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Naive Deconvolution Theory

Module by: Chris Lamontagne, Bryce Luna. E-mail the authors

Summary: Theory behind deconvolution of recorded sound to reproduce an input.

There are many characteristics of a room that determine the impulse response of a room. The physical dimensions of the room and the wall surfaces are crucial in predicting how sound reacts. Using basic properties of geometry, we can predict the paths that sound waves will travel on, even as they bounce off walls. The walls themselves have certain reflection coefficients that describe the power of the reflected signal with respect to the signal in contact with the wall. So it appears that with the dimensions of the room and the reflection coefficients of the walls in the room it is possible to generate an impulse response for that room. Using a simple tape measure we recorded the height, width, and length of Duncan 1075 and a Will Rice dorm room, and used a MATLAB program called Room Impulse Response to find the approximate impulse response of these two rooms. We decided to take two samples in each room, leaving us with four theoretical impulse responses.

Figure 1
Theoretical Impulse in Duncan - LeftTheoretical Impulse in Duncan - Right
(a) (b)
Theoretical Impulse in Duncan - Left (DuncanTheoryLeft.png)Theoretical Impulse in Duncan - Right (DuncanTheoryRight.png)

Figure 2
Theoretical Impulse in Will Rice - LeftTheoretical Impulse in Will Rice - Right
(a) (b)
Theoretical Impulse in Will Rice - Left (WillRiceTheoryLeft.png)Theoretical Impulse in Will Rice - Right (WillRiceTheoryRight.png)

Clearly these will not be incredibly accurate, as they assume a completely rectangular, and empty, room. Neither of these rooms were completely rectangular, and they were also not empty. However, this gives us a good approximation of the impulse response. The signals decay significantly as time increases, which is expected. When we record the actual response using an approximate impulse, this data will help determine if we have an accurate measurement.

Once we have the impulse response of each room, we must find an appropriate signal to deconvolve. We chose a piano tune, as a piece of music should have a more simple frequency response than speech. After recording the impulse response and the input, we theoretically have enough data to reconstruct the signal using deconvolution. The recorded output is the convolution of the input with the system. y t = x t * h t y t = x t * h t y t x t h t The recorded output has a frequency spectrum defined by Y jw = X jw H jw Y jw = X jw H jw y t x t h t Using simple algebra, we can solve for the input frequency coefficients: X jw = Y jw / H jw X jw = Y jw / H jw y t x t h t We have H(jw), the impulse response, and Y(jw), the FFT of the recorded signal. Thus we can find X(jw), the frequency spectrum of the input signal, and by taking the inverse FFT we are left with the input signal x(t).

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