Help with synthesizing sound?

Hello, I'm having major trouble on one of the problems for my university assignment so if anyone could help me out that'd be awesome:

For the problem we are supposed to use the 'spectral envelope to synthesize an example of each vowel of pitch 150 Hz and duration 5 seconds. Display the spectrum of the synthetic vowel and confirm that it is a reasonable approximation to the real vowel.'

Do I need to use the fourier synthesis equation for this? And if so, since there's a lot of vowels I need to do, is there some sort of function I can do to make it quick?

[Abigail]

Just to note, I have the vowel already recorded at 2 seconds and a sampling rate of 10000. I've also been given more information, but I still don't know exactly what to do: I need to synthesize the vowel using the equation: A sin(2*pi*f0t)+ A1 sin(2*pi*2f0t)... etc.

Then, I need to write a function that synthesizes a vowel sound of pitch f0 for a duration of D seconds from a smoothed spectrum S.

Answer 1

[Image Analyst]

See an example of where I synthesize a sound in the file attached below.