Phasors and Beamforming in Signal Processing
Explore the concepts of phasors, beamforming, and signal processing through a series of informative images and explanations. Learn about conjugate match beamforming, maximizing responses to desired signals, phase relationships, and more, all illustrated with helpful visuals and detailed insights.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Dougs Aha Moment on Conjugate Match Beamforming Doug Hayman
Phasors By McFadden, Strauss Eddy & Irwin for Desilu Productions - eBay itemphoto frontphoto back, Public Domain, https://commons.wikimedia.org/w/index.p hp?curid=20656372 www.netanimations.net/Reflecti ons-in-water-animated-gifs.htm By Gonfer at English Wikipedia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?c urid=11313700
Phasors 2 ??? = Re ???????? Im ??? = ?cos ?? + ? ? ? y = ???? Re
?1+ + ?? ??= ??? ? = ?1
Two element beamformer ??? = ?1?1cos ?? + ??1+ ??1 +?2?2cos ?? + ??2+ ??2 By Gonfer - en wikipedia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid =7026837
Maximize response to desired signal ??? = ?1?1cos ?? + ??1+ ??1 a2 +?2?2cos ?? + ??2+ ??2 ? = 1 Weights need a constraint Limit on total power into next stage Most cases gain increases also increase noise Zero added power a1 ?1+ + ?? ??= ??? ? = ?1
Phase relationship ?1+ + ?? ??= ??? Im ? = ?1 ?? To maximize S make ?? all align with real axis. ?2 ?1 ?2 ?1 Re ?
Phase relationship Note the convention of multiplying by the conjugate of the weight vector implies the phases Im ??1= ??1 ??2= ??2 ?1 ?2 ?1 ?2 Re ?
Magnitude ?1+ + ?? ??= ??? ? = ?1 With aligned phases, let ?? = ?? & ?? = ?? Deal with just magnitudes ? = ?1 Aha! That looks familiar Dot products are projections ? ? = ? ? cos? ?1+ + ?? ??= ?1?1+ + ????= ? ?
What is a projection? a2 ? ? ? ? = ? is maximized when ? and ? are aligned ? cos? ? a1 ?1 ? ?1= ?1 ? ?1 = ? = 1
Put it all together Recall phases were made equal ?1= ?1 ?,?2= ?2 ? Or compactly ? ? ? = And ??? ? ? = ??? = = ?
