Chad Lillian and Dr. Andrew D. Pollington, Mathematics
Recruitment of loudness is a hearing impairment in which certain frequency ranges are not as audible as others. If a common hearing aid is used to remedy this problem all frequencies will be amplified and there will be no advantage. The solution to this problem is a hearing aid that only selectively amplifies those frequencies that are less audible.
There are many ways to decompose a signal, such as sound, into its constituent frequencies. Perhaps the oldest and most common is the Fourier transform. The Fourier transform breaks a signal into a sequence of sine and cosine functions with different frequencies. The original signal, or a part of it, may then be reconstructed by combining all or selecting part of the sine and cosine functions.
In the past decade wavelet analysis, a new field of mathematics related to Fourier analysis, has gained recognition as being superior to Fourier analysis in several ways. Wavelet analysis was chosen because it promises to be more efficient than Fourier analysis and because wavelets can be custom designed to perform certain tasks.
The main problems that needed to be addressed in constructing wavelets for the hearing aid were 1) good frequency response and 2) vanishing moments.
The most basic form of a wavelet is a 2-band wavelet, a 2- band wavelet breaks a signal into two parts. Good frequency response means that the wavelet will act like a high pass and a low pass filter (i.e. The high pass part contains only the high frequencies and the low pass part only contains low frequencies).
Certain applications of wavelets need a certain number of vanishing moments. Wavelets with a certain number of vanishing moments have a certain order of regularity or smoothness. For sound only two vanishing moments are needed.
The first consideration is how to decompose the signal into the desired frequency ranges. An M-band wavelet (an M-band wavelet breaks signals into M parts) could be used to break the signal into the 16 desired bands. This, however, would require a 32-band wavelet for it to have the necessary resolution to isolate the smaller bands at the lower end of the spectrum.
One of the advantages of using wavelets is something known as multi-resolution analysis. Multi-resolution analysis is a process by which a wavelet can operate on a signal iteratively, thus an M-band wavelet can act like an N*M-band wavelet where N is the number of iterations.
The method that appears to be most promising is a combination of multi-resolution analysis and M-band wavelets. Two wavelets will be used a 2-band and a 3-band, the M k AkA T kj 0jI 0j 0 if jg0 1 if j 0 [A I A]a[B I B] [AB AB 2AB I A BAB] signal will first be processed by the 2 band wavelet the high pass part will then be processed by the 3-band wavelet, the low pass part will then be processed by the 2-band wavelet and the process repeated until all of the desired frequency ranges have been isolated.
Wavelets are commonly represented in matrix form. To be classified as a wavelet matrix the matrix must satisfy certain restrictions. One of these restrictions is known as the shifted orthogonality condition. To satisfy the shifted orthogonality condition a matrix must be composed of several square matrices which have the following properties.
If [A A A A … A ] is a wavelet 0 l 2 3 m matrix where each Aj is a square matrix then to satisfy the shifted orthogonality condition the following must be true:
The first method that was investigated for the construction of wavelets was the Pollen product .1,2 The Pollen product defines a method of matrix multiplication which is similar to polynomial multiplication.
The Pollen product:
The advantage of the Pollen product is that when A and B are ‘symmetric projectors’ the result is guaranteed to be a wavelet matrix. There are, however, several disadvantages in that it is difficult to guarantee a certain number of vanishing moments or a certain frequency response by the choice of symmetric projectors.
The next method that was studied is from Daubechies.3 This method makes the restrictions of frequency response and vanishing moments easier to incorporate into the design of the wavelet.
The Daubechies method begins with choosing the desired number of vanishing moments and then creates a formula which can be optimized to form the desired frequency response. This appears to be the method which will give the best results.
The questions that remain to be answered are 1) how many coefficients will be needed in the wavelet matrix to attain the desired frequency response 2) how to choose the coefficients and 3) how to extend this knowledge from the 2-band case to the 3-band case.
- Kautsky, Jaroslav and Radka Turcajova. 1994. pp 117-134. In Wavelets: Theory, Algorithms, and Applications. Academic Press.
- Kautsky, Jaroslav and Radka Turcajova. 1995. Linear Algebra and its Applications 222:241-260.
- Daubechies, Ingrid. 1992. Ten Lectures on Wavelets. Capital City Press, Montpelier, Vermont.