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For a graduate math course I chose to do my final project on recovering undersampled audio files using compressed sensing. The first step of this problem is to determine an effective basis to represent signals such that they are approximately sparse. Then a measurement matrix is chosen and finally with this we can then solve the basis pursuit problem shown in the image to the left.
Several transforms were tested:
1) Discrete Fourier Transform (DFT)
2) Short Time Discrete Fourier Trans. (STFT)
3) Discrete Cosine Transform (DCT)
4) Short Time Discrete Cosine Trans. (STCT)
STCT was determined to sparsify the collection of audio signals the most as indicated by the fastest decay of the magnitude of the ordered coefficients relative to the other transforms.
Two audio clips are used, vocal as well as pure instrumental.
Restricted identity measurement matrix is used to undersample the audio clips.
Two audio clips are used, vocal as well as pure instrumental.
Bernouilli measurement matrix is used to undersample the audio clips.
Two audio clips are used, vocal as well as pure instrumental.
Gaussian measurement matrix is used to undersample the audio clips.
INSTRUMENTAL
Conclusions:
1) The STCT is observed to best sparsify the audio signals studied. Given that this transform will sparsify signals made up of pure tones, instrumental signals are better sparsified than vocal signals as most their makeup is of pure tones.
2) When compressed sensing is applied to vocal signals, the reconstruction inaudibly differs from the original when 60% of the original signal is measured. In the case of the instrumental signals, only 30% of the original signal had to be measured to produce seemingly identical quality to the original.
VOCAL
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