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Консультация № 161679
01.03.2009, 11:59
0.00 руб.
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Подскажите пожалуйста:
Два сообщения содержат одинаковое количество символов, количество информации в первой тексте в 1,5 раза больше, чем во втором. Сколько символов содержат алфавиты с помощью которых записаны сообщения, если известно, что число символов в каждом алфавите не перевышает 10 и на каждый символ приходится целое число битов?
Два сообщения содержат одинаковое количество символов, количество информации в первой тексте в 1,5 раза больше, чем во втором. Сколько символов содержат алфавиты с помощью которых записаны сообщения, если известно, что число символов в каждом алфавите не перевышает 10 и на каждый символ приходится целое число битов?
Обсуждение
Неизвестный
02.03.2009, 15:39
общий
это ответ
Здравствуйте, Sunshine!
Пусть каждое сообщение содержит X символов, а в первом и втором алфавитах число символов равно К1 и К2 соответственно.
Формула хартли для определения I количества информации на один символ I = log2(K) (где К - число символов в алфавите, а log2 - двоичный логарифм)
Количество информации, которое содержит сообщение, закодированное с помощью знаковой системы, равно количеству информации, которое несет один знак, умноженному на число знаков в сообщении
Тогда по условию имеем:
xlog2(k1)=1,5*xlog2(k2)
log2(k1)=1,5*log2(k2)
2*log2(k1)=3*log2(k2)
log2(k1^2)=log2(k2^3)
k1^2=k2^3
Получили соотношение, которым связаны количества символов в алфавитах.
Далее. В условии сказано, что символов в каждом алфавите не более 10, причем на каждый приходится целое число бит. Для К<=10 двоичный логарифм принимает целые значения только в трех случаях: если к=2, если к=4 и если к=8.
Легко убедиться, что 8^2 = 4^3 = 64, а это значит, что в первом алфавите содержится 8 символов, а во втором алфавите содержится 4 символа.
Все.
Рад был помочь.
Пусть каждое сообщение содержит X символов, а в первом и втором алфавитах число символов равно К1 и К2 соответственно.
Формула хартли для определения I количества информации на один символ I = log2(K) (где К - число символов в алфавите, а log2 - двоичный логарифм)
Количество информации, которое содержит сообщение, закодированное с помощью знаковой системы, равно количеству информации, которое несет один знак, умноженному на число знаков в сообщении
Тогда по условию имеем:
xlog2(k1)=1,5*xlog2(k2)
log2(k1)=1,5*log2(k2)
2*log2(k1)=3*log2(k2)
log2(k1^2)=log2(k2^3)
k1^2=k2^3
Получили соотношение, которым связаны количества символов в алфавитах.
Далее. В условии сказано, что символов в каждом алфавите не более 10, причем на каждый приходится целое число бит. Для К<=10 двоичный логарифм принимает целые значения только в трех случаях: если к=2, если к=4 и если к=8.
Легко убедиться, что 8^2 = 4^3 = 64, а это значит, что в первом алфавите содержится 8 символов, а во втором алфавите содержится 4 символа.
Все.
Рад был помочь.
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