidal Wave

Trochoidal waveforms are difficult to manipulate mathematically and irregular waves are analysed for their sinusoidal components. Taking the x-axis in the still water surface, the same as the mid-height of the wave, and z-axis vertically down, the wave surface height at x and time t can be written as (Figure 9.5):

Figure 9.5. Profile of sinusoidal wave.

$z=\frac{H\sin (qx+\omega t)}{2}$

In this equation q=2π/λ is termed the wave number and ω=2π/T is known as the wave frequency. T is the wave period. The principal characteristics of the wave, including the wave velocity, C, are

$C=\frac{\lambda }{T}=\frac{\omega }{q};T^2=\frac{2\pi \lambda }{g};{\omega }^2=\frac{2\pi g}{\lambda }\mathrm{and}{C}^2=\frac{g\lambda }{2\pi }$

As with trochoidal waves water particles in the wave move in circular orbits, the radii of which decrease with depth in accordance with

$r=\frac{1}{2}H\exp (–qz)$

From this it is seen that for depth λ/2 the orbit radius is only 0.02H which can normally be ignored.

The average total energy per unit area of wave system is ρgH ²/8, the potential and kinetic energies each being half of this figure. The energy of the wave system is transmitted at half the speed of advance of the waves. The front of the wave system moves at the speed of energy transmission so the component waves, travelling at twice this speed, will 'disappear' through the wave front.

For more information on sinusoidal waves, including proofs of the above relationships, the reader should refer to a standard text on hydrodynamics.

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Signal processing and feature extraction

Yaguo Lei , in Intelligent Fault Diagnosis and Remaining Useful Life Prediction of Rotating Machinery, 2017

2.4.1.1 Fourier Series

The basic concept of FT is to decompose a signal as the summation of several sinusoidal components and almost all signals are decomposed in this way. This Fourier's original analysis, namely Fourier series, is applied to process the signals with finite length. In mechanical vibration analysis, it is used to process periodic signals which are generated by a machine rotating at a constant speed. Thus, for any periodic signal x(t) with the period T it satisfies

(2.16) $x\left(t\right)=x\left(t+nT\right)$

where n is an integer, it can be expanded as

(2.17) $x\left(t\right)=\frac{a_0}{2}+\sum _{k=1}^{\infty }{a}_k\cos \left(k{\omega }_{0}t\right)+\sum _{k=1}^{\infty }{b}_k\sin \left(k{\omega }_{0}t\right)$

where ω ₀ denotes the fundamental angular frequency. The coefficients of the cosine and sine components can be described as

(2.18) $a_k=\frac{2}{T}{\int }_{-T/2}^{T/2}x\left(t\right)\cos \left(k{\omega }_{0}t\right)dt$

(2.19) $b_k=\frac{2}{T}{\int }_{-T/2}^{T/2}x\left(t\right)\sin \left(k{\omega }_{0}t\right)dt$

Therefore, a given periodic signal can be decomposed into sine and cosine terms dependent on an arbitrary time assignment of zero time, but all frequency components at frequency ω _k (=kω ₀) are given by

(2.20) $a_k\cos \left({\omega }_{k}t\right)+b_k\sin \left({\omega }_{k}t\right)$

which is also written as

(2.21) $C_k\cos \left({\omega }_{k}t+{\phi }_k\right)$

where $C_k=\sqrt{a_k^2+b_k^2}$ and φ _k  = tan^–1(b _k /a _k ). Eq. (2.21) makes it clearer that the sinusoid component possesses constant amplitude and phase angle at the arbitrarily defined zero time. Here, zero time only affects the initial phase φ _k . Eq. (2.21) can also be expressed in an exponential form

(2.22) $\frac{C_k}{2}\left\{\exp \left[\text{j}\left({\omega }_{k}t+{\phi }_k\right)\right]+\exp \left[-\text{j}\left({\omega }_{k}t+{\phi }_k\right)\right]\right\}$

which is illustrated as two rotating vectors with length C _k /2, one rotating at angular frequency ω _k with initial phase φ _k and the other rotating at angular frequency –ω _k with initial phase –φ _k , as illustrated in Fig. 2.10 (Randall, 1987).

Figure 2.10. Representation of a sinusoid component as a sum of two rotating vectors.

(A) A sinusoid component, (B) corresponding two rotating vectors.

According to this illustration of Fourier analysis, x(t) can be expressed as a sum of two rotating vectors and further the Eq. (2.17) is replaced as

(2.23) $x\left(t\right)=\sum _{k=-\infty }^{\infty }{A}_k\exp \left(\text{j}{\omega }_{k}t\right)$

where the coefficient A _k including the phase shift is

(2.24) $A_k=\frac{C_k}{2}\exp \left(\text{j}{\phi }_k\right)$

In addition, the coefficient A _k can also be calculated by

(2.25) $A_k=\frac{1}{T}{\int }_{-T/2}^{T/2}x\left(t\right)\exp \left(-\text{j}{\omega }_{k}t\right)dt$

Its physical implication is that the multiplication by exp(–jω _k t) would subtract angular frequency ω _k from each component. It suggests that the one originally rotating at ω _k would be stopped in the position with zero time, which can be extracted by the integral. However, all other components still rotate at some other positive or negative multiples of ω _k . Each frequency component A _k , denotes the position and value of the corresponding rotating vector at zero time. To obtain its position at an arbitrary time t, we can make it rotate at angular frequency ω _k by multiplying an exponential term exp(jω _k t) and then all frequency components are summed via Eq. (2.23).

Fourier series is interpreted as a sum of rotating vectors with each of amplitude half C _k /2, which produces a two-sided spectrum because the component in each positive frequency is accompanied by its complex conjugate at negative frequency. The same result can also be achieved by retaining the positive frequency components only, but doubling their length to C _k and then taking the projection of each vector on real axis as illustrated in Fig. 2.11B (Randall, 2011). A signal with a one-sided spectrum like this is complex, for example, an analytic signal, but later it is interpreted that the projection on the imaginary axis is the Hilbert transform (HT) of the real part.

Figure 2.11. (A) Equivalence of the vector sum of positive and negative frequency components and (B) projection on the real axis of a positive frequency component.

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Basics of Imaging Theory and Statistics

CHIEN-MIN KAO , ... XIAOCHUAN PAN , in Emission Tomography, 2004

B Discrete Fourier Transform

When working with a sampled image, it is useful to make use of a discrete counterpart to the Fourier transform known as the discrete Fourier transform (DFT). In the DFT, a finite 2D array

(41) $\begin{array}{c}{f}_{mn},m=0,\mathrm{\dots },M-1,\\ n=0,\mathrm{\dots },N-1\end{array}$

is considered to constitute one period of a periodic 2D array. (Without loss of generality, we assume that both M and N are even numbers in the following discussion.) The DFT of this periodic array is defined as

(42) $F_{mn}=\frac{1}{\sqrt{MN}}\underset{m^{\prime }=0}{\overset{M-1}{\Sigma }}\underset{n^{\prime }=0}{\overset{N-1}{\Sigma }}{f}_{m^{\prime }{n}^{\prime }}{e}^{-j2\pi m{m}^{\prime }/M}{e}^{-j2\pi n{n}^{\prime }/N}$

It is not difficult to see that F_mn is also a periodic 2D array with the same period as f_mn. Therefore, it suffices to compute F_mn for one period and by convention the period given by 0≤ m ≤ M-1 and 0≤ n ≤ N-1 is typically used. The inverse DFT is given by

(43) $f_{mn}=\frac{1}{\sqrt{MN}}\underset{m^{\prime }=0}{\overset{M-1}{\Sigma }}\underset{n^{\prime }=0}{\overset{N-1}{\Sigma }}{F}_{m^{\prime }{n}^{\prime }}{e}^{j2\pi m{m}^{\prime }/M}{e}^{-j2\pi n{n}^{\prime }/N}$

As in the continuous case, this equation states that one can decompose a 2D finite (or periodic) image array f_mn into sinusoidal components exp{j2π mm′/M} exp{j2πnn′/N} and the discrete Fourier transform of f_mn , denoted by F_mn , gives the amplitude in this decomposition. Therefore, one can also speak of a finite (or periodic) image array and its spectrum interchangeably. ¹

(44) $F_{mn}=\underset{m^{\prime }=0}{\overset{M-1}{\Sigma }}\underset{n^{\prime }=0}{\overset{N-1}{\Sigma }}{f}_{m^{\prime }{n}^{\prime }}{e}^{-j2\pi m{m}^{\prime }/M}{e}^{-j2\pi n{n}^{\prime }/N}$

(45) $f_{mn}=\frac{1}{MN}\underset{m^{\prime }=0}{\overset{M-1}{\Sigma }}\underset{n^{\prime }=0}{\overset{N-1}{\Sigma }}{F}_{m^{\prime }{n}^{\prime }}{e}^{j2\pi m{m}^{\prime }/M}{e}^{j2\pi n{n}^{\prime }/N}$

It is noted that, because of the periodic structure, larger values of the subscript indices m and n on F_mn do not indicate higher frequency components. In fact, because exp{j2πmm′/M} = exp{j2π(m - M)m′/M}, the DFT component F_mn with M/2 < m < M is actually equal to F_m′n with m′ = m - M, which is negative and smaller than m in magnitude. The same consideration also applies to the n index. Therefore, we conclude that F_mn with m = M/2 and n = N/2 is the highest frequency component in the period given by 0≤m≤M-1 and 0 ≤ n ≤ N-1. It is interesting to note that this periodic structure of F_mn is quite similar to that of F_s(v_x,v_y), with the discrete indexes m = M/2 and n = N/2 in F_mn playing the role of v_x = v^{(s )} _x /2 and v_y = v ^(s) _y /2 in F_s(v_x,v_y). The relationship between F_mn and F_s(v_x,v_y) is made clear in the next section.

Many properties of the Fourier transform also hold for the discrete Fourier transform. For example, one can show that

(46) $\underset{m=0}{\overset{M-1}{\Sigma }}\underset{n=0}{\overset{N-1}{\Sigma }}{f}_{mn}{g}_{mn}^{*}=\underset{m=0}{\overset{M-1}{\Sigma }}\underset{n=0}{\overset{N-1}{\Sigma }}{F}_{mn}{G}_{mn}^{*}$

and in particular,

(47) $\underset{m=0}{\overset{M-1}{\Sigma }}\underset{n=0}{\overset{N-1}{\Sigma }}|f_{mn}{|}^2=\underset{m=0}{\overset{M-1}{\Sigma }}\underset{n=0}{\overset{N-1}{\Sigma }}|F_{mn}{|}^2$

Therefore, ${\left|F_{mn}\right|}^2$ can be interpreted as the energy spectrum of f_mn. In addition, the convolution theorem becomes

(48) $g_{mn}=h_{mn}*f_{mn}\iff G_{mn}=H_{mn}{F}_{mn}$

where G_mn, H_mn , and F_mn are the DFTs of g_mn, h_mn , and h_mn , respectively, and * denotes the circular convolution defined by

(49) $h_{mn}*f_{mn}=\underset{i=0}{\overset{M-1}{\Sigma }}\underset{j=0}{\overset{N-1}{\Sigma }}{h}_{ij}{f}_{(m-i)(n-j)}$

where, as previously mentioned, all arrays are implicitly assumed to be periodic. As in the continuous case, h_mn and H_mn are, respectively, the point spread function and the system transfer function of a discrete, linear, periodically shift-invariant system.

Finally, we note that DFTs for image arrays of special sizes can be computed by use of fast Fourier transform (FFT) algorithms (Press et al., 1992). These algorithms are computationally efficient, making frequency-space-based image operations practical.

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The Radar System – Technical Principles

Alan Bole , ... Andy Norris , in Radar and ARPA Manual (Third Edition), 2014

2.6.4.1 The Mixer Principle

It can be shown that if two sinusoidal signals of differing frequencies are mixed, the resultant complex signal consists of a number of sinusoidal components. One has a frequency which is equal to the difference between the two frequencies that were mixed and which is known as the beat frequency. Other sinusoidal components generated include the sum of the mixed frequencies, twice the higher frequency and twice the lower frequency, and numerous even higher frequency components. The basic concept is more correctly known as the heterodyne principle and the radar receiver is said to be of the superheterodyne type, commonly abbreviated to superhet receivers. The resultant signal is said to have been down-converted because the main mixed component within the process is at a lower frequency than the original. This can be considerably lower by a factor of many tens of seconds.

The principle is applied in the radar receiver by mixing the incoming signal, consisting of bursts of electromagnetic energy at the transmitted frequency, with a continuous low power RF signal generated by a device known as the local oscillator. The resulting waveform at the output of the mixer will contain, among others, the component whose frequency is equal to the difference between that of the transmitted signal and that of the local oscillator. This signal is used as the input to the IF amplifier. This amplifier is designed so that it will respond only to that component of the mixer output which lies at the chosen 'beat' frequency. As a result, all the other higher frequency components generated by the mixing process are rejected. Thus the IF section of the receiver deals with pulses whose envelope resembles the shape originally imparted by the transmitter, as modified by targets, and which encloses bursts of oscillations at a frequency which is sufficiently low to be easily amplified and further processed.

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Dynamic Modelling of Vehicle Suspension

James Balkwill , in Performance Vehicle Dynamics, 2018

8.2.3.3 Harmonic Road Profile

In the previous section, we approximated the road profile to be a constant amplitude sine wave of increasing frequency. Clearly, this must be incorrect. One would not be surprised to find a road surface had components with amplitudes of a metre or more if the wavelength of these components were in the hundreds of metres. If, however, the wavelength was just a few metres, then an amplitude of a metre would be more of a low wall than a bump. Thus, it is clear that real roads have amplitudes that depend on their wavelengths. Long are large, short are small. But what does a real road profile actually look like? We need to answer this first and then develop a profile that is representative. It would be unwieldy to have to use input profiles lasting as long as an entire journey so we are seeking a representative wave of acceptably short duration to use as an input to the SDOF. This process starts with the profile of real roads.

The earliest known contracts involving the levelness of a road date back to the Roman Empire where sticks were used to measure the local road surface height relative to a long planks on the road surface. This is the idea behind the early wheeled profilometer shown in Fig. 8.15.

Fig. 8.15. An early design of profilometer.

Clearly, methods such as this have limitations; the whole vehicle will rise and fall a little as is moves forwards. It is also restricted to recording vibrations of wavelengths rather less than the vehicle itself. The modern approach is to measure vertical acceleration at the upright of a test vehicle and then, by using a validated dynamic model of the suspension, determine what road profile must have been experienced in order to produce the acceleration log recorded.

A possible road profile is shown in Fig. 8.16 above, in which the vertical axis is in millimetres and the horizontal one in metres. For the purposes of our analysis, we seek a wave that accurately represents the one shown but is practical and simple function. Ideally, it should have an analytical wave form of acceptably short duration that it can be used for practical tests on a four-post test rig where it is possible compare theoretical estimates of expected suspension performance with actual measured values.

Fig. 8.16. Example road profile.

A function is needed that is swept through the frequency domain. A car travelling over a road surface receives inputs that simulate its suspension in the range 0.5–40   Hz. Much below this range and the body simply moves up and down with the road surface with negligible movement in the suspension. Much above this range and the tyre simply accommodates all the movement without communicating it to the upright at all.

Given the earlier comments on how the longer wavelengths must have larger amplitudes, we should expect our function that represents the road surface to look something like

The function shown in Fig. 8.17 is of a sinusoidal wave in which the amplitude decreases exponentially and the frequency increases quadratically with advancing time. It is a guess for a waveform that represents the road. To evaluate whether the waveform does indeed represent the road, we shall make use of the 'power spectral density' concept.

Fig. 8.17. Expected shape for road profile approximation.

Power Spectral Density

One approach familiar to many engineers is that of Fourier analysis. In this technique, one can decompose a waveform into its component sinusoidal components. For example, if we have a waveform consisting of just three sinusoidal components and three different frequencies, then in the time domain, we would see a wavy graph containing the three components. In the frequency domain, after using a Fourier transform, we would have an output line that was zero except at the three constituent frequencies, where there would be a peak with a height corresponding to the amplitude of that wave in the original waveform at that frequency.

We could use this technique to assess whether a proposed road profile is representative of the real road by taking the Fourier transform of the road and of the approximation, and if the resulting transforms are very similar, then the approximation must be valid since they both contain equal quantities of each component, even if their overall shapes are very different.

This idea works very well for waveforms that consist of a large number of oscillations at any given frequency. However, we are seeking a waveform that of its very nature will have a rapid frequency sweep; we don't want the wave to be any longer than necessary.

Decomposing a swept frequency wave into its frequency components is problematic mathematically. Consider a function such as that in Fig. 8.17. If we now try to estimate the amplitude of the component that has a frequency of exactly 0.3   Hz, there clearly must be a component at this frequency since at times between 0 and 10   s just one wave is completed (much less than 0.3   Hz), yet between 40 and 50   s, around 10 cycles are completed (much more than 0.3   Hz). However, what would be the amplitude of the frequency component at exactly 0.3   Hz? If we were to find the point at which the waveform was at exactly 0.3   Hz, there would still be a problem because within just the one cycle, the frequency would have increased. Each individual cycle is faster at the end than the start so perhaps in theory at least, the amplitude of any given precise frequency is zero! Instead, we need to be able to consider frequencies that lie within a given range and not at a given value. To do this, the method of power spectral density is used.

Fig. 8.18 above shows the power spectral density for two roads, 'A' and 'B'. We interpret the graph in the following way. We start by choosing a frequency interval, for example, between 1 and 2   Hz, as shown. The amplitude of the wavelengths between these two frequencies is given by the square root of the area of the rectangle shown. This result comes from lengthy treatments of frequency analysis not reproduced here. Instead, it is quoted here because of its importance. It allows us to move to a waveform that faithfully represents the road without actually being the road but instead a much simpler and easy to manipulate function.

Fig. 8.18. Graph based on ISO 0608, road PSD versus wave number.

In Fig. 8.19 below, a waveform that offers a very good approximation to a typical road surface is shown. It is a good approximation in the sense that when its PSD is compared with that of a real road, they agree well.

Fig. 8.19. Constant peak velocity sinusoidal approximation to a road surface.

The input amplitude waveform consists of a sinusoidal wave of constant peak velocity. Such a signal is easy to generate analytically. A simple version of the velocity waveform is given by a sinusoidal waveform of the following form:

$\stackrel{.}{x}\left(t\right)=sin\left\{k_1\frac{\left(k_2^T-1\right)}{Ln\left(k_2\right)}\right\}$

where

$\begin{array}{l}{k}_1=2\pi f_0\\ k_2=1.087629\end{array}$

and f ₀ is the frequency at which the sweep starts.

A very marked difference between this waveform and the swept sine one used earlier is that in this case, the velocity has a constant peak value. Thus, the damper will be stroked at the same maximum value at every frequency. In the case of the swept sine wave of constant amplitude, the peak velocity raises with frequency. Here, it is constant and takes a value of around 80   mm/s. The corresponding input amplitude starts at around 40   mm.

We may finally modify the wave a little to make it more practical, particularly for use with a four-post test rig.

Fig. 8.20 shows a well-prepared four-post rig input waveform. The wave starts with a dwell at zero, and the oscillatory component builds up rapidly from this. The first couple of cycles are the same amplitude, and this is enough to get the vehicle through the transient associated with the rapid initial evolution of the amplitude. After this, two important factors are at work. The first is that the amplitude decays. This decay is a specially selected function above and permits a wave whose duration is acceptably short, in this case, just 60   s. The second affect at work is that the frequency increases as described above. At around 58   s, the function is seen to have a fade out, so that the rig actuators can be guaranteed to be at rest when the input stops.

Fig. 8.20. A well-prepared four-post rig input function.

For a single-degree-of-freedom system, we need only model the vertical input applied directly to the 'tyre'. In a real vehicle, the situation is much more complex. Each wheel could move independently, and the vehicle be stimulated to vibrate in four modes, heave, pitch, roll and warp. We shall deal with these cases in due course.

Such a waveform cannot be fed into an SDoF model and then analysed analytically, as the resulting maths is simply insoluble. Instead, we shall have to make use of numerical computing to solve the suspension modelling problems from now on. This was an inevitable occurrence as we move towards two degrees of freedom and above, the equations are bound to become too unwieldy to use at some point. Computer packages able to do this sort of analysis easily include the multibody codes such as Adams, the symbolic modellers such as MatLab's Simulink or even one's own spreadsheets using a numerical approach rather than an analytical one.

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Optical performance monitoring based on RF pilot tones

Paul K.J. Park , Yun C. Chung , in Optical Performance Monitoring, 2010

9.2.1 Operating principle

Figure 9.1 shows that an optical signal is generated from node A and then transmitted to node C via node B. A pilot tone (i.e., a small sinusoidal component) is added to the optical signal at node A. At node B, this pilot tone can be extracted by using a low-speed photodiode and used for optical performance monitoring. It should be noted that in a dynamically configurable WDM network, pilot tones can also be used to monitor the optical paths of a WDM signal. This is because the pilot tone is bound to follow the corresponding WDM signal to wherever in the network once the tone is attached. Thus, we can monitor the optical path of each WDM signal only by tracking the tone frequency.

Figure 9.1. Pilot-tone-based optical performance monitoring technique. ²

For using the pilot tone in practical systems, it is necessary that pilot tones should be added into and extracted from WDM signals anywhere in the network. Figure 9.2 shows typical techniques used for the generation and detection of pilot tones. ² A pilot tone could be generated by dithering the laser's bias current (Figure 9.2(a)), ⁵ the bias voltage of the amplitude modulator (Figure 9.2(b)), ⁹ or the phase modulator (Figure 9.2(c)). ¹⁰ These techniques would require a slight modification of the existing transmitter and help in suppressing the stimulated Brillouin scattering. ¹⁴ For the detection of pilot tones, a technique based on the fast Fourier transform (FFT) can be utilized as shown in Figure 9.2(d). ⁵ This technique is attractive since every pilot tone added to WDM signals can be detected simultaneously without any scanning mechanism. In case of ~GHz tone frequency, a tunable electrical bandpass filter or a tunable local oscillator can be used for the tone detection, as shown in Figures 9.2(e) and (f), respectively.

Figure 9.2. Pilot-tone generation and detection methods. (a) Adding a small sinusoidal current to the laser's bias current. (b) Dithering bias voltage of external modulator. (c) PM tone generation by using phase modulator. (d) Pilot-tone detection using FFT. (e) Using tunable electrical bandpass filter. (f) Using tunable local oscillator for the down-conversion of tone frequency. LD, laser diode; AM, amplitude modulator; PM, phase modulator; PD, photodetector; A/D, analog-to-digital converter; FFT, fast Fourier transform; BPF, tunable bandpass filter; RFD, radio frequency power detector; LOSC, tunable local oscillator. ²

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Source: https://www.sciencedirect.com/topics/engineering/sinusoidal-component

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