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Acoustic phonetics studies speech as a physical sound wave. A microphone records a signal related to changing sound pressure; analysis asks how its amplitude, timing and frequency content vary. A waveform displays variation over time, whereas a spectrogram displays the distribution of signal energy across both time and frequency.

For an original comparison, record a sustained vowel followed by a sustained hiss. Inspect each waveform at the same time scale and compare their spectrograms. Approximately repeated vocal cycles can give a harmonic structure, while the hiss provides an example of a noisier excitation. Broad spectral concentrations associated with the vocal tract are formants; they should be distinguished from the fundamental frequency of a repeating source. The spectrogram's analysis-window duration affects which temporal and spectral details are visible.^[ref]

These are measurements of the signal, rather than direct readings of a listener's perceived pitch or a speaker's tongue position. Relating the signal to perception and production requires auditory phonetics and articulatory phonetics.

Canadian raising names a pronunciation pattern in which the starting portion of the diphthongs conventionally transcribed /aɪ/ and /aʊ/ is raised in particular contexts, especially before a voiceless syllable coda. The pattern is associated with Canadian English, while related effects also occur in other North American varieties.^[ref]

Comparing lout and loud in controlled contexts can illustrate a voiceless-versus-voiced following consonant, but an acoustic test should compare repeated tokens and their formant trajectories. Sky Onosson's dissertation separates changes in trajectory from the shortening of vowels before voiceless consonants; its abstract reports different cross-dialect patterns for the two diphthongs. A single midpoint or duration can therefore miss part of the phenomenon. Raising refers to vowel quality, not simply to a higher voice fundamental frequency.

The current distortion factor is the ratio $D_I=I_1/I_{\mathrm{rms}}$, where $I_1$ is the root mean square magnitude of the fundamental electric current component and $I_{\mathrm{rms}}>0$ is the root mean square magnitude of the complete electric current. Its square is the fraction of the current's mean square supplied by the fundamental component. This ratio is distinct from displacement power factor, which measures the relative phase of the fundamental potential difference and current.

Use a Fourier series at fundamental frequency $\omega /(2\pi )$:Here $I_0$ is the DC component and the $I_n$ are RMS magnitudes of harmonic components. Averaging over a complete period, orthogonality and the Parseval identity giveFor purely sinusoidal electric potential difference $v(t)=\sqrt{2}V\cos \omega t$, only the fundamental electric current contributes to active power:Thus a displacement power factor of one can coexist with a lower total power factor.

For $I_1>0$, use fundamental-normalized current total harmonic distortion, ${\mathrm{THD}}_I=\sqrt{{\sum }_{n\ge 2}{I}_n^2}/I_1$. The relation between current distortion factor and THD with DC current isWith zero DC component, this reduces to $D_I=(1+{\mathrm{THD}}_I^2{)}^{-1/2}$. The definition of RMS here includes the DC component. For example, a 10 ampere fundamental and 3 ampere DC component, with no higher harmonic components, have zero total harmonic distortion but $D_I=10/\sqrt{109}\approx 0.9578$.

For an original zero-mean waveform, take a 230 volt RMS sinusoidal supply, $I_1=10\mathrm{A}$, $\cos {\varphi }_1=0.8$, and RMS third/fifth harmonic components of 6 and 3 amperes. ThenThe active power is 1840 watts, apparent power is about 2769.57 volt-amperes and total power factor is about 0.6644. This is an illustrative waveform model, not a measurement of a particular rectifier.

Terminology requires care. Overview of Functions: Harmonic Measurements by Schneider Electric uses distortion factor for the ratio of power factor to the fundamental cosine. Power Quality Considerations by Schneider Electric uses the label for harmonic content relative to the fundamental, or total harmonic distortion. The defining ratio identifies the intended quantity.

If the supply also contains harmonic components, use the harmonic active power sum:with matching voltage/current phase conventions. The fundamental-only factorization above assumes a purely sinusoidal supply.

A diphthong is a vowel whose quality changes during its production, conventionally treated as one dynamic vowel unit. A monophthong has relatively stable quality instead. The vowel in English toy, often transcribed [ɔɪ], moves from a back rounded quality toward a high front unrounded quality.^[ref]

A recording can represent that change through formant trajectories. For an invented illustration, let normalized vowel time be $0\le u\le 1$ and setin hertz. These are stipulated linear interpolation tracks, not measurements of toy. Their endpoints differ even if fundamental frequency stays fixed, separating changing vowel quality from voice pitch. Real trajectories need not be linear, and identifying a single diphthong rather than a sequence of separate vowels requires phonological evidence as well as an acoustic curve.

A formant is a resonant spectral feature associated with the filtering of speech by the vocal tract. Its centre frequency and bandwidth describe different aspects of the feature; the first few formants are conventionally labeled $F_1,F_2,\dots$. Praat represents estimated formant structure as time-sampled centre frequencies and bandwidths.^[ref]

A simple physical model treats the vocal tract as a uniform tube, closed at one end and open at the other. Under those ideal boundary conditions its resonance frequencies arewhere $c$ is the speed of sound and $L$ the tube length. For $L=0.17\mathrm{m}$ and $c=340\mathrm{m}{\mathrm{s}}^{-1}$, the first three model resonances are 500, 1500 and 2500 hertz. A real vocal tract has a varying cross-section, losses and moving articulators, so this illustrates the mechanism rather than predicting every vowel.

These resonances are distinct from the fundamental frequency of vocal fold vibration. The source-filter model separates the repeating excitation from the filter's spectral envelope, explaining how a speaker can change vowel quality without making the same change in pitch.

The lowest positive normal mode frequency of an oscillating system is its fundamental frequency. For an ideal uniform string of length $L$ with fixed ends and phase speed $c$, $f_1=c/(2L)$. A spatial finite difference approximation has its own lowest positive normal mode frequency, approaching this continuous value as the grid is refined.

Gunnar Fant developed acoustic models connecting the physical production of speech to its measurable signal. His Acoustic Theory of Speech Production by Gunnar Fant relates vocal-tract geometry to acoustic behavior, including calculations based on X-ray studies of articulations. The original book dates to 1960; the inspected publisher preview is its 1970 second printing.^[ref]

His source-filter model of speech production separates excitation from the filtering of the vocal tract. Changing a filter's resonances can move formants without requiring the same change in the fundamental frequency of the source. This makes speech synthesis an experimental tool: a researcher can vary model parameters and compare the generated signal with recorded speech. KTH's account explicitly describes his synthesis work as a means of testing production hypotheses.^[ref]

KTH's curriculum vitae records 1919–2009 and his leadership of the Speech Transmission Laboratory from 1951 to 1966.^[ref]

The harmonic active power sum expresses average electrical power as the sum of contributions from matching components of periodic terminal electric potential difference and electric current. With a common fundamental frequency and root mean square magnitudes for nonzero harmonic components, writeHere $V_0,I_0$ are the signed DC components. The passive sign convention makes the average product absorbed active power. Averaging over a complete fundamental period givesFor finite sums this follows from the cosine-product identity: products of different integer harmonic components average to zero, and matching components give half the cosine of their relative phase. For square-integrable periodic waveforms, the corresponding Fourier series identity follows by continuity of the inner product in the mean-square norm. The orthogonality is a property of the averaging interval and basis, not an assumption that the physical sources are independent.

For an original example, set $V_0=10\mathrm{V}$, $I_0=2\mathrm{A}$, $V_1=100\mathrm{V}$, $I_1=3\mathrm{A}$, and fundamental relative phase $\pi /3$. Add third-harmonic magnitudes $V_3=20\mathrm{V}$, $I_3=1\mathrm{A}$ with relative phase $\pi$. ThenThe third-harmonic contribution is $-20\mathrm{W}$, even though the total absorbed active power is positive. The root mean square values satisfy $V_{\mathrm{rms}}^2=10500{\mathrm{V}}^2$ and $I_{\mathrm{rms}}^2=14{\mathrm{A}}^2$, so apparent power is about $383.41\mathrm{VA}$ and total power factor about $0.3912$.

For a purely sinusoidal supply with no DC component, only $V_1$ survives, giving $P=V_1{I}_1\cos ({\alpha }_1-{\varphi }_1)$ even if the electric current is distorted. Current in other harmonic components can still increase conductor heating. The usual one-frequency complex electrical power construction is derived in the primary MIT treatment; the sum above applies that average-product reasoning separately to orthogonal components.^[ref]

A proposed measurement records the two waveforms synchronously and compares their directly averaged product with the resolved sum. A non-integer-period window, unresolved harmonic components, channel timing error or changing waveform can spoil the simple finite-record equality. The formula assumes a shared periodic model; multiplying separately measured RMS magnitudes gives apparent power instead.

A harmonic component is a sinusoid whose frequency is a positive integer multiple of a specified fundamental frequency: $f_n=n{f}_1$. The fundamental itself is the first harmonic component. In a periodic Fourier series with period $T=1/f_1$,the constant $a_0$ is a DC component, and the positive-frequency sinusoidal terms are the harmonic components. A component may have zero amplitude; a missing fundamental does not require the spacing of the remaining components to change. An overtone is a component above the lowest mode, but physical overtones need not be exact integer multiples.

For an original generation model, apply $v_{\mathrm{in}}(t)=1\mathrm{V}\cos (2\pi 1000t)$ to a memoryless nonlinear device with $v_{\mathrm{out}}=v_{\mathrm{in}}+\beta v_{\mathrm{in}}^2$ and $\beta =0.2{\mathrm{V}}^{-1}$. The identity ${\cos }^{2}x=(1+\cos 2x)/2$ givesThe nonlinearity produces both a DC offset and a second harmonic component; they are different parts of the frequency spectrum. The second harmonic component's peak amplitude is $0.1\mathrm{V}$, while the fundamental's is $1\mathrm{V}$.

A linear time-invariant device in sinusoidal steady state cannot create this extra frequency from a single sinusoidal input; a nonlinear response, a different input component, or an unexcluded transient needs to be considered. In an actual measurement, the source's own harmonic components and the receiver's frequency response must also be accounted for. Anritsu's primary measurement procedure separately acquires the fundamental and selected harmonic components, with measurement calibration covering their frequencies.^[ref] A discrete Fourier transform can instead separate a suitably sampled record; its bin amplitudes still depend on normalization and record length.^[ref]

A monophthong is a vowel with relatively stable quality through its central portion, in contrast with a diphthong's substantial movement between qualities. Relatively stable does not mean physically unchanging: neighboring sounds, vocal tract movement and the analysis interval can all affect a recording.^[ref]

For a stipulated acoustic illustration, hold the first two formants at $F_1=300$ and $F_2=2200$ hertz, while varying fundamental frequency from 100 to 150 hertz. The model changes voice pitch without requiring the same change in the formant pattern. Conversely, changing formants can change vowel quality while fundamental frequency stays fixed.

An experimental analysis can record a sustained vowel and compare several points within a consistently chosen interval. Changes concentrated near consonant boundaries need not make the whole segment a diphthong. The distinction therefore combines a phonetic trajectory with an analysis of the language's sound units.

For zero damping the conserved mechanical energy isFor $\gamma <0$, the potential grows to infinity in both directions, so every nonstationary finite-energy trajectory is trapped between two turning points and is periodic; the zero-energy solution is stationary. The case $\gamma =0$ is the harmonic oscillator. The negative-$\gamma$ comparison extends the parameter range initially stated in the source.

For $\gamma >0$, the origin lies in a local potential well, with barriers at $x=\pm {\gamma }^{-1/2}$ of height $1/(4\gamma )$. A trajectory launched from the origin oscillates precisely when its initial energy is below that barrier:At equality the trajectory approaches a barrier equilibrium asymptotically, rather than being periodic; for example the positive branch is $x={\gamma }^{-1/2}\tanh (t/\sqrt{2})$. Above the barrier the trajectory escapes the well. The source's inequality without the absolute value tacitly takes $v\ge 0$; for signed initial velocity the energy criterion requires $|v|$.

Next set $\beta =\gamma =0$ and assume $\alpha >0$. The energy $H=({\dot{x}}^2+x^2)/2$ satisfies $\dot{H}=-\alpha x^2{\dot{x}}^2\le 0$. This bounds the complete trajectory. The zero-dissipation set is the union of $x=0$ and $\dot{x}=0$, and its only invariant trajectory is the origin: a nonzero point on either axis immediately leaves it under the differential equation. The LaSalle invariance principle therefore gives $\boxed{x(t)\to 0,\dot{x}(t)\to 0}$. If $\alpha =0$ as well, damping is absent and that conclusion fails for nonzero $v$; the positive order-one damping assumption is essential.

For the weakly nonlinear periodic motion, write $\gamma =\alpha \kappa$, set $\tau =\omega t$, $\omega =1+\alpha {\omega }_1+\cdots$, and expand $x=A\cos \tau +\alpha x_1+\cdots$. The first correction obeysResonant forcing at the fundamental frequency would generate secular terms. Eliminating both coefficients yieldsConsequently the maximum displacement is $\boxed{2\sqrt{\beta }+O(\alpha )}$ for fixed positive $\beta$. A possible third-harmonic correction is $x_1=-A^3\sin 3\tau /32-\kappa A^3\cos 3\tau /32$, up to homogeneous amplitude and phase choices. The leading amplitude also balances averaged damping: its slow evolution is $\dot{A}=\alpha A(\beta /2-A^2/8)$, whose nonzero fixed point is attracting. The case $\beta =0$ is a degenerate zero-amplitude limit, rather than a nonzero limit cycle.

Each supplied eigenvector gives relative displacements at $x=1,2,3$ in a standing wave. Insert zero at $x=0,4$ to describe the whole sampled string. A real normal mode is $u_j(t)=C{v}_j\cos (\omega t+\varphi )$: its shape stays fixed while its amplitude oscillates.

These semidiscrete fixed-string normal modes are samples of $\sin (k\pi x/4)$, with $k=3,2,1$ in the order listed. More precisely, their interior vectors are $v_j^{(k)}=2^{-1/2}\sin (k\pi j/4)$, which have unit Euclidean norm and form an orthonormal basis of ${\mathbb{R}}^3$. Their eigenvalues and angular frequencies obeyThus larger absolute value of a negative eigenvalue means faster oscillation, rather than faster decay. The numerical angular frequencies approximate the continuous fixed-string values $k\pi c/L$; the finite difference spatial approximation changes them. Only three independent interior normal modes exist on this grid, and any interior displacement or initial velocity vector can be expanded in this orthonormal basis.

Paul Boersma is a phonetician who co-created Praat with David Weenink. His work connects acoustic measurements to computational methods, including the estimation of fundamental frequency from sampled speech.^[ref]

His 1993 periodicity-analysis paper examines how signal sampling and a finite analysis window affect estimates based on autocorrelation. Repetition produces peaks at nonzero time lags; locating a suitable peak gives a candidate period, while the relative peak height contains information about periodicity. Correcting the influence of the analysis window is important before interpreting those peaks as properties of the original signal.^[ref]

Homepage: www.fon.hum.uva.nl/paul/

Phonetics studies the physical production, transmission and perception of speech. Articulatory phonetics examines the moving structures that shape the airflow; acoustic phonetics examines the resulting sound waves; auditory phonetics examines how a listener perceives them. These are different measurements along the same communication chain.

During voiced speech, airflow from the lungs drives vocal fold vibration. Movements of the tongue, lips and other parts of the vocal tract shape the emerging sound. This separates the production of a vibrating source from the articulation of recognizable speech, and both from the language expressed by it.^[ref]

A simple acoustic comparison records the same spoken phrase twice, once as a statement and once as a question. The sound pressure time series and its spectrogram allow changes in fundamental frequency, segment duration and pauses to be compared. The comparison investigates physical differences in prosody; the written words alone do not specify those measurements.

A pitch contour describes how perceived pitch varies over an utterance. For speech with voicing, a common measurable correlate is the fundamental frequency $F_0(t)$ of vocal fold vibration. A rising estimated $F_0(t)$ indicates increasing repetition rate; stretches without sufficiently periodic vibration need not have a meaningful estimate.^[ref]

For an ideal periodic segment whose repetitions are separated by $T=0.01\mathrm{s}$, $F_0=1/T=100\mathrm{Hz}$. Detecting repetition at $2T$ instead produces an erroneous 50 hertz estimate, illustrating why a visually smooth pitch contour can still contain an octave error. Noise, the analysis interval and the allowed frequency range affect the estimate. Current Praat documentation distinguishes methods intended for speech intonation from methods intended for other periodicity measurements.^[ref]

A pitch contour is not the same as a formant contour: changing the fundamental frequency changes the repetition rate, whereas formants describe the resonant spectral structure shaped by the vocal tract.

Praat is software for analyzing, synthesizing, annotating and manipulating speech, created by Paul Boersma and David Weenink. It makes the connection between a recorded acoustic signal and measurable speech properties explicit: a waveform, a spectrogram, a pitch contour and estimated formants describe different aspects of the same recording.^[ref]

For a sustained vowel, one can compare the repeating structure of the waveform with the estimated fundamental frequency, then examine how the frequency spectrum is shaped by the vocal tract. A change in vowel quality need not be a change in pitch. Annotations can align intervals with phonemes or syllables without treating every acoustic boundary as an unambiguous linguistic boundary.

Measurements depend on the analysis settings and the signal. In particular, a displayed pitch contour can contain octave errors, while estimated formants depend on the analysis model. Inspecting the recorded sound and analysis assumptions is therefore part of the experiment, rather than accepting every displayed trajectory as ground truth.

Prosody comprises patterns of pitch, timing and emphasis extending across the individual segments of speech. Speech intonation concerns the changing pitch pattern, while speech rhythm concerns the temporal organization. The same sequence of words can be spoken with different prosody.

For acoustic analysis, a recording is examined over time. A spectrogram can reveal changes in the fundamental frequency of voiced portions; segment durations and pauses provide timing measurements. The perceived pitch and emphasis are perceptual judgments, so they should be distinguished from the measured frequency and sound pressure. Phonetics relates the recorded signal to its production and perception.

Quefrency is the independent coordinate of a cepstrum. When a cepstrum is obtained by an inverse Fourier transform of a logarithmic magnitude spectrum expressed as a function of frequency in hertz, its quefrency coordinate has units of time. A repeated pattern across frequency can consequently appear at the corresponding reciprocal spacing on the quefrency axis.^[ref]

For an $N$-point discrete Fourier transform with sampling rate $f_s$, bin spacing is $\Delta f=f_s/N$. The inverse-transform coordinate at index $n$ has quefrencyWith $f_s=8000\mathrm{Hz}$, index 40 represents $5\mathrm{ms}$. An ideal echo delayed by 40 samples generates spectral ripples spaced by $1/(0.005\mathrm{s})=200\mathrm{Hz}$ and a cepstrum contribution at that $5\mathrm{ms}$ delay, with further contributions at its multiples. The cepstrum's real-valued convention can also produce mirrored peaks at negative quefrencies.

A $5\mathrm{ms}$ quefrency peak is therefore not a $5\mathrm{ms}$-period oscillation of the original waveform by definition. Depending on the signal, it might reflect an echo or a fundamental frequency near $200\mathrm{Hz}$. Interpretation requires a model of how the spectral repetition arose.

Speech intonation is the pattern of pitch variation across an utterance. Its acoustic description commonly uses a pitch contour, tracking the estimated fundamental frequency through voiced portions of speech. The pitch contour describes one component of prosody; it does not by itself encode every distinction conveyed by emphasis, pauses or timing.

To investigate it, record the same phrase with different intended emphasis, then align the recordings by their words and compare the pitch contours. Inspect the underlying waveform as well as the automatic estimate: a tracking error can resemble an abrupt jump. The analysis range matters, and a low pitch floor requires a longer time window, trading the ability to detect low fundamental frequencies against the resolution of rapid changes.^[ref]

An estimated fundamental frequency should not be read as a measured vocal oscillation during a genuinely unvoiced interval. Nor does a rise have a universal meaning independently of the language and conversational setting.

Speech synthesis generates an artificial speech signal from a specified representation. Text-to-speech starts from writing, while a research synthesizer can instead take a phonetic specification or directly controlled acoustic parameters. The distinction separates the input representation from the mechanism producing the sound.

In a source-filter model of speech production, a researcher can set an excitation's fundamental frequency and independently adjust the filter's formants. Holding the first parameter fixed while moving a formant tests whether listeners' responses depend on the spectral change rather than a change in voice pitch. Gunnar Fant's work at KTH used synthesis to test hypotheses about speech production.^[ref] Generated stimuli permit controlled comparisons in speech perception, but intelligibility and similarity to a natural speaker are different outcomes. Recording the parameter settings makes a synthesis experiment reproducible.