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The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency. [1]


Any signal that can be represented as an amplitude that varies with time has a corresponding frequency spectrum. This includes familiar concepts such as visible light (color), musical notes, radio/TV channels, and even the regular rotation of the earth. When these physical phenomena are represented in the form of a frequency spectrum, certain physical descriptions of their internal processes become much simpler. Often, the frequency spectrum clearly shows harmonics, visible as distinct spikes or lines, that provide insight into the mechanisms that generate the entire signal.

Electromagnetic emission spectrum of Iron in the visible region.

A source of light can have many colors mixed together and in different amounts (intensities). A rainbow, or prism, sends the different frequencies in different directions, making them individually visible at different angles. A graph of the intensity plotted against the frequency (showing the amount of each color) is the frequency spectrum of the light. When all the visible frequencies are present in equal amounts, the effect is the "color" white, and the spectrum is a flat line. Therefore, flat-line spectrums in general are often referred to as white, whether they represent light or something else.

Similarly, a source of sound can have many different frequencies mixed together. Each frequency stimulates a different length receptor in our ears. When only one length is predominantly stimulated, we hear a note. A steady hissing sound or a sudden crash stimulates all the receptors, so we say that it contains some amounts of all frequencies in our audible range. Things in our environment that we refer to as noise often comprise many different frequencies. Therefore, when the sound spectrum is flat, it is called white noise. This term carries over into other types of spectrums than sound.

Each broadcast radio and TV station transmits a wave on an assigned frequency domain (aka channel). A radio antenna adds them all together into a single function of amplitude (voltage) vs. time. The radio tuner picks out one channel at a time (like each of the receptors in our ears). Some channels are stronger than others. If we made a graph of the strength of each channel vs. the frequency of the tuner, it would be the frequency spectrum of the antenna signal.

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Spectrum analysis

Example of voice waveform and its frequency spectrum
A triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental frequency component is at 220 Hz (A2).

Spectrum analysis is the technical process of decomposing a complex signal into simpler parts.Analysis means decomposing something complex into simpler, more basic parts. As described above, 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, or intensities), 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. Periodic functions (such as sin(t)) are particularly well-suited for this sub-division, but it can be applied to any deterministic function in the form of an integral.

The Fourier transform of a function produces a frequency spectrum which contains all of the information about the original signal, but in a different form. This means that the original function can be completely reconstructed (synthesized) by an inverse Fourier transform. For perfect reconstruction, the spectrum analyzer must preserve both the amplitude and phase of each frequency component. These two pieces of information can be represented as a 2-dimensional vector, as a complex number, or as magnitude (amplitude) and phase in polar coordinates. When the amplitude is squared, the resulting plot is referred to as a power spectrum.

Because of reversibility, the Fourier transform is called a representation of the function, in terms of frequency instead of time, thus, it is a frequency domain representation. Linear operations that could be performed in the time domain have counterparts that can often be performed more easily in the frequency domain. Frequency analysis is also improves understanding and interpretation of the effects of various time-domain operations, both linear and non-linear. For instance, only non-linear operations can create new frequencies in the spectrum.

The Fourier transform of a stochastic (random) waveform (noise) is also random. Some kind of averaging is required in order to create a clear picture of the underlying frequency content (frequency distribution). Typically, the data is divided into time-segments of a chosen duration, and transforms are performed on each one. Then the magnitude or (usually) squared-magnitude components of the transforms are summed into an average transform. This is a very common operation performed on digitally sampled time-domain data, using the discrete Fourier transform. This type of processing is called Welch's method. When the result is flat, it is commonly referred to as white noise. However, such processing techniques often reveal spectral content even among data which appears noisy in the time domain.

Physical acoustics of music

Acoustic spectrogram of the note G played on a Piano. In this spectrogram, the vertical axis represents frequency linearly extending from 0 to 10 kHz, and the horizontal axis represents time over an interval of 1.5 seconds. Generated with Fatpigdog's PC based Real Time FFT Spectrum Analyzer. Click below to hear the G Piano Note:
68448 pinkyfinger Piano G.ogg

Sound spectrum is one of the determinants of the timbre or quality of a sound or note. It is the relative strength of pitches called harmonics and partials (collectively overtones) at various frequencies usually above the fundamental frequency, which is the actual note named (eg. an A).

The spectrum analyzer is an instrument which can be used to convert the sound wave of the musical note into a visual display of the constituent frequencies. This visual display is referred to as an acoustic spectrogram. Software based audio spectrum analyzers are available at low cost, providing easy access not only to industry professionals, but also to academicians, students and the lay hobbyist. The acoustic spectrogram generated by the spectrum analyzer provides an acoustic signature of the musical note. In addition to revealing the fundamental frequency and its overtones, the spectrogram is also useful for analysis of the temporal attack, decay, sustain, and release of the musical note.

References

  1. ^ Alexander, Charles; Sadiku, Matthew (2004). Fundamentals of Electric Circuits (Second ed.). McGraw-Hill. p. 761. ISBN 0-07-249350-X. http://www.amazon.com/Fundamentals-Electric-Circuits-Charles-Alexander/dp/007249350X. "The frequency spectrum of a signal consists of the plots of the amplitudes and phases of the harmonics versus frequency."  

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