Lab 2 (15 points) Additive synthesis: The sum of K sinusoids, can be expressed as K-1...

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Lab 2 (15 points) Additive synthesis: The sum of K sinusoids, can be expressed as K-1 K-1 y(t) = Yk (t) = Ak cos(2xt + k) k=0 k=0 where Ak, fk, and k are the peak amplitude, frequency (Hz), and initial phase (rad) of the kth sinusoidal component of frequency fk. K is the total number of sinusoidal components. Creating such a sum of sinusoids is known as additive synthesis. a) Write a function that generates N samples of a sinusoid knowing its amplitude, A, frequency, f, phase, phi, given the sampling rate, Fs. Use this function to create, plot (only 0.01 s), and listen to one-second-long, unit-amplitude, zero-phase, 440 Hz cosine wave sampled at 44100 Hz. b) Write a function that uses the function above to generate the sum of K sinusoids. This function should take lists/vectors of amplitudes, frequencies, and phases (each of the same length) as well as the number of samples and the sampling rate and should return the sum of the sinusoids. Use your function to create the sum of K = 5 sinusoids with frequencies fk = [220, 660, 1100, 1540, 1980] and amplitudes Ak = [1, 1/3, 1/5, 1/7, 1/9], all with zero phase. Use a sampling rate of 44.1kHz and return a one-second segment. Plot (only 0.01 s) and listen to it. How does it sound? What waveform does this approximate? c) Use the functions above to create a signal that is comprised of each of the sinusoids from b) (starting with 220Hz) presented in a sequence 1 second long and added to the previous sinusoid (e.g., 220Hz for 1 second, then add the 660Hz sinusoid for 1 second, ..., finally add the 1980 Hz sinusoid for 1 second). Plot (full duration) and listen to it. Plot at t = [0.1, 1.1, 2.1, 3.1, 4.1] seconds for 0.02 s. How does the signal evolve through time? How does it sound? d) Generate the same waveform as in b) but with randomized phase. Does it sound the same as b)? Does it look the same? Lab 2 (15 points) Additive synthesis: The sum of K sinusoids, can be expressed as K-1 K-1 y(t) = Yk (t) = Ak cos(2xt + k) k=0 k=0 where Ak, fk, and k are the peak amplitude, frequency (Hz), and initial phase (rad) of the kth sinusoidal component of frequency fk. K is the total number of sinusoidal components. Creating such a sum of sinusoids is known as additive synthesis. a) Write a function that generates N samples of a sinusoid knowing its amplitude, A, frequency, f, phase, phi, given the sampling rate, Fs. Use this function to create, plot (only 0.01 s), and listen to one-second-long, unit-amplitude, zero-phase, 440 Hz cosine wave sampled at 44100 Hz. b) Write a function that uses the function above to generate the sum of K sinusoids. This function should take lists/vectors of amplitudes, frequencies, and phases (each of the same length) as well as the number of samples and the sampling rate and should return the sum of the sinusoids. Use your function to create the sum of K = 5 sinusoids with frequencies fk = [220, 660, 1100, 1540, 1980] and amplitudes Ak = [1, 1/3, 1/5, 1/7, 1/9], all with zero phase. Use a sampling rate of 44.1kHz and return a one-second segment. Plot (only 0.01 s) and listen to it. How does it sound? What waveform does this approximate? c) Use the functions above to create a signal that is comprised of each of the sinusoids from b) (starting with 220Hz) presented in a sequence 1 second long and added to the previous sinusoid (e.g., 220Hz for 1 second, then add the 660Hz sinusoid for 1 second, ..., finally add the 1980 Hz sinusoid for 1 second). Plot (full duration) and listen to it. Plot at t = [0.1, 1.1, 2.1, 3.1, 4.1] seconds for 0.02 s. How does the signal evolve through time? How does it sound? d) Generate the same waveform as in b) but with randomized phase. Does it sound the same as b)? Does it look the same?