# Spectrogram

Typical spectrogram of the spoken words "nineteenth century". The lower frequencies are more dense because it is a male voice. The legend to the right shows that the color intensity increases with the density.

A spectrogram, or sonogram, is a visual representation of the spectrum of frequencies in a sound.[1] Spectrograms are sometimes called spectral waterfalls, voiceprints, or voicegrams.

Spectrograms can be used to identify spoken words phonetically, and to analyse the various calls of animals. They are used extensively in the development of the fields of music, sonar, radar, and speech processing,[2] seismology, etc.

The instrument that generates a spectrogram is called a spectrograph. The sample outputs on the right show a select block of frequencies going up the vertical axis, and time on the horizontal axis.

Spectrogram of the actual recording of this violin playing. Note the harmonics occurring at whole-number multiples of the fundamental frequency. Note the fourteen draws of the bow, and the visual differences in the tones.

## Format

3D surface spectrogram of a part from a music piece.

The most common format is a graph with two geometric dimensions: the horizontal axis represents time, the vertical axis is frequency; a third dimension indicating the amplitude of a particular frequency at a particular time is represented by the intensity or colour of each point in the image.

There are many variations of format: sometimes the vertical and horizontal axes are switched, so time runs up and down; sometimes the amplitude is represented as the height of a 3D surface instead of color or intensity. The frequency and amplitude axes can be either linear or logarithmic, depending on what the graph is being used for. Audio would usually be represented with a logarithmic amplitude axis (probably in decibels, or dB), and frequency would be linear to emphasize harmonic relationships, or logarithmic to emphasize musical, tonal relationships.

## Generation

Spectrogram of a male voice saying 'tatata'.
Spectrogram of an FM signal. In this case the signal frequency is modulated with a sinusoidal frequency vs. time profile

Spectrograms are usually created in one of two ways: approximated as a filterbank that results from a series of bandpass filters (this was the only way before the advent of modern digital signal processing), or calculated from the time signal using the short-time Fourier transform (STFT). These two methods actually form two different Time-Frequency Distributions, but are equivalent under some conditions.

The bandpass filters method usually uses analog processing to divide the input signal into frequency bands; the magnitude of each filter's output controls a transducer that records the spectrogram as an image on paper.[3]

Creating a spectrogram using the STFT is usually a digital process. Digitally sampled data, in the time domain, is broken up into chunks, which usually overlap, and Fourier transformed to calculate the magnitude of the frequency spectrum for each chunk. Each chunk then corresponds to a vertical line in the image; a measurement of magnitude versus frequency for a specific moment in time. The spectrums or time plots are then "laid side by side" to form the image or a three-dimensional surface.[4]

The spectrogram of a signal s(t) can be estimated by computing the squared magnitude of the STFT of the signal s(t), as follows:[5]

$\mathrm{spectrogram}(t,\omega)=\left|\mathrm{STFT}(t,\omega)\right|^2$

## Applications

Spectrogram of dolphin vocalizations; chirps, clicks and harmonizing are visible as inverted Vs, vertical lines and horizontal striations respectively
Great Tit : song
Spectrogram of Great Tit song
• Early analog spectrograms were applied to a wide range of areas including the study of bird calls, with current research continuing using modern digital equipment[6] and applied to all animal sounds. Contemporary use of the digital spectrogram is especially useful for studying frequency modulation (FM) in animal calls. Specifically, the distinguishing characteristics of FM chirps, broadband clicks, and social harmonizing are most easily visualized with the spectrogram.
• Spectrograms are useful in assisting in overcoming speech defects and in speech training for the portion of the population that is profoundly deaf[7]
• The studies of phonetics and speech synthesis are often facilitated through the use of spectrograms.[8][9]
• By reversing the process of producing a spectrogram, it is possible to create a signal whose spectrogram is an arbitrary image. This technique can be used to hide a picture in a piece of audio and has been employed by several electronic music artists.[10] See also steganography.
• Some modern music is created using spectrograms as an intermediate medium; changing the intensity of different frequencies over time, or even creating new ones, by drawing them and then inverse transforming. See Audio timescale-pitch modification and Phase vocoder.
• Spectrograms can be used to analyze the results of passing a test signal through a signal processor such as a filter in order to check its performance.[11]
• High definition spectrograms are used in the development of RF and microwave systems[12]
• Spectrograms are now used to display S-parameters measured with vector network analyzers[13]
• The US Geological Survey now provides real-time spectrogram displays from seismic stations[14]

## Limitations and resynthesis

From the formula above, it appears that a spectrogram contains no information about the exact phase of the signal that it represents. For this reason, it is not possible to reverse the process and generate a copy of the original signal from a spectrogram, though in situations where the exact initial phase is unimportant it may be possible to generate a useful approximation of the original signal. The Analysis & Resynthesis Sound Spectrograph [1] is an example of a computer program that attempts to do this. The Pattern Playback was an early speech synthesizer, designed at Haskins Laboratories in the late 1940s, that converted pictures of the acoustic patterns of speech (spectrograms) back into sound.

In fact, there is some phase information in the spectrogram, but it appears in another form, as time delay (or group delay) which is the dual of the Instantaneous Frequency; an experiment explaining and relating these two concepts is described in.[15]

## References

1. ^ Haykin, Simon (1991). Advances in Spectrum Analysis and Array Processing. Prentice-Hall. ISBN 0-13-007444-6.
2. ^ JL Flanagan, Speech Analysis, Synthesis and Perception, Springer- Verlag, New York, 1972
3. ^ Illustration of an electro-mechanical spectrograph
4. ^ Spectrogram definition
5. ^ STFT spectrogram details
6. ^ Bird Songs and Spectrograms of Southern Tuscany
7. ^ A wearable tactile sensory aid for profoundly deaf children