The Whole Story Behind Blind Adaptive Equalizers/ Blind Deconvolution

The Bayesian Approach

Author(s): Monika Pinchas

Pp: 43-67 (25)

DOI: 10.2174/978160805352011201010043

* (Excluding Mailing and Handling)

Abstract

In this chapter the Bayesian approach for blind deconvolution is explained. We cover the dierent proposed expressions for the conditional expectation known in the literature and explain in detail the advantages and disadvantages of each ob- tained expression.

In the literature, the convolutional noise probability density function (pdf) is modeled as a Gaussian pdf. In this chapter, a new closed-form approximated ex- pression is derived for the conditional expectation based on a new model for the convolutional noise pdf. The new derived expression for the conditional expecta- tion, is based on the Maximum Entropy approach, Edgeworth expansion, Laplace integral method and is valid for the noiseless, real valued and two independent quadrature carrier case. According to simulation results carried out for signal to noise ratio (SNR) SNR = 30 [dB], the new derived expression leads to improved equalization performance for the 16QAM input constellation case.


Keywords: Bayesian approach, conditional expectation, Maximum Entropy, Edgeworth ex- pansion, Laplace integral, blind deconvolution, nonlinearity function, Shannon's entropy, Lagrange multipliers, orthogonal expansion, Hermite polynomials

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      • 13. app/Models/UserAccess/UserAccess.php:1266
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      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Trial Access" as access_base_text, CASE j.type WHEN 111 THEN true ELSE false END as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_trail` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 309725698 and j.to_ip >=309725698 )) and ((j.from_date <= 1739847391 and j.to_date >= 1739847391 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 3384) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Subscribed" as access_base_text, false as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_corporate` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 309725698 and j.to_ip >=309725698 )) and ((j.from_date <= 1739847391 and j.to_date >= 1739847391 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 3384) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Token Based Access" as access_base_text, false as terms_popup, "R" as content_download_type, j.uc_id as restricted_user_access_key, "v" as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_token` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 309725698 and j.to_ip >=309725698 )) and ((j.from_date <= 1739847391 and j.to_date >= 1739847391 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 3384)
      690μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
      Backtrace
      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 1026 and `b`.`banner_from_date` <= 1739847391 and `b`.`banner_to_date` >= 1739847391 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.09msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 1026
      • 3: 1739847391
      • 4: 1739847391
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='ec27867ec95ed1236a67d746189fbc9d'
      230μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    11Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        vsBWnu7pNB8sZwk9fpZBzW36Y07Xqjiz51OQ4XrM
        uc_cart_id
        ec27867ec95ed1236a67d746189fbc9d
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/3384" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/3384
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6IkZ2bTk2dzlCM2U1NXEwS2tZTUp5WkE9PSIsInZhbHVlIjoiVWRNU013QitCZ2VJd0RuQWZUcytucWdFSHU5NmJ4NVBvT1A1TWJOeXZGcz0iLCJtYWMiOiI4NzZlZDY2YTM1ZWFmMzM4OGU5ZTM5NjdhNGZjODljNTlmMGFhYmJiZGIzNDZmMjAwMWE2MWZkZmQwOGQ1NDM4IiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6IkZ2bTk2dzlCM2U1NXEwS2tZTUp5WkE9PSIsInZhbHVlIjoiVWRNU013QitCZ2VJd0RuQWZUcytucWdFSHU5NmJ4NVBvT1A1TWJOeXZGcz0iLCJtYWMiOiI4NzZlZDY2YTM1ZWFm" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
        array:1 [ "wayfindertarget" => null ]
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Tue, 18 Feb 2025 02:56:31 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InR5c2RWWGpUdGlya1FTUE9iWnhqR2c9PSIsInZhbHVlIjoiWFJqaVJlU0Uxd0NWMEFkblpkQ3JrL2ViczIvM1hBdmN3R3VPdlJ3RSt4bjNaQllUMEVHcElvclZycGthZEEwQ2o1MkIySVFydU8xcHc5L3pNSGt4eElWZFRIZXJ4ai96VlZGUENPNHRIbVBGenYyWE5IWG15UmZFbWNheWwzUGIiLCJtYWMiOiJiMjdjOThmMjRhMDYwOGY5NzcyYzBiOTVlOGYyOTQ3ODM3YmUyZWQ4NGEzMTUxZDA2MmVkYmUwMDFiMmYwMjkxIiwidGFnIjoiIn0%3D; expires=Thu, 20 Feb 2025 14:56:31 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6InR5c2RWWGpUdGlya1FTUE9iWnhqR2c9PSIsInZhbHVlIjoiWFJqaVJlU0Uxd0NWMEFkblpkQ3JrL2ViczIvM1hBdmN3R3VPdlJ3RSt4bjNaQllUMEVHcElvclZycGthZEEwQ2o1MkIyS" 1 => "alphaeurekaselectnet_session=eyJpdiI6IlVpZktYNHpJbk1WMkFrMWVMZVFHcEE9PSIsInZhbHVlIjoiaWVCckpHZU1kWEZocDMrK1AwY0tRU0pYaVQ3MFltU25lb1pjcCtlbC9xLzROb1pGMkkvcmVYTUsrNG1vNUFqeU9xU1dVMDl0ZC9EQVVoQUM1VUhaQWl3blhUek1wS1dDWVA4UjNYT1ZBWFQwalZwR1FLTC9ralhZWkx1S3BQa3ciLCJtYWMiOiJmYzkzYTVhMzk5ZDM4OTA1ODE2YzQ3OGYyYzdkMjhkZmI3YmY1NWRjYTNkZWQ2MjM2ZjhiMzAzMWMzMzFhODUwIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6IlVpZktYNHpJbk1WMkFrMWVMZVFHcEE9PSIsInZhbHVlIjoiaWVCckpHZU1kWEZocDMrK1AwY0tRU0pYaVQ3MFltU25lb1pjcCtlbC9xLzROb1pGMkkvcmVYTUs" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6InR5c2RWWGpUdGlya1FTUE9iWnhqR2c9PSIsInZhbHVlIjoiWFJqaVJlU0Uxd0NWMEFkblpkQ3JrL2ViczIvM1hBdmN3R3VPdlJ3RSt4bjNaQllUMEVHcElvclZycGthZEEwQ2o1MkIySVFydU8xcHc5L3pNSGt4eElWZFRIZXJ4ai96VlZGUENPNHRIbVBGenYyWE5IWG15UmZFbWNheWwzUGIiLCJtYWMiOiJiMjdjOThmMjRhMDYwOGY5NzcyYzBiOTVlOGYyOTQ3ODM3YmUyZWQ4NGEzMTUxZDA2MmVkYmUwMDFiMmYwMjkxIiwidGFnIjoiIn0%3D; expires=Thu, 20-Feb-2025 14:56:31 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6InR5c2RWWGpUdGlya1FTUE9iWnhqR2c9PSIsInZhbHVlIjoiWFJqaVJlU0Uxd0NWMEFkblpkQ3JrL2ViczIvM1hBdmN3R3VPdlJ3RSt4bjNaQllUMEVHcElvclZycGthZEEwQ2o1MkIyS" 1 => "alphaeurekaselectnet_session=eyJpdiI6IlVpZktYNHpJbk1WMkFrMWVMZVFHcEE9PSIsInZhbHVlIjoiaWVCckpHZU1kWEZocDMrK1AwY0tRU0pYaVQ3MFltU25lb1pjcCtlbC9xLzROb1pGMkkvcmVYTUsrNG1vNUFqeU9xU1dVMDl0ZC9EQVVoQUM1VUhaQWl3blhUek1wS1dDWVA4UjNYT1ZBWFQwalZwR1FLTC9ralhZWkx1S3BQa3ciLCJtYWMiOiJmYzkzYTVhMzk5ZDM4OTA1ODE2YzQ3OGYyYzdkMjhkZmI3YmY1NWRjYTNkZWQ2MjM2ZjhiMzAzMWMzMzFhODUwIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6IlVpZktYNHpJbk1WMkFrMWVMZVFHcEE9PSIsInZhbHVlIjoiaWVCckpHZU1kWEZocDMrK1AwY0tRU0pYaVQ3MFltU25lb1pjcCtlbC9xLzROb1pGMkkvcmVYTUs" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "vsBWnu7pNB8sZwk9fpZBzW36Y07Xqjiz51OQ4XrM" "uc_cart_id" => "ec27867ec95ed1236a67d746189fbc9d" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/3384" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-18 02:56:31GET/chapter/33841121913