{"id":4500,"date":"2017-04-27T09:06:16","date_gmt":"2017-04-27T07:06:16","guid":{"rendered":"http:\/\/www.tonmeister.ca\/wordpress\/?p=4500"},"modified":"2018-05-29T07:40:18","modified_gmt":"2018-05-29T05:40:18","slug":"probability-density-functions-part-3","status":"publish","type":"post","link":"http:\/\/www.tonmeister.ca\/wordpress\/2017\/04\/27\/probability-density-functions-part-3\/","title":{"rendered":"Probability Density Functions, Part 3"},"content":{"rendered":"

In the last posting<\/a>, I talked about the effects of a bandpass filter on the probability density function (PDF) of an audio signal. This left the open issue of other filter types. So, below is the continuation of the discussion…<\/p>\n

I made\u00a0noise signals (length 2^16 samples, fs=2^16) with different PDFs, and filtered them as if I were building a three-way loudspeaker with a 4th order Linkwitz-Riley crossover (without including the compensation for the natural responses of the drivers). The crossover frequencies were 200 Hz and 2 kHz (which are just representative, arbitrary values).<\/p>\n

So, the filter magnitude responses looked like Figure 1.<\/p>\n

\"\"
Fig 1: Magnitude responses of the three filter banks used to process the noise signals.<\/figcaption><\/figure>\n

The resulting effects on the probability distribution functions are shown below. (Check the last posting for plots of the PDFs of the full-band signals – however note that I made new noise signals, so the magnitude responses won’t match directly.)<\/p>\n

The magnitude responses shown in the plots below have been\u00a01\/3-octave smoothed – otherwise they look really<\/em>\u00a0noisy.<\/p>\n

 <\/p>\n

\"\"
Fig 2: PDFs of a noise signal with a rectangular distribution that has been split into the three bands shown in Figure 1. Note the\u00a0DC offset of the signal, visible in the low-pass output’s PDF.<\/figcaption><\/figure>\n

 <\/p>\n

\"\"
Fig 3: PDFs of a noise signal with a linear distribution that has been split into the three bands shown in Figure 1.\u00a0Note the\u00a0DC offset of the signal, visible in the low-pass output’s PDF.<\/figcaption><\/figure>\n

 <\/p>\n

\"\"
Fig 4: PDFs of a noise signal with a triangular distribution that has been split into the three bands shown in Figure 1.<\/figcaption><\/figure>\n

 <\/p>\n

\"\"
Fig 5: PDFs of a noise signal with an exponential distribution that has been split into the three bands shown in Figure 1.\u00a0Note the\u00a0DC offset of the signal, visible in the low-pass output’s PDF.<\/figcaption><\/figure>\n

 <\/p>\n

\"\"
Fig 6: PDFs of a noise signal with a Laplacian distribution that has been split into the three bands shown in Figure 1.<\/figcaption><\/figure>\n

 <\/p>\n

\"\"
Fig 7: PDFs of a noise signal with a Gaussian distribution that has been split into the three bands shown in Figure 1.<\/figcaption><\/figure>\n

 <\/p>\n

Post-script<\/h2>\n

This posting has a Part 1 that you’ll find here<\/a> and a Part 2 that you’ll find here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

In the last posting, I talked about the effects of a bandpass filter on the probability density function (PDF) of an audio signal. This left the open issue of other filter types. So, below is the continuation of the discussion… I made\u00a0noise signals (length 2^16 samples, fs=2^16) with different PDFs, and filtered them as if […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[4,43,5,42],"tags":[],"class_list":["post-4500","post","type-post","status-publish","format-standard","hentry","category-audio","category-dsp","category-loudspeakers","category-math"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p48hIM-1aA","_links":{"self":[{"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/posts\/4500","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/comments?post=4500"}],"version-history":[{"count":8,"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/posts\/4500\/revisions"}],"predecessor-version":[{"id":4520,"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/posts\/4500\/revisions\/4520"}],"wp:attachment":[{"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/media?parent=4500"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/categories?post=4500"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.tonmeister.ca\/wordpress\/wp-json\/wp\/v2\/tags?post=4500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}