#### Fourier Analysis and Transforms

Spectrum analysis, also referred to as frequency domain analysis or spectral density estimation, is the technical process of decomposing a complex signal into simpler parts.  As described, many physical processes are best described as a sum of many individual frequency components.  Any process that quantifies the various amounts (e.g. amplitudes, powers, intensities, or phases), versus frequency can be called spectrum analysis.

Spectrum analysis can be performed on the entire signal.  Alternatively, a signal can be broken into short segments (sometimes called frames), and spectrum analysis may be applied to these individual segments. One of the analysis methods used commonly with hearing aid designers is Fourier analysis and its resultant Fourier transform.

Fourier analysis is named for Joseph Fourier (Jean Baptiste Joseph Fourier), a French mathematician and physicist who lived between 1768 and 1830.  He showed that any repetitive waveform can be broken down into a series of sine waves of appropriate amplitudes and phases.

Typical everyday sounds are said to be complex because they contain a range of frequency components.  This is true of individual speech sounds and individual notes in music as well as various clatters, beeps, thumps, etc., that populate our auditory world.  When components at higher frequencies are removed from such sounds they can seem muffled, or if lower frequencies are missing they can sound tinny.  Individually, a single frequency component from a complex sound is a pure tone with a particular frequency and amplitude.  Therefore, different complex sounds can be obtained by adding together pure tones with different frequencies and amplitudes.  One complex sound is shown in Figure 1, along with the three individual pure tones that constitute this sample.  Adding together the pressures of each of the pure-tone components at each point in time forms the waveform of the complex sound.

##### Fourier Analysis

Any time-varying (complex) sound can be analyzed to find its frequency components.  Additionally, any complex wave can be constructed by adding together sine waves of appropriate frequency, amplitude, and phase.  Fourier analysis is a technique that is used to determine which sine waves constitute a given signal, i.e., to deconstruct the signal into its individual sine waves.  The result is expressed as sine wave amplitude as a function of frequency (Figure 1).  The process of decomposing a periodic function into its constituent sine or cosine waves is called Fourier analysis.  (I recall having to perform Fourier analysis with pencil and paper mathematically during my doctoral studies in a course titled “Mathematics for Hearing Scientists” – not fun.  We must have had a masochist for an instructor, especially when it could have been done rapidly and simply electronically.  But, I guess that he wanted us to learn what it was really about).

Figure 1. Fourier analysis, illustrating that any complex wave form can be shown to consist of a series of individual sine waves.

In the processing of audio signals (although it can be used for radio waves, light waves, seismic waves, and even images), Fourier analysis can isolate individual components of a continuous complex waveform, and concentrate them for easier detection and/or removal.  This operation is often used in contemporary digital hearing aids for noise cancellation, equalization, feedback elimination, etc.

Fourier Transform

As we know that sound is just a combination of sine functions, we can assign each sine-function, and therefore the original sound, to a distinct energy or power spectrum which gives us the energy/amplitude per frequency.  This process is called Fourier transform.  Thus, we gain a description of sound as a series of energies at distinct frequencies (Figure 2).

Figure 2. Power spectra of simple and complex periodic sounds are shown on the right along with their waveforms on the left. These are called periodic sounds because their waveforms repeat. Their spectra are vertical lines whose left-right position represents the frequency of a component (pure tone), while the line’s height represents the power of a component in amplitude (in dB).

In signal processing, the Fourier transform commonly converts/transforms a continuous time domain representation (i.e., speech sample) and maps it into a frequency domain representation (known as the frequency spectrum), and vice versa. That is, it takes a function from the time domain into the frequency domain.  It is a decomposition of a function into sinusoids of different frequencies (Fourier analysis).  In Fourier analysis each value of the function is usually expressed as a complex number that can be interpreted as having a magnitude and a phase component.  The term “Fourier transform” refers to both the transform operation and to the complex-valued function it produces.  When it produces a continuous function of frequency, it is known as a frequency distribution, as shown in Figure 3.  One function is transformed into another, and the operation is reversible.  In the case of a periodic function (musical tone; speech), but not necessarily sinusoidal, the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients.  In the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.

Figure 3. Illustration of the Fourier transform. This action converts/transforms a continuous time domain representation into a frequency domain frequency spectrum.

FFT

In practice, nearly all software and electronic devices that generate frequency spectra apply a fast Fourier transform (FFT), which is a specific mathematical approximation to the full integral solution.  Formally stated, the FFT is a method for computing the discrete Fourier transform of a sampled signal.  It is essentially an efficient algorithm to compute the Discrete Fourier Transform (DFT) and its inverse.  Many distinct FFT algorithms exist, involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory.  These are beyond the depth of this blog.

For certain signals, a Fourier transform can be performed analytically with calculus.  For arbitrary signals, the signal must first be digitized, and a Discrete Fourier Transform (DFT) performed.  The standard numerical algorithm used for the DFT is called the Fast Fourier Transform (FFT) or Discrete FFT (DFFT).  Due to limitations inherent in digitization and numerical algorithms, the FFT will result in an approximation to the spectrum.

FFT algorithms are so commonly employed to compute DFTs that the term “FFT” is often used to mean “DFT” in colloquial settings.  Formally, there is a clear distinction.  DFT refers to a mathematical transformation or function, regardless of how it is computed, whereas FFT refers to a specific family of algorithms for computing DFTs.  The FFT has been described as the most important numerical algorithm of our lifetime.

Summary

For those in the discipline of hearing, one important application of the FFT is for the analysis of sound.  It is important to assess the frequency distribution of the power in a sound because the human ear exercises that capacity in the hearing process.

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