In my problem there exists a vector x = [x1,x2, ... x_N] of length N, and I have M measurements y_k = x .* e_k, 1 <= k <= M, where e_k is non-gaussian multiplicative noise. Currently, I denoise by taking logs of each y_k and averaging the M replicates. In the ideal case (ie additive mean 0 gaussian noise) this would improve SNR by a factor of sqrt(M). However, since log(e_k) is not mean 0 gaussian noise, averaging is probably not the best approach. This is the first time I've encountered an ensemble of signals. Does anyone know where I can look for more information on this problem?
Ensemble averaging for multiplicative noise
Started by ●January 4, 2012
Reply by ●January 4, 20122012-01-04
On Jan 4, 9:21=A0am, "ryan1234567890" <compton.ryan@n_o_s_p_a_m.gmail.com> wrote:> In my problem there exists a vector > > x =3D [x1,x2, ... x_N] > > of length N, and I have M measurements > > y_k =3D x .* e_k, 1 <=3D k <=3D M, > > where e_k is non-gaussian multiplicative noise. > > Currently, I denoise by taking logs of each y_k and averaging the M > replicates. In the ideal case (ie additive mean 0 gaussian noise) this > would improve SNR by a factor of sqrt(M). However, since log(e_k) is not > mean 0 gaussian noise, averaging is probably not the best approach. > > This is the first time I've encountered an ensemble of signals. Does anyo=ne> know where I can look for more information on this problem?Mulitplicative noise often results in lognormal distributions. You may wish to look them up. These show up often eg., salary distributions, sizes of rocks after being crushed, received intensity of multiply reflected waves where the direct path is occluded. Whole books have been written on the topic. One on my shelf is "The Lognormal Distribution with special reference to its uses in economics" by Aitchison and Brown - Cambridge University Press. Also all of chapter 14 in "Continuous Univariate Distributions Volume I" by Johnson,Kotz, and Balakrishnan is devoted to the Lognormal. You may find this in you university library. If you are into stats, this 3 volume set is a wonderful reference. Depeding on how your original data was distributed before the multiplicative noise was added in, you may also find Benford's law applicable as it is the distribution of distributions. Also it is one where the logarithms of the data are uniformly distributed. IHTH, Clay
Reply by ●January 19, 20122012-01-19
>On Jan 4, 9:21=A0am, "ryan1234567890" ><compton.ryan@n_o_s_p_a_m.gmail.com> wrote: >> In my problem there exists a vector >> >> x =3D [x1,x2, ... x_N] >> >> of length N, and I have M measurements >> >> y_k =3D x .* e_k, 1 <=3D k <=3D M, >> >> where e_k is non-gaussian multiplicative noise. >> >> Currently, I denoise by taking logs of each y_k and averaging the M >> replicates. In the ideal case (ie additive mean 0 gaussian noise) this >> would improve SNR by a factor of sqrt(M). However, since log(e_k) isnot>> mean 0 gaussian noise, averaging is probably not the best approach. >> >> This is the first time I've encountered an ensemble of signals. Doesanyo=>ne >> know where I can look for more information on this problem? > >Mulitplicative noise often results in lognormal distributions. You may >wish to look them up. These show up often eg., salary distributions, >sizes of rocks after being crushed, received intensity of multiply >reflected waves where the direct path is occluded. Whole books have >been written on the topic. One on my shelf is "The Lognormal >Distribution with special reference to its uses in economics" by >Aitchison and Brown - Cambridge University Press. > >Also all of chapter 14 in "Continuous Univariate Distributions Volume >I" by Johnson,Kotz, and Balakrishnan is devoted to the Lognormal. You >may find this in you university library. If you are into stats, this 3 >volume set is a wonderful reference. > > >Depeding on how your original data was distributed before the >multiplicative noise was added in, you may also find Benford's law >applicable as it is the distribution of distributions. Also it is one >where the logarithms of the data are uniformly distributed. > >IHTH, >Clay > >So, the noise is multiplicative Gaussian and I can take log to end up with an additive lognormal denoising problem. Any suggestions when the noise is multiplicative non Gaussian?
