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انجام پروژه های مهندسی برق - آشنایی با (Random Matrix theory(RMT و کاربرد های آن
آشنایی با (Random Matrix theory(RMT و کاربرد های آن
چهارشنبه 23 دی 1394 ساعت 03:11 ب.ظ | نوشته ‌شده به دست رسول نیک بخت | ( نظرات )
در طول سالیان گذشته روابط مربوط به توزیع احتمال ماتریس های مختلف (ماتریس کواریانس، ماتریس Wishartو غیره ) و  تحت شرایط مختلف ( مولفه های با نویز گوسی، مولفه با سطر های و ستون های مستقل از هم، مولفه های با ستون های مستقل از هم)  مورد بررسی قرار گرفته  است. نکته ای که در مورد این ماتریس ها بسیار جالب می باشد این است که با افزایش ابعاد ماتریس  تصادفی، می توان روابط ساده شده ای را برای توزیع احتمال آن  یا تابعی از آن و یا یا مقادیر ویژه آن پیدا کرد. 
اگر چه این روابط برای حالتی محاسبه می شوند که ابعاد ماتریس ها به بی نهایت میل می کند ولی برای مقادیر کوچک نیز جواب های مناسبی اریه می دهد( به عنوان مثال یک ماتریس با درایه های گوسی iid برای تعداد سطر و ستون در حد 10 الی 100 با تقریب خوبی از روابط حالت حدی پیروی می کند). بنابراین می توان از این تئوری در علوم بسیاری نظیر فیزیک، اقتصاد، مهندسی برق و ... استفاده کرد.
در این نوشتار یک مثال پایه همراه با شبیه سازی Matlab مورد بررسی قرار می گیرد.
 
 در شکل زیر مقادیر ویژه ماتریس کواریانس با رابطه توزیع احتمال تئوری آن که از تئوری RMT به دست آمده مقایسه شده است. در این رابطه N طول فریم موردی بررسی و L برابر تعداد فریم ها یا بلوک ها می باشد. همچنین y برابر نسبت N به L می باشد.

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مرتبط با: Random Matrix Theory , MATLAB ,


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