Project 2 Fall 2009
Vowel Synthesis (corrected)

Download proj2-09-fix.xlsm from ASSIGNMENT DETAILS

A.  Introduction

The project for this section of the course expands on the information and skills developed in Project 1, Problem Set 2.2, and the use of decibels (Section 1.4).  The main project focuses on the synthesis of human vowel sounds rather than sine waves (Project 1) or musical instruments (PS2.2).  The synthesis will be of the vowel /a/, the first sound in the phrase "Ah, how nice!"  The synthesis will be based on information gained from the Fourier analysis of an /a/ vowel produced by a female talker.  In an optional exercise, you may speak into a microphone attached to the computer and produce your own /a/ vowel, record it, insert it into the Excel spreadsheet, and then compare it to the female vowel and the synthesized vowel.

B.  Background

A vowel can be synthesized from the harmonics of a specified fundamental frequency.  The amplitude of the individual harmonics may be obtained from Fourier analysis of the original vowel.  In general, a specific sound can be generated from the first few harmonics of a fundamental frequency, F0 as follows:

yH(n) = AH · sin (2 · π · (H · F0) · n/Fs + ф)

where

AH is the amplitude of each harmonic
H is the harmonic number (H = 1, 2, . . ., 7)
F0 is the fundamental frequency (F0 = 220 Hz for a female talker)
Fs is the sample frequency in samples/second (Fs = 11025 Hz)
n is the sample number (n = 0, 1, 2, 3, . . .)
ф is the starting phase in radians (ф = 0)  

Because we assume ф = 0, it can be dropped from the equation above

The Excel spreadsheet proj2-09-fix.xlsm provides a number of different worksheet templates.   On the first sheet, labeled "sawtooth," the template provides an example of the first three harmonics of a sawtooth wave, which is used to represent the human glottal source (as well as oboes).  Included in this sheet is the sum of these three waveforms in an array named "saw3," as well as the amplitude spectrum of "saw3" obtained via Fourier analysis.  On the sheet labeled "ah," the female /a/ vowel is included.

For this project, you are to use Fourier analysis of the female /a/ on sheet "ah" to determine the amplitudes of the first seven harmonics of the sound, AH, H = 1, . . .,7.  This may be done by carefully placing the cursor over the peaks in the graph or by examining the column of numbers used to create the graph.  Next, you should synthesize an /a/ vowel by summing sinusoids with the appropriate amplitude values (determined in the previous step). 

C. Follow the steps below and the Guidelines for Writing Project Reports to complete Project 2.

Make sure the "Analysis ToolPak " and the "Analysis ToolPak VBA" Add-In are turned on in your version of Excel. In Excel 2003, this is on the "Tools > Add-Ins" menu. On Excel 2007, this is through Office Button->Excel Options->Addins->Manage Excel Addins->Go-> .

  1. Building on Problem Set 2.2, synthesize a sawtooth signal for the glottal source using seven harmonics.  The first three harmonics are provided in the template with the amplitude 5000. Change the amplitude 4000. To examine how the magnitude spectrum changes with amplitude changes, a new Fast Fourier analysis should be calculated using  the "NewFFT" macro. To do so, highlight the column containing the data samples, and then click on the NewFFT button.  Determine the amplitudes of the three harmonics from the magnitude FFT spectrum by identifying the value of points on the graph using the mouse and compare values for the two different amplitudes. Use the equation above and harmonic amplitudes corresponding to a fundamental with an amplitude of 4000 (AH = 4000/H, H = 1, . . .,7) to generate a sawtooth waveform with 7 harmonics. The speaker about "saw 7" already has the sound of the 7 harmonic sawtooth. Use "NewFFT" to calculate the FFT of the 7-harmonic sawtooth and print the chart of this amplitude spectrum.

  2. Decompose the sawtooth and the /a/ vowel into their Fourier components using the "NewFFT" macro button and print the chart of this amplitude spectrum. Determine the amplitudes of the first seven harmonics (RH, H = 1, . . ., 7) from the amplitude spectrum by identifying the value of points on the graph using the mouse.  Determine the frequencies of those components and the fundamental frequency, Fo.

  3. To synthesize the /a/ vowel, copy the worksheet containing the 7 sawtooth components onto a new sheet. Change the frequency of the harmonics to reflect the fundamental frequency determined above.  Then change the amplitudes of each of the sinusoidal (Fourier) components of the sawtooth wave, AH, with the amplitudes, RH, from the real /a/ vowel that you obtained above.  Sum the new harmonic components to synthesize a new /a/ vowel. The yellow cells with the speaker and "Synthetic ah" on the ah sheet already has the sound of the synthtetic /a/ vowel. Graph and print the first 100 to 200 points of the waveform. 

  4. Graduate Student Work: Using a microphone (available in SPHS 164) and Adobe Audition, speak and record your own /a/ vowel for comparison.  To record your voice or sounds in the room, follow the instructions below:

    • Adobe Audition in Edit view. Use record button, select sample freq. = 11025 Hz, mono, 16-bit, OK and say /a/ for 1 second, press stop. Under File menu, save to Desktop
    • In Excel, use Insert-Clipart. Press Organize Clips button. In new window, under File menu-Add clips->On My own and get .wav file from Desktop. In clip art window, put file name and click Go button. Double click on .wav icon and it will appear in spreadsheet.
    • Check to see that the sound file was installed in your Excel worksheet by clicking on the speaker icon in Excel.

D.  Report Questions for Project 2

  1. Listen and compare the real and synthetic /a/ vowels and the sawtooth wave with seven harmonics. They all have 7 harmonics.  Describe the differences you hear.  Provide some possible explanations for those differences.

  2. For the sawtooth waveform, the values of the amplitudes of each harmonic, AH, were specified when you synthesized the 7 component sawtooth. You also used the FFT to decompose the 7-component sawtooth and create the amplitude spectrum graph. Compare the amplitudes calculated by the FFT for the 7 harmonics with the original AH amplitudes and explain why they are not the same.

  3. Explain the concepts of Fourier analysis and synthesis in your own words.  Do Fourier analysis and Fourier synthesis make intuitive sense for periodic complex sounds?

  4. In Excel, calculate the amplitude spectrum of the /a/ vowel in terms of decibels.  Go to the worksheet containing the amplitude spectrum of your synthesized /a/ vowel.   The the H column has the frequency values and the I column has the amplitude values. Convert amplitude to decibels in column J using the formula dB = 20 · log10(amplitude), where the digital reference for dB is 1, or 1 bit (see Section 1.5).

  5. Plot the first 200 points of the decibel amplitude spectrum.  Compare the representation of the harmonics in linear amplitude to those in dB by examining the corresponding data columns.  Describe the effect of the log transformation on the harmonics with lower amplitudes (the valleys).

  6. Graduate Student Work: The /a/ vowels analyzed and synthesized in this section were for a female speaker.  How does they compare to your own /a/ vowel that you recorded? Males have fundamental frequencies about one octave lower than that of females.  Alter the synthesized vowel for a male fundamental frequency.  Calculate the amplitude for your synthesized /a/ spectrum in dB, and compare it to the one for the female speaker. The spectrum should look "incomplete;" explain why.

Upload completed Project 2 Excel file and your report file to your Module 2 drop box. Turn in your printed report with the printed graphs.