Sum-of-sines

The quantity x k) in Equation (8.20) is the experimentally observed absorption, like that in Figure 8.32, after subtraction of the smoothly declining background. What is left is a sum of sine waves of which we require the wavelengths which can be related to Rj, provided the phase factor 6j k) is known. This process of obtaining wavelengths from a superposition of... 

The shorter the wavelength, the faster its decay. Mineral scale heterogeneities in rocks disappear long before meter-scale or even larger heterogeneities. This concept can be extended to any arbitrary combination of periodic functions in Section 2.6, we have already met the idea that any function bounded over an interval can be expanded as a sum of sine and cosine functions. Shorter wavelengths will decay much faster... 

The trouble is now that the source term does not include the sum of sines, so we will use a trick resting on the Leibniz s rule for differentiating integrals. A particular solution of the diffusion equation with radiogenic accumulation is... 

Figure 20-24 A curve to be decomposed into a sum of sine and cosine terms by Fourier analysis.

Fourier analysis is a procedure in which a curve is decomposed into a sum of sine and cosine terms, called a Fourier series. To analyze the curve in Figure 20-24, which spans the interval xx = 0 to x2 = 10, the Fourier series has the form. 

Equation 20-10 says that the value of y for any value of x can be expressed by an infinite sum of sine and cosine waves. Successive terms correspond to waves with increasing frequency. 

Fourier series Infinite sum of sine and cosine terms that add to give a particular function in a particular interval. 

I will write Fourier series in this form throughout the remainder of the book. This kind of equation is compact and handy, but quite opaque at first encounter. Take the time now to look at this equation carefully and think about what it represents. Whenever you see an equation like this, just remember that it is a Fourier series, a sum of sine and cosine wave equations, with the full sum representing some complicated wave. The hth term in the series, Fh 1 ni hx, can be expanded to Fj/cos 2ir(hx) + i sin 2-tt(/ )], making plain that the hth term is a simple wave of amplitude Fh, frequency h, and implicit phase [Pg.88]

A similar analysis can be made for quasi-periodic signals which consist of a sum of sine waves with slowly-varying amplitude and instantaneous frequency each of which is assumed to pass through a single filter. 

In addition to FM signals with nested modulators, Justice also considered a class of signals modeled by a harmonic sum of sine waves with a slowly varying amplitude, typical of many speech and music sounds... 

It has been assumed that AM and FM control functions can be selected, based on experience and musical knowledge, to create a variety of instrumental-like sounds with a specific timbre. Nevertheless, a more formal approach is desired in the AM-FM analysis and synthesis. Justice [Justice, 1979] addressed the problem of finding an analysis technique which can yield the parameters of a given FM signal he also investigated the use of the FM synthesis model in representing a signal which consists of a sum of sine waves. 

The electrical output signal from a conventional scanning spectrometer usually takes the form of an amplitude-time response, e.g. absorbance vs. wavelength. All such signals, no matter how complex, may be represented as a sum of sine and cosine waves. The continuous fimction of composite frequencies is called a Fourier integral. The conversion of amplitude-time, t, information into amplitude-frequency, w, information is known as a Fourier transformation. The relation between the two forms is given by... 

In general, the perturbing signal may have an arbitrary form. However, in practice, the most often used perturbation signals are ° (l) pulse, (2) noise, and (3) sum of sine waves. 

Figure 9. FFT analysis of the sum of sine wave perturbation left side, no optimization right side, optimization of phases, (a) Perturbation voltage in the time domain, (b) Perturbation voltage in the frequency domain, (c) Complex plane plots of simulated impedance spectra with 5% noise added to the current response. Solid lines show response without noise.

Any mathematical function that repeats in a regular periodic manner can be represented as the sum of sine and cosine functions of appropriate amplitudes and phases. The periodicities of the terms are the appropriate integral fractions of the... 

Since the wave functions are band limited, Eq. (23) equals the sum of sine functions i >(g) = 2" 2-1) iKnA<7)sinc[A max(<7 — Aq)]. The relation is a consequence of the fact that the sine function is the Fourier transform of a band (rectangle). See also Eq. (18). 

Fig. 1.13 In a Fourier series a periodic Junction is expressed as a sum of sine and cosine functions...

Fourier analysis permits any continuous curve, such as a complex spectmm of intensity peaks and valleys as a function of wavelength or frequency, to be expressed as a sum of sine or cosine waves varying with time. Conversely, if the data can be acquired as the equivalent sum of these sine and cosine waves, it can be Fourier transformed into the spectrum curve. This requires data acquisition in digital form, substantial computing power, and efficient software algorithms, all now readily available at the level of current generation personal computers. The computerized instmments employing this approach are called FT spectrometers—FTIR, FTNMR, and FTMS instruments, for example. 

Harmonic analysis, or Fourier analysis, is the decomposition of a periodic function into a sum of simple periodic components. In particular, Fourier series are expansions of periodic functions /(x) in terms of an infinite sum of sines and cosines of the form... 

Taylor series expansions, as described above, provide a very general method for representing a large class of mathematical functions. For the special case of periodic functions, a powerful alternative method is expansion in an infinite sum of sines and cosines, known as a trigonometric series or Fourier series. A periodic function is one that repeats in value when its argument is increased by multiples of a constant L, called the period or wavelength. For example. 

More complex periodic responses require the sums or differences of sines and cosines with different periods. Fourier series is a means to express any periodic response in terms of the sum of sine and cosine waves with fundamental and harmonic frequencies (Figure 4.3.4). 

FIGURE 4.3.4 A complex waveform expressed as the sum of sines and cosines. At the top is aramp function expressed as the sum (shown as a dotted line) of the fundamental, second, and third harmonics (shown individually as solid lines). At the bottom is the same function composed of ten harmonics. The more harmonics are used, the closer will be the representation to the actual waveform. 

The only waveform containing just one frequency is the sine wave. A periodic waveform can be created by a sum of sine waves, each being a harmonic component of the sine wave at the fundamental frequency determined by the period. This is illustrated in Figure 8.6(a), showing the sum of a fundamental and its third and fifth harmonic... 

The idea is to use a sum of sine waves with different predetermined frequencies for excitation. A multisine excitation signal Sexc(t) with frequencies fr to fr of k sine wave components can be expressed as... 

The simple physical interpretation of eqn [4] is that any arbitrary (not necessarily periodic ) function f t) can be expanded as an integral ( sums ) of sine and cosine functions, with F(v) interpreted as the amplitudes of the waves. The necessary amplitudes F(v) can be obtained from eqn [1], which thus represents the frequency analysis of the arbitrary function, f t). In other words, eqn [1] analyses the function f t) in terms of its frequency components and eqn [2] puts the components back together again to recreate the function. Notice that if f t) is an even function (i.e., f — t)=f t)), then the cosine transform suffices (i.e., only first term on the right hand side of eqn [4] need be retained), but this is rarely the case in practice. 

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