Sine wave complex numbers

Some Algebraic Rules for Sine Wave Complex Numbers... 

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

Theoretically, the perturbation can be an arbitrary, more or less complex, function of time. However, only a limited number of functions have been shown to be of practical importance. These are known as step, pulse, double-step, double-pulse, periodic square wave and periodic sine wave. A survey of the most common techniques is found in Table 2. 

Figure 4.2b is a presentation of the FID of the decoupled 13C NMR spectrum of cholesterol. Figure 4.2c is an expanded, small section of the FID from Figure 4.2b. The complex FID is the result of a number of overlapping sine-waves and interfering (beat) patterns. A series of repetitive pulses, signal acquisitions, and relaxation delays builds the signal. Fourier transform by the computer converts the accumulated FID (a time domain spectrum) to the decoupled, frequency-domain spectrum of cholesterol (at 150.9 MHz in CDC13). See Figure 4.1b.

The raw data or FID is a series of intensity values collected as a function of time time-domain data. A single proton signal, for example, would give a simple sine wave in time with a particular frequency corresponding to the chemical shift of that proton. This signal dies out gradually as the protons recover from the pulse and relax. To convert this time-domain data into a spectrum, we perform a mathematical calculation called the Fourier transform (FT), which essentially looks at the sine wave and analyzes it to determine the frequency. This frequency then appears as a peak in the spectrum, which is a plot in frequency domain of the same data (Fig. 3.27). If there are many different types of protons with different chemical shifts, the FID will be a complex sum of a number of decaying sine waves with different frequencies and amplitudes. The FT extracts the information about each of the frequencies ... 

Almost all molecular systems of interest give rise to many NMR frequencies, so that FIDs are typically complex interference patterns of a number of sine waves of differing amplitude and often of differing decay constants, T2. The unraveling of these patterns and the display of amplitude as a function of frequency (the spectrum) are usually carried out by a Fourier transformation. 

The mathematics of treating sine waves is simplified by the introduction of imaginary numbers strictly G is complex, G = G + iG". 

For a linear system, the current response will be a sine wave of the same frequency as the excitation signal, but shifted in phase. Since the impedance is the ratio of two sine waves, it is a complex number that can be represented by an amplitude and a phase shift or as the sum of real and imaginary components, Z(co) = Z ( )) + jZ (co). 

In addition to vectors in the space domain, vectors may also be defined in the time domain, in particular rotating vectors with a fixed initial point at the origin of a Cartesian coordinate system. A variable such as the electric field strength may be a vector both in time and space. Every space vector may also be a time vector, and often it is not clear what sort of vector an author actually is dealing with. Vectors in the time domain are used for sine waves when the maxima do not occur simultaneously. These two-dimensional (planar) time vectors are more conveniently represented by complex numbers. 

When a complex number is multiplied by j, the phase angle is increased by 90° (7r/2). When a complex number is divided by j, the phase angle is decreased by 90° (7r/2). di/dt = ijw (i is complex sine wave). 

The simple sine waves used for illustration reveal their periodicity very clearly. Normal sounds, however, are much more complex, being combinations of several such pure tones of different frequencies and perhaps additional transient sound components that punctuate the more sustained elements. For example, speech is a mixture of approximately periodic vowel sounds and staccato consonant sounds. Complex sounds can also be periodic the repeated wave pattern is just more intricate, as is shown in Fig. 1.105(a). The period identified as Ti appHes to the fundamental frequency of the sound wave, the component that normally is related to the characteristic pitch of the sound. Higher-frequency components of the complex wave are also periodic, but because they are typically lower in amplitude, that aspect tends to be disguised in the summation of several such components of different frequency. If, however, the sound wave were analyzed, or broken down into its constituent parts, a different picture emerges Fig. 1.105(b), (c), and (d). In this example, the analysis shows that the components are all harmonics, or whole-number multiples, of the fundamental frequency the higher-frequency components all have multiples of entire cycles within the period of the fundamental. 

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