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ECE 598: The Speech Chain

ECE 598: The Speech Chain. Lecture 7: Fourier Transform; Speech Sources and Filters. Today. Fourier Series (Discrete Fourier Transform) Sawtooth wave example Finding the coefficients Spectrogram Frequency Response Speech Source-Filter Theory Speech Sources Transient Frication

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ECE 598: The Speech Chain

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  1. ECE 598: The Speech Chain Lecture 7: Fourier Transform; Speech Sources and Filters

  2. Today • Fourier Series (Discrete Fourier Transform) • Sawtooth wave example • Finding the coefficients • Spectrogram • Frequency Response • Speech Source-Filter Theory • Speech Sources • Transient • Frication • Aspiration • Voicing • Formant Transitions

  3. Topic #1: Discrete Fourier Transform

  4. Sawtooth Wave • Sawtooth first harmonic: • Amplitude: 0.64 • Phase: 0.52p radians

  5. Sawtooth Second Harmonic • Sawtooth second harmonic: • Amplitude: 0.32 • Phase: 0.53p radians

  6. Sawtooth Third Harmonic • Sawtooth third harmonic: • Amplitude: 0.21 • Phase: 0.55p radians

  7. Sawtooth: Calculating the Fourier Transform • Average over one full period of (125Hz sawtooth * 125Hz cosine) = -0.03

  8. Sawtooth: Calculating the Fourier Transform • Average over one full period of (125Hz sawtooth * 125Hz sine) = 0.636

  9. Sawtooth: Amplitude and Phase of the First Harmonic • First Fourier Coefficient of the Sawtooth Wave: • -0.03 + 0.636j • |-0.03 + 0.636j| = 0.64 • phase(-0.03 + 0.636j) = 0.52p • X1 = -0.03+0.636j = 0.64 ej0.52p • First Harmonic: • h1(t) = 0.64 ej(250p t+0.52p) • Re{h1(t)} = 0.64 cos(250pt+0.52p)

  10. Sawtooth: Calculating the Fourier Transform • Average over one full period of (125Hz sawtooth * 250Hz cosine) = -0.03

  11. Sawtooth: Calculating the Fourier Transform • Average over one full period of (125Hz sawtooth * 250Hz sine) = -0.3173

  12. Sawtooth: Amplitude and Phase of the Second Harmonic • Second Fourier Coefficient of the Sawtooth Wave: • -0.03 + 0.3173j • |-0.03 + 0.3173j| = 0.32 • phase(-0.03 + 0.3173j) = 0.53p • X1 = -0.03+0.3173j = 0.32 ej0.53p • First Harmonic: • h2(t) = 0.32 ej(500p t+0.53p) • Re{h2(t)} = 0.64 cos(250pt+0.53p)

  13. How to Reconstruct a Sawtooth from its Harmonics • At sampling frequency of FS=8000Hz (8000 samples/second), the period of a 125Hz sawtooth is • (8000 samples/second)/(125 cycles/second) = 64 samples • The “0”th harmonic is DC (0*125Hz = 0 Hz) • So we need to add up 63 harmonics: • x(t) = h1(t) + h2(t) + … + h63(t)

  14. Why Does This Work??? • It’s a property of sines and cosines • They are a BASIS SET, which means that… • Let <> mean “average over one full period:” <cos2(mw0t)> = 0.5 for any integer m <cos(mw0t)cos(nw0t)> = 0 if m ≠ n above two properties also hold for sine <cos(mw0t) sin(nw0t)> = 0 • w0 = 2pF0 = 2p/T0 is the “fundamental frequency” in radians/second

  15. Topic #2: Spectrogram

  16. Spectrogram Frequency (Hz) Time (seconds)

  17. How to Create a Spectrogram • Divide the waveform into overlapping window • Example: window length = 6ms • Example: window spacing = one per 2ms • Result: neighboring windows overlap by 4ms • Fourier transform each frame to create the “short-time Fourier transform” Xk(t) • Read: kth frequency bin, taken from the frame at time t seconds • Could also write X(t,f) = Fourier coefficient from the frame at time=t seconds, from the frequency bin centered at f Hz. • In pixel (k,t), plot 20*log10(abs(Xk(t)) • The Fourier transform expressed in decibels!

  18. Reading a Spectrogram Turbulence (Frication or Aspiration) Stop Nasals Vowel Frequency (Hz) Time (seconds) Glide (Extreme Formant Frequencies) Each vertical stripe is a glottal closure instant (BANG – high energy at all frequencies) Inter-bang period = pitch period ≈ 10 ms

  19. Topic #3: Frequency ResponseandSpeech Source-Filter Theory

  20. Frequency Response • Any sound can be windowed • Call the window length “N” • Its windows can be Fourier transformed: x(t) = h1(t) + h2(t) + … x(t) = X1ejw0t + X2e2jw0t + … • What happens when you filter a cosine? Input = ejw0t --- Output = H(w0)ejw0t Input = e2jw0t --- Output = H(2w0)e2jw0t …

  21. Frequency Response • If x(t) goes through any filter: x(t) → ANY FILTER → x(t) • Then we can find y(t) using freq response! x(t) = X1ejw0t + X2e2jw0t + … y(t) = H(w0)X1ejw0t + H(2w0)X2e2jw0t + …

  22. Frequency Response • In general, we can write • Y(w) = H(w)X(w)

  23. Examples of Filters • Low-Pass Filter: • Eliminates all frequency components above some cutoff frequency • Result sounds muffled • High-Pass Filter: • Eliminates all frequency components below some cutoff frequency • Result sounds fizzy • Band-Pass Filter: • Eliminates all components except those between f1 and f2 • Result: sounds like an old radio broadcast • Band-Stop Filter: • Eliminates only the frequency components between f1 and f2 • Result: usually hard to hear that anything’s happened!

  24. Examples of Filters • A Room: • Adds echoes to the signal • A Quarter-Wave Resonator: • Emphasizes frequency components at f=(c/4L)+(nc/2L), for all integer n • Components at other frequencies are passed without any emphasis • A Vocal Tract: • Emphasizes any part of the input that is near a formant frequency • Any energy at other frequencies is passed without emphasis

  25. The Source-Filter Model of Speech Production • Input: Weak Sources • Glottal closure • Turbulence (frication, aspiration) • Filter: The Vocal Tract • Energy near formant frequency is emphasized by a factor of 10 or a factor of 100 • Energy at other frequencies is passed without emphasis • Speech = (Vocal Tract Transfer Function) X (Excitation) • S(w) = T(w) E(w)

  26. Topic #4: Speech Sources.Presented as:Events in the Release of a Stop Consonant

  27. Events in the Release of a Stop “Burst” = transient + frication (the part of the spectrogram whose transfer function has poles only at the front cavity resonance frequencies, not at the back cavity resonances).

  28. Pre-voicing during Closure To make a voiced stop in most European languages: Tongue root is relaxed, allowing it to expandm so that vocal folds can continue to vibrating for a little while after oral closure. Result is a low-frequency “voice bar” that may continue well into closure. In English, closure voicing is typical of read speech, but not casual speech. “the bug”

  29. Burst = Transient and FricationFrication = Turbulence at a Constriction Turbulence striking an obstacle makes noise Front cavity resonance frequency: FR = c/4Lf The fricative “sh” (this ship)

  30. Aspiration = Turbulence at the Glottis (like /h/). Transfer Function During Aspiration…

  31. “the mom” Formant Transitions: Labial Consonants “the bug”

  32. “the supper” Formant Transitions: Alveolar Consonants “the tug”

  33. “the shoe” Formant Transitions: Post-alveolar Consonants “the zsazsa”

  34. “the gut” Formant Transitions: Velar Consonants “sing a song”

  35. Formant Transitions: A Perceptual Study The study: (1) Synthesize speech with different formant patterns, (2) record subject responses. Delattre, Liberman and Cooper, J. Acoust. Soc. Am. 1955.

  36. Perception of Formant Transitions

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