Multiple-Frequency Fourier Analysis

Impedance transfer functions may be determined through use of an input signal containing more than a single frequency. Such signals may be a tailored multi-sine 

Remember 7.2 The input and output signals used to generate impedance spectra are functions of time, not frequency. Thefrequency dependence of the impedance results from the processing of time-domain signals. 

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Although other descriptions are possible, the mathematical concept that matches more closely the intuitive notion of smoothness is the frequency content of the function. Smooth functions are sluggish and coarse and characterized by very gradual changes on the value of the output as we scan the input space. This, in a Fourier analysis of the function, corresponds to high content of low frequencies. Furthermore, we expect the frequency content of the approximating function to vary with the position in the input space. Many functions contain high-frequency features dispersed in the input space that are very important to capture. The tool used to describe the function will have to support local features of multiple resolutions (variable frequencies) within the input space. 

In spite of the frequency-shifted excitation, the quantized PIP inevitably excites multiple sidebands located at n/At ( = 1, 2,...) from the centre band. An attempt was made16 to calculate the excitation profile of multiple bands created by a PIP of a constant RF field strength, using an approximate method based on the Fourier analysis. The accuracy of the method relies partly on the linear response of the spin system, which is, unfortunately, not true in most cases except for a small angle excitation. In addition, the spins inside a magnet consitute a quantum system, which is sensitive not only to the strengths but also to the phases of the RF fields. Any classical description is doomed to failure if the quantum nature of the spin system emerges. 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

But we also know that any repetitive waveform, of almost arbitrary shape, can be decomposed into a sum of several sine (and cosine) waveforms of frequencies that are multiples of the basic repetition frequency f ( the fundamental frequency ). That is what Fourier analysis is all about. Note that though we do get an infinite series of terms, it is... 

The use of FFT (fast Fourier transform) spectral analysis can be the perfect complement to the single sine analysis technique. The FFT can perform very fast measurements since it is not necessary to sweep the stimulus frequency in order to measure multiple frequencies. With the introduction of DSP technology, very fast... 

Square waves are made up of an infinite number of sine waves the fundamental frequency plus mostly odd harmonics. Fourier analysis shows that a square wave consists of the fundamental sine wave plus sine waves that are odd multiples of the fundamental. Figure 20.56 illustrates the mechanisms involved. The fundamental frequency contributes about 81.7% of the square wave. The third harmonic contributes about 9.02%, and the fifth harmonic about 3.24%. Higher harmonics contribute less to the shape of the square wave. As outlined here, approximately 94% of the square wave is derived from the fundamental and the third and fifth harmonics. Inaccuracies introduced by the instrument are typically 2% or less. The user s ability to read a conventional CRT display may contribute another 4% error. Thus, a total of 6% error may be introduced by the instrument and the operator. A 96% accurate reproduction of a square wave... 

In the present state-of-the-art equipment, it is possible to measure and plot the electrochemical impedance automatically. Electronic circuitry is designed to generate the frequency sweep of a desired resolution over the range of interest. The generator can be programmed to sweep from a maximum to a minimum frequency in a number of required frequency steps. The commonly used modem equipment AC impedance measurement techniques can be subdivided into two main groups - single sine (lock-in amplification and frequency response analysis) and multiple sine techniques such as fast Fourier transforms (FFT). 

Periodic waves are usually a superposition of sinusoidal components with a stronger one (the fundamental) and weaker ones (the harmonics), whose frequencies are multiples of that of the first. Each sinusoidal component is characterized by an amplitude A, a frequency v (or wavelength X = v/v, where v is the velocity of the wave, about 340 m/s for sound waves in standard atmosphere and 300,000 km/s for light in vacuum) and a phase ( ). In music, the frequency of the fundamental determines the pitch of the note, its amplitude, the strength or intensity of the note, and the harmonic pattern, the tone [21]. It results from Fourier analysis of sounds that the tempo of a piece and durations of the notes also play a role in the harmonic patterns. 

Spectral modelling techniques are the legacy of the Fourier analysis theory. Originally developed in the nineteenth century, Fourier analysis considers that a pitched sound is made up of various sinusoidal components, where the frequencies of higher components are integral multiples of the frequency of the lowest component. The pitch of a musical note is then assumed to be determined by the lowest component, normally referred to as the fundamental frequency. In this case, timbre is the result of the presence of specific components and their relative amplitudes, as if it were the result of a chord over a prominently loud fundamental with notes played at different volumes. Despite the fact that not all interesting musical sounds have a clear pitch and the pitch of a sound may not necessarily correspond to the lower component of its spectrum, Fourier analysis still constitutes one of the pillars of acoustics and music. 

The previous discussions of the signal are nicely illustrated by an extremely simple model analysis using real fields and signals for two Lorenzian resonances at frequencies a and b. The sample is irradiated with two very short pulses whose spectra are flat. The real generated field from the sample is the real part of Eq. (21) or Eq. (33) with T set equal to zero for convenience since is in any case a multiplicative factor. In time-domain interferometry, this is measured directly along the indicated time axes as described above. In spectral interferometry the real generated field along with a real local oscillator field, delayed by time d, is dispersed (i.e., Fourier-transformed) by a monochromator, then squared by the detection to yield a spectrum on the array detector at each value of t ... 

The modem methods of taking NMR spectra involve the use of very short radio frequency pulses (of variable duration from 1 to 200 ms) instead of a continuous signal as in older NMR. This requires full automation of the test, the Fourier transform analysis, data storage and multiple scan capability. With the scalar (low power, ca. 4 kHz) and dipolar (about 45 kHz) decoupling, magic angle spinning and cross polarization methods one can obtain spectra of solid samples with resolution similar to those known for liquids. The spectra provide precise information on the... 

An alternative numerical method for resolving complicated signals is to analyze the frequency spectrum of the curves. The so-called Fourier transform analysis (FT analysis) approximates the sum of sine and cosine functions for empirically generated signals. First, 2 m points of supports, spaced equidistantly, are chosen (Fig. 2-2). We recommend choosing a multiple of four for 2 m and using the values 12, 24, 36, 72,. .. 

New-generation spectrometers use a different and much faster recording technique, in which a pulse of electromagnetic radiation that covers the entire frequency range under santiny (NMR, IR, UV) is used to obtain the whole spectrum instantly. Moreover, rather than simple absorption, as in traditional CW instruments, the decay of the absorption event with time is recorded, a procedure that requires a more elaborate computer analysis, called Fourier transform (FT), after the French mathematician Joseph Fourier (1768-1830). Apart from the speed of this technique, multiple pulse accumulation of the same spectrum allows for much higher sensitivity, which is of great value when only small amounts of sample are available. 

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