Well, FFT breaks a signal (in time domain) down to its frequency components. After FFT analysis you know how much energy the signal has in each analysed frequency "bin".

You wrote, that FFT would break a signal down into its "wave components". I would rather say "sinusoidal components" or frequency components. Because you can restore the signal in time domain by summarizing all these sinusoidal components with different amplitudes and phases.

Anyway - I think here is the problem: The LPC Analysis does not do that. LPC does not break a signal down into its frequency components. LPC calculates the "envelope" of the spectrum. You won't get all the single frequency components as with the FFT, but only a few maxima. It's kind of a "smooth" version of the FFT spectrum.

You wrote:

LPC also produces a fundamental and a series of formant frequencies with the same concept, that each of the formants represents a component waveform with significant energy. The fundamental and formant series can be convoluted back producing the perceived signals. Maybe I am wrong somewhere?

This is not true, LPC follows a different concept. LPC does not "produce a fundamental". The formant frequencies do not have an amplitude and a phase as the FFT frequencies. The formant frequencies also have a bandwidth! It really is a different thing. It's like comparing apples and oranges.

Have a look at:

http://clas.mq.edu.au/acoustics/frequency/spectral.html

There you will see some fine and detailed FFT spectra - and also smoothed LPC spectra.

Also you may find that the first component in the FFT analysis usually is NOT the fundamental of the analysed sound. That also means that not every component has to be a harmonic of the fundamental.

I hope the website with the lecture slides bring some clarity ...

Thank you for the succinct explanation on the difference between FFT and LPC.

When I first posted this question, I admit I was not clear of the difference between harmonics and formants. This would have led to misuse of terms. Since then it has become clearer. Despite the theoretical input I have yet to provide an explanation to the discrepancies in my observation.

Firstly I had measured voice signals using a condenser microphone connected through a 4 channel DAQ. This signal (Fs 44100 Hz, single channel, 16 bit) was filtered (band-passed) at 50Hz and 5000Hz. When I perform FFT (FFT size 44100) on this signal (i.e the phoneme /im/), I got the result of the FFT as shown in Fig 1. [All figures as a pdf file in attachment]. The highest visible harmonic frequency was 593Hz. From theory, for vowel /i/, I am expecting some signals at F2 (2200Hz) and F3 (2700). I could see some ‘bleeps’ in the FFT at frequency range between 1800 – 2500 Hz as shown in Fig 2. I am not sure how significant are these ‘bleeps’.

In order to get a better picture, I analyzed the signals using PRAAT. I have been so far unsuccessful to save the signals in a PRAAT readable file format except in wav format. Fig 3 and 4 show the same signal in wav format. Except for a slight shift in the formant values from the theoretical expectation, I suppose they are more or less acceptable. If you are wondering why I did not continue the analysis in Labview, I would need additional toolkit which is costly.

Then I realized I overlooked two issues. Firstly the PRAAT spectral slice gives y-axis scale in dB whereas the voice signal waveform was measured in labview, into EU values, Pascal (Pa). Using the online calculator at http://www.sengpielaudio.com, 54.2 and 18.2 dB were equal to 0.010 and 0.00016 Pa respectively. In Fig 1, the peak amplitude at 96Hz was 0.46 Pa, which was way too high from 0.0010 Pa. The second issue which I had overlooked was this. When I saved the waveform in wav format I have to normalize the amplitude to +1 and -1. This step would had falsely reduce the amplitude, erroneously leading to underestimated dB. I am currently trying to save the waveform in txt format (without the normalization). This I have successfully done as in (Fig 5). Prior to this, I had tried but failed to save in PRAAT readable binaries. I am currently studying on PRAAT script regarding string manipulation in order to get PRAAT to read this txt file.

Once I have overcome this step, I could get a better picture what is going on. Nevertheless I am wondering if the dark wide-bands seen in voice spectrogram which gives almost similar dark tone to F0, F1 and F2 is a misrepresentation. Perhaps the logarithmic manipulation during conversion from Pa to dB had ‘shrunken’ the difference in energy level between frequencies at the F0 region compared to at the formants. Actually at the very beginning, I was expecting an FFT result that may look like in Fig 6. In fact I even thought my FFT analysis (Fig 1) was wrong.

Thanks for the reference website at MacQuarie. Apparently the FFT which I did gave me a line spectra graph. What I was looking for is actually in the two dimensional spectral (frequency & intensity) graph.

Finally, I have managed to link the two. This became obvious when I converted the sound pressure to sound pressure level (20log([Signal]Pa/20microPa)) dB. The FFT for Fig 1 is similar to the one I have included in the previous entry. Fig 2 is the same FFT analysis but with conversion of the y-axis from sound pressure to sound pressure level. (All done within Labview) Although we could not see any signals at 4 to 10kHz in Fig 1, yet we could see fluctuating peaks in Fig 2. Now I could see the peak of the harmonics increases and decreases with peaks corresponding to the formant frequencies as detailed in Fig 3. This was what I was looking for. Although the values are not exactly the same, they are more or less.

Solved. Thanks to all who had contributed.

"formants are just the peaks of the intonation contour"?

Well, this is not true. You can measure the intonation contour in the time domain where you will have some peaks and valleys.

Formants are peaks in the *spectral* domain. And for the display/measurement of the spectral domain you should use LPC and not FFT. With LPC-spectra you can identify the peaks easier than with FFT-spectra.

I thought I have understood formant until I bumped to an explanation from a different perspective. It all started with chapter 6, The Source-Filter Theory of Vowels from the book Principles of Voice Production by Ingo Titze. My confusion is related to the quarter-wave resonance tube.

According to the quarter wave resonance tube, the formant waves resonate within the tube and the formant frequencies can be derived mathematically. Fn = (2n-1)(c/4L) where n=1,2,3….; c being the speed of sound and L the length of the tube. In a 17.5cm long tube mimicking an average male vocal tract, the lowest four formant frequencies are F1 =500Hz, F2 = 1,500Hz, F3 = 2,500Hz, and F4 = 3,500Hz. These values appear to correspond somewhat with the open vowel /ɑ/

At these frequencies the waves are suppose to resonate within the tube creating a standing wave with no acoustic energy loss at the open end (ideal situation).

Now why are the formant frequencies derived from LPC of voice spectrogram from voice recorded outside the mouth [not in the oral cavity :) show peaks (elevated energy; dB) rather that dips!!!

Formant frequency and harmonic frequency are independent of each other (at least in a first approximation). Formant frequencies derive from the vocal tract, and harmonic frequencies derive from the vocal chord. There is no reason why they should be the same, other than coincidence. Do this experiment. Say the vowel /a/ and change the pitch of your voice singing from low to high. The harmonics will become more widely spaces (the fundamental frequency will increase as your pitch goes up), but unless you have changed the shape of the vowel and your vocal tract, there should be no change in formant frequency.

Don't forget that a "formant frequency" is not really a frequency in the same sense that a harmonic is. It is only the theoretical center frequency of a wide band at the center of a theoretical bandpass filter. The bandwidth can be wider or narrower depending upon the Q of the filter. Different vowels, different accents, etc can form filters with higher or lower Q's, i.e. narrower or wider bands, the extreme being the formants used by Tuvan "harmonic" singers. Thus rather than thinking ofa formant as having "a" frequency, it is more accurate to think of it as a region.

This is an update on the quarter wave resonance tube I put up on 14 Sept 13. In the end I had to post the question to the author, Dr Ingo Tietze.

The following is his email reply. I suppose there is no harm in me sharing his reply.

"Thank you for your question. The spectrogram shows peaks at the formants for radiated sound because this radiated sound is maximum at the formants. The reflection at the lips is not 100%, but more like 90 - 95%. Hence, sound escapes at the end of the tube, making the acoustic length of the tube not exactly a quarter wavelength. If you compare the pressure outside the tubecto the pressure inside the mouth, it is much smaller.

Best,

Dr. Titze"

LPC is a model based spectrum, which is designed to suppress the harmonics or the multiples of the fundamental frequency of vibration (pitch) of the vocal folds. Also, in speech software, such as PRATT, the signal is preemphasized (equivalent to a high pass filter or differentiation) to boost the higher frequencies, because in all natural signals, the higher frequencies have lower energy. The derivation of LPC is based on maximizing the spectral match, which, by its design, gives priority to peaks, rather than valleys. Thus, to get better estimates of higher formants through LPC, we need to boost the higher frequencies. The formants are the resonant frequencies of the different "cavities" of the vocal tract and are in no way connected to the pitch or fundamental frequencey of vocal folds, as Marshall Chasin has already mentioned. Having said this, as per the source-filter model, the power spectrum of the vocal folds signal is multiplied by the power spectrum of the vocal tract frequency shaping, and hence, the harmonics that overlap or are very close to the formant frequencies, will definitely be amplified because of spectral peaks of the vocal tract. So, when one reads the formant frequencies approximately from the DFT spectrum, it is likely that the value read from the spectrum is one of the harmonics of the pitch that is very close to the formant frequency. Further, there are approximations in the LPC model, as pointed out by Paul Poletti, and hence all these estimates are only true to the extent the model is correct. Also, when you change the model order significantly in LPC estimation, the estimated formants can change marginally.

AGR

Another critical point to remember is that the ultimate success of any algorithm which purports to extract the overall formant curve from an input signal depends completely on the density of the spectral content; the denser, the better. In terms of the human voice, this means low pitch. This is the only way to get each formant region populated with a lot of content in order to observe the true shape of the localized peaks and dips with the highest resolution possible. Every time the voice goes up an octave, you lose more than half the content, first because by definition a tone an octave higher has only 50% of the harmonic content of the lower, and second, due to the fact that formant regions are essentially fixed and do not move upward with the fundamental tone, you lose the mostly densely populated octave of spectral content, the highest, as it has now been transposed to a region above that which is affected by formants. Male voices are the easiest to analyze and return the most reliable results. Female voices become more problematic as the basic pitch rises, and children’s voices are the worst. That’s why a bright male voice is the best for public announcements, it has the richest content and will therefore remain the most intelligible in noisy environments. People think just the opposite, since female voices are brighter, but since they have significantly less spectral content, you lose vowel intelligibility. I’ve noticed the difference countless times in airports, train, and subway stations. Listeners with some higher frequency hearing loss will also do better with male voices.

Formants are vocal tract resonances, and it sometimes happens that there is a partial right at that frequency and then the spectrum envelope peak will appear at the formant frequency. Otherwise not.

@Johan, Abu Dzarr, A.G. Ramakrishnan, thanks for this privilege to concretize my current lessons on experimental phonetics. Your points are valid, and I have learnt something new.