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Гомоморфная обработка речи (реферат)

Язык: русский
Формат: реферат
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18

БЕЛОРУССКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ

ИНФОРМАТИКИ И РАДИОЭЛЕКТРОНИКИ

факультет телекоммуникаций

кафедра Сетей и устройств телекоммуникаций

РЕФЕРАТ

На тему:

«Гомоморфная обработка речи»

МИНСК, 2008

1 Структурные схемы гомоморфной обработки и анализа речевых сигналов

Одно из основных предположений состоит в том, что речевой сигнал
трактуется как сигнал на выходе линейной системы с медленно
изменяющимися параметрами. Это предположение позволяет считать, что на
коротких сегментах речевой сигнал можно рассматривать как сигнал на
выходе линейной системы с постоянными параметрами, возбуждаемой либо
последовательностью импульсов, либо случайным шумом.. Поскольку сигнал
возбуждения и импульсная характеристика фильтра взаимодействуют через
операцию свертки, задача анализа речи может рассматриваться как задача
разделения компонент, участвующих в операции свертки. Такая задача
иногда называется задачей обратной свертки.

Гомоморфные относительно свертки системы. Гомоморфные относительно
свертки системы удовлетворяют обобщенному принципу суперпозиции. Принцип
суперпозиции для линейных систем можно представить в виде следующих
соотношений
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 (1)

picscalex100010009000003990200000200b10000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c028002
80111300000026060f001c00ffffffff00004e0010000000c0ffffffa6ffffff40110000
260200000b00000026060f000c004d617468547970650000700005000000090200000002
050000001402bf0145011c000000fb02f7fde30000000000900100000001000200105379
6d626f6c0000a9090ad92ced1200d89ff377e19ff3772020f57752096623040000002d01
000009000000320a00000000010000005b790000050000001402bf0126051c000000fb02
f7fde300000000009001000000010002001053796d626f6c0000a0090a342ced1200d89f
f377e19ff3772020f57752096623040000002d01010004000000f001000009000000320a
00000000010000005d790000050000001402bf01d6081c000000fb02f7fde30000000000
9001000000010002001053796d626f6c0000a9090ada2ced1200d89ff377e19ff3772020
f57752096623040000002d01000004000000f001010009000000320a0000000001000000
5b790000050000001402bf01090c1c000000fb02f7fde300000000009001000000010002
001053796d626f6c0000a0090a352ced1200d89ff377e19ff3772020f577520966230400
00002d01010004000000f001000009000000320a00000000010000005d79000005000000
1402a0014b031c000000fb0280fe0000000000009001000000000002001054696d657320
4e657720526f6d616e00d89ff377e19ff3772020f57752096623040000002d0100000400
0000f001010010000000320a00000000060000002829282928294f0194054f01f7034f01
0003050000001402a00146001c000000fb0280fe00000000000090010100000000020010
54696d6573204e657720526f6d616e00d89ff377e19ff3772020f5775209662304000000
2d01010004000000f001000018000000320a000000000b0000004c61786e614c786e6179
6e009201c0003d014203c000a4013d014203c00044010003050000001402a001ec051c00
0000fb0280fe0000000000009001000000010002001053796d626f6c0000a9090adc88e9
1200d89ff377e19ff3772020f57752096623040000002d01000004000000f00101000a00
0000320a00000000020000003d3de3060003b100000026060f0057014d61746854797065
55554b01050100050244534d543500011357696e416c6c4261736963436f646550616765
7300110554696d6573204e657720526f6d616e00110353796d626f6c001105436f757269
6572204e65770011044d5420457874726100120008212f458f442f4150f4100f475f4150
f21f1e4150f4150f4100f445f425f48f425f4100f4100f435f4100f48f45f42a5f48f48f
4100f4100f40f48f417f48f4100f412a5f445f45f45f45f45f410f0c0100010001020202
0200020001010100030001000400000a01000200834c0003000303000100020083610002
0083780002008228000200836e000200822900000200965b000200965d00000204863d00
3d02008361000200834c0003000303000100020083780002008228000200836e00020082
2900000200965b000200965d00000204863d003d02008361000200837900020082280002
00836e0002008229000000650b00000026060f000c00ffffffff01000000000000001c00
0000fb021000070000000000bc02000000cc0102022253797374656d0023520966230000
0a0021008a010000000001000000d4f31200040000002d01010004000000f00100000300
00000000 (2)

где L – линейный оператор.

Принцип суперпозиции устанавливает, что если сигнал на входе является
линейной комбинацией элементарных сигналов, то и сигнал на выходе будет
представлен в виде линейной комбинации соответствующих сигналов.

Прямым следствием принципа суперпозиции является тот факт, что сигнал на
выходе линейной системы может быть представлен в виде дискретной
свертки:

picscalex100010009000003d00200000200cc0000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c024004
a0161300000026060f001c00ffffffff00004e0010000000c0ffffffafffffff60160000
ef0300000b00000026060f000c004d617468547970650000e00005000000090200000002
050000001402800218011c000000fb0280fe000000000000900100000000000200105469
6d6573204e657720526f6d616e00d89ff377e19ff3772020f577800c662c040000002d01
000019000000320a000000000c000000282928292828292928292829500127056c037e00
500156017e00ca025001c80150010003050000001402ec036f041c000000fb0222ff0000
000000009001010000000002001054696d6573204e657720526f6d616e00d89ff377e19f
f3772020f577800c662c040000002d01010004000000f001000009000000320a00000000
010000006b79bc0105000000140280025e001c000000fb0280fe00000000000090010100
00000002001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577800c
662c040000002d01000004000000f001010018000000320a000000000b000000796e686e
6b786b686e786e294401270550011602e6013e014e035001da013e010003050000001402
fa002d051c000000fb0222ff0000000000009001000000010002001053796d626f6c0000
730c0aaf94d91200d89ff377e19ff3772020f577800c662c040000002d01010004000000
f001000009000000320a0000000001000000a579bc01050000001402ec03ef040c000000
320a00000000030000003d2da56387007900bc01050000001402800240031c000000fb02
80fe0000000000009001000000010002001053796d626f6c00007a0c0a6494d91200d89f
f377e19ff3772020f577800c662c040000002d01000004000000f00101000c000000320a
00000000030000003d2d3dafdb055a060003050000001402d802b2041c000000fb02c0fd
0000000000009001000000010002001053796d626f6c0000730c0ab094d91200d89ff377
e19ff3772020f577800c662c040000002d01010004000000f001000009000000320a0000
000001000000e579800405000000140280023b131c000000fb0280fe0000000000009001
00000001000200104d54204578747261000c0a6598d91200d89ff377e19ff3772020f577
800c662c040000002d01000004000000f001010009000000320a00000000010000006779
0003cc00000026060f008e014d6174685479706555558201050100050244534d54350001
1357696e416c6c4261736963436f6465506167657300110554696d6573204e657720526f
6d616e00110353796d626f6c001105436f75726965724d43590011044d54204578747261
00120008212f458f442f4150f4100f475f4150f21f1e4150f4150f4100f445f425f48f42
5f4100f4100f435f4100f48f45f42a5f48f48f4100f4100f40f48f417f48f4100f412a5f
445f45f45f45f45f410f0c01000100010202020200020001010100030001000400000a01
00020083790002008228000200836e0002008229000204863d003d030010700001000200
83680002008228000200836e0002048612222d0200836b00020082290002008228000200
83780002008228000200836b00020082290002008229000204863d003d02008368000200
8228000200836e00020082290002048b8fe967020083780002008228000200836e000200
822900000b01000200836b000204863d003d02048612222d0204861e22a5000100020486
1e22a5000d0204861122e50000000b00000026060f000c00ffffffff0100000000000000
1c000000fb021000070000000000bc02000000cc0102022253797374656d002c800c662c
00000a0021008a010000000001000000e4e31200040000002d01010004000000f0010000
030000000000 (3)

гдеpicscalex100010009000003500100000200770000000000050000000201010000000
50000000102ffffff00050000002e0119000000050000000b0200000000050000000c020
00200031300000026060f001c00ffffffff00004e0010000000c0ffffffc6ffffffc0020
000c60100000b00000026060f000c004d617468547970650000500005000000090200000
0020500000014026001ff001c000000fb0280fe000000000000900100000000000200105
4696d6573204e657720526f6d616e00d89ff377e19ff3772020f577b70e6617040000002
d0100000a000000320a000000000200000028294f01000305000000140260013a001c000
000fb0280fe0000000000009001010000000002001054696d6573204e657720526f6d616
e00d89ff377e19ff3772020f577b70e6617040000002d01010004000000f00100000a000
000320a0000000002000000686e4f0100037700000026060f00e4004d617468547970655
555d800050100050244534d543500001357696e416c6c4261736963436f6465506167657
300110554696d6573204e657720526f6d616e00110353796d626f6c001105436f7572696
572204e65770011044d5420457874726100120008212f458f442f4150f4100f475f4150f
21f1e4150f4150f4100f445f425f48f425f4100f4100f435f4100f48f45f42a5f48f48f4
100f4100f40f48f417f48f4100f412a5f445f45f45f45f45f410f0c01000100010202020
200020001010100030001000400000a0100020083680002008228000200836e000200822
90000000b00000026060f000c00ffffffff01000000000000001c000000fb02100007000
0000000bc02000000cc0102022253797374656d0017b70e661700000a0021008a0100000
0000000000070f31200040000002d01000004000000f0010100030000000000 –
импульсный отклик линейной системы.

Этот принцип иллюстрируется на рис. 1, где символ « + » на входе и
выходе означает, что аддитивная комбинация сигналов на входе приводит к
аддитивной комбинации выходных сигналов.
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02fefeff0202020202020202020202020202020202020202020202020202020202020202
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а) б)

Рис. 1. Представление линейной системы, для которой выполняется принцип
суперпозиции (а)и гомоморфной относительно свертки (б)

Символ « * » означает свертку в дискретном времени. По аналогии с
принципом суперпозиции для обычных линейных систем определим класс
систем, удовлетворяющих обобщенному принципу суперпозиции, в котором
сложение заменяется сверткой (легко показать, что свертка обладает
такими же алгебраическими свойствами, как и сложение:
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 (4)

Системы, обладающие свойством (4), названы гомоморфными относительно
свертки системами. Эта терминология объясняется тем, что данные
преобразования оказываются гомоморфными преобразованиями линейного
векторного пространства. При изображении таких систем (рис. 1, б)
операцию свертки представляют в явном виде на входе и выходе системы.
Гомоморфный фильтр является гомоморфной системой, обладающей тем
свойством, что одна компонента (выделяемая) проходит через эту систему
без изменений, а другая – устраняется. В соотношении (4), например, если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 – нежелательная компонента, то необходимо потребовать, чтобы
выход, соответствующий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020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
02020202020202020202020202020202040000002701ffff030000000000

Рис. 2. Каноническая форма системы, гомоморфной относительно свертки

Третий блок является обратным первому, т. е. преобразует сигналы,
представленные в виде суммы, в сигналы, представленные в виде свертки.
Важность данного канонического представления заключается в том, что
разработка гомоморфной системы сводится к разработке линейной системы.
Блок, обозначаемый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и называемый характеристическим блоком гомоморфной относительно свертки
системы, фиксирован при каноническом представлении (рис. 2). Очевидно,
что обратное преобразование также фиксировано. Характеристическая
система для гомоморфной обратной свертки подчиняется обобщенному
принципу суперпозиции, в котором операция на входе – свертка, а на
выходе – обычное сложение. Свойства характеристической системы
определяются выражением
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 (5)

Аналогично обратная характеристическая система удовлетворяет соотношению
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(6)

Математическое описание характеристической системы определяется
требованиями к выходному сигналу. Если на входе имеется сигнал свертки,
то
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(7)

и z-преобразование входного сигнала имеет вид
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Рис. 3. Представление системы, гомоморфной относительно свертки в
частотной области

Этот подход основан на том, что логарифм произведения равен сумме
логарифмов сомножителей, т. е.

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Если необходимо представлять сигналы во временной, а не в частотной
области, то характеристическая система примет вид, представленный на
рис. 4.

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02aaaa3f0202020202020202020202020202020202020202020202020202020202020202
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1c0c02020202020202020202020202020202020202020202020202020202020202020202
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02fefefe0202020202020202020202020202020202020202020202020202020202020202
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0e17020202020202020202020e1702020202020202020202020202020202020202020202
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0e1702020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
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02fefefe0202020202020202020202020202020202020202020202020202020202020202
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001202020202020202020202020202020202020202020202020202020202020202020202
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020202020202020202020202020202020202020202020202020202020202020202020202
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02fefefe0202020202020202020202020202020202020202020202020202020202020202
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020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
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020202020202020202020202020202020202020202020202020202020202020202020202
02fefefe0202020202020202020202020202020202020202020202020202020202020202
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020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
02020202020202020202020202020202020202020202020202fefefe0202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
02020202020202020202020202fefefe0202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
02fefefe0202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020304050202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202020202020202020202020202020202020202020202020202020202020202
020202020202060703020202020202020202020202020202021515e9040000002701ffff
030000000000

Рис.5. Представление характеристической системы, обратной гомоморфной
системе

Представление прямой и обратной характеристических систем зависит от
справедливости соотношения (9). Таким образом, логарифм должен быть
определен так, чтобы логарифм произвеления равнялся сумме логарифмов
сомножителей. Это тривиально для действительных положительных величин.
Однако в общем случае z-преобразование имеет комплексный характер и
вопрос единственности логарифма комплексной случайной величины
чрезвычайно важен. С точки зрения вычислений целесообразно рассмотреть
случай, когда (9) справедливо на единичной окружности, т. е. для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.

Для решаемых задач цифровой обработки вполне подходит определение
логарифма в виде
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 (10)

В этом соотношении действительная часть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 не вызывает
трудностей. Проблема единственности возникает при определении мнимой
части (т.е.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), которая представляет собой фазовый угол
z-преобразования, вычисленного на единичной окружности. Одним из
подходов к решению проблемы единственности является предположение, что
фазовый угол представляет собой непрерывную нечетную функцию. В этих
условиях уравнение (9) справедливо.

С учетом возможности вычисления комплексного логарифма, удовлетворяющего
(9), обратное преобразование комплексного логарифма преобразования Фурье
входного сигнала, являющееся выходом характеристической системы для
свертки, имеет вид

picscalex100010009000003d00300000400ce0000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c028004
20101300000026060f001c00ffffffff00004e0010000000c0ffffffc0ffffffe00f0000
400400000b00000026060f000c004d617468547970650000f00008000000fa0200001300
000000000002040000002d01000005000000140240025704050000001302400236060500
0000090200000002050000001402ad01e7041c000000fb0280fe00000000000090010000
00000002001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577cb0a
6698040000002d01010009000000320a000000000100000031790003050000001402a002
ff001c000000fb0280fe0000000000009001000000000002001054696d6573204e657720
526f6d616e00d89ff377e19ff3772020f577cb0a6698040000002d01020004000000f001
01000d000000320a0000000004000000282928294f01db064d020003050000001402ce03
6f041c000000fb0280fe0000000000009001000000000002001054696d6573204e657720
526f6d616e00d89ff377e19ff3772020f577cb0a6698040000002d01010004000000f001
020009000000320a000000000100000032790003050000001402f4015d0a1c000000fb02
22ff0000000000009001010000000002001054696d6573204e657720526f6d616e00d89f
f377e19ff3772020f577cb0a6698040000002d01020004000000f00101000c000000320a
000000000300000069696e842f02db00bc01050000001402a0024c001c000000fb0280fe
0000000000009001010000000002001054696d6573204e657720526f6d616e00d89ff377
e19ff3772020f577cb0a6698040000002d01010004000000f001020010000000320a0000
000006000000786e586565643d017406b0012f0222020003050000001402e900bc061c00
0000fb0222ff0000000000009001010000010002001053796d626f6c00008a080a47d4e5
1200d89ff377e19ff3772020f577cb0a6698040000002d01020004000000f00101000900
0000320a00000000010000007079bc01050000001402f401930a0a000000320a00000000
0200000077772f02bc010500000014023d04030709000000320a00000000010000007077
bc01050000001402a002d20e1c000000fb0280fe00000000000090010100000100020010
53796d626f6c00009f090a86d4e51200d89ff377e19ff3772020f577cb0a669804000000
2d01010004000000f001020009000000320a000000000100000077770003050000001402
ce031d0509000000320a0000000001000000707700030500000014023d0484061c000000
fb0222ff0000000000009001000000010002001053796d626f6c00008a080a48d4e51200
d89ff377e19ff3772020f577cb0a6698040000002d01020004000000f001010009000000
320a00000000010000002d77bc01050000001402a00226031c000000fb0280fe00000000
00009001000000010002001053796d626f6c00009f090a87d4e51200d89ff377e19ff377
2020f577cb0a6698040000002d01010004000000f001020009000000320a000000000100
00003d7700030500000014022a03c1061c000000fb02c0fd000000000000900100000001
0002001053796d626f6c00008a080a49d4e51200d89ff377e19ff3772020f577cb0a6698
040000002d01020004000000f001010009000000320a0000000001000000f27780040500
000014023a0260081c000000fb0280fe000000000000900100000001000200104d542045
7874726100090a88d4e51200d89ff377e19ff3772020f577cb0a6698040000002d010100
04000000f001020009000000320a0000000001000000267700030500000014028e026d00
09000000320a000000000100000026770003ce00000026060f0091014d61746854797065
55558501050100050244534d543500001357696e416c6c4261736963436f646550616765
7300110554696d6573204e657720526f6d616e00110353796d626f6c001105436f757269
6572204e65770011044d5420457874726100120008212f458f442f4150f4100f475f4150
f21f1e4150f4150f4100f445f425f48f425f4100f4100f435f4100f48f45f42a5f48f48f
4100f4100f40f48f417f48f4100f412a5f445f45f45f45f45f410f0c0100010001020202
0200020001010100030001000400000a010002018378000600020002008228000200836e
0002008229000204863d003d03000b0000010002008831000001000200883200020484c0
0370000003000f710001000201835800060002000200822800020083650003001c00000b
010101000200836900020484c9037700000a0200822900000b010002048612222d020484
c00370000100020484c00370000d0204862b22f2000a020083650003001c00000b010101
000200836900020484c903770200836e0000000a0200836400020484c903770000000b00
000026060f000c00ffffffff010000000000000008000000fa0200000000000000000000
040000002d0102001c000000fb021000070000000000bc02000000cc0102022253797374
656d0000cb0a669800000a0021008a0100000000ffffffff24f01200040000002d010300
04000000f0010100030000000000 (11)

Выход характеристической системы назван «комплексным кепстром» (термин
«кепстр» является в настоящее время общепринятым для обозначения
обратного преобразования Фурье логарифма спектра мощности сигнала;
термин «комплексный кепстр» означает, что применяется комплексный
логарифм).

Термин «кепстр» используется для величины
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(12)

Последовательность с(п) представляет собой четную часть комплексного
кепстра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: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.

Таким образом, определена характеристическая система для гомоморфной
свертки и каноническая форма всех гомоморфных систем относительно
свертки. Все системы этого класса отличаются только линейной частью.
Выбор линейной системы определяется свойствами входного сигнала.
Следовательно, для правильного построения линейной системы необходимо
прежде всего определить вид и структуру сигнала на выходе
характеристической системы, т.е. рассмотреть свойства комплексного
кепстра для типичных входных сигналов.

2. Комплексный кепстр речи

Модели сигналов, с одной стороны, и методы анализа во временной области
– с другой, можно объединить и эффективно использовать в теории
гомоморфной фильтрации речи. Вспомним, что модель речеобразования
обязательно состоит из линейной системы с медленно изменяющимися во
времени параметрами и сигнала возбуждения в виде последовательности
импульсов или белого шума. Поэтому короткий сегмент вокализованного
речевого сигнала целесообразно рассматривать как результат воздействия
сигнала возбуждения в виде последовательности импульсов на линейную
систему с постоянными параметрами. Аналогично короткий сегмент
невокализованного сигнала можно представить как результат возбуждения
линейной системы с постоянными параметрами случайным шумом.

Короткий сегмент вокализованной речи можно представить в виде
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(13)

где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 –
периодическая импульсная последовательность с периодом Np отсчетов;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 – импульсная характеристика линейной системы, отражающая
эффект формы источника возбуждения g(n), импульсную характеристику
речевого тракта
picscalex100010009000003820100000200780000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c020002
20031300000026060f001c00ffffffff00004e0010000000c0ffffffc6ffffffe0020000
c60100000b00000026060f000c004d617468547970650000500005000000090200000002
050000001402600106011c000000fb0280fe000000000000900100000000000200105469
6d6573204e657720526f6d616e00d89ff377e19ff3772020f577030a66c7040000002d01
00000a000000320a0000000002000000282950010003050000001402600190011c000000
fb0280fe0000000000009001010000000002001054696d6573204e657720526f6d616e00
d89ff377e19ff3772020f577030a66c7040000002d01010004000000f001000009000000
320a00000000010000006e2900030500000014026001feff1c000000fb0280fe00000000
00009001010000010002001053796d626f6c0000b1110aca88e91200d89ff377e19ff377
2020f577030a66c7040000002d01000004000000f001010009000000320a000000000100
00006e2900037800000026060f00e5004d617468547970655555d900050100050244534d
543500001357696e416c6c4261736963436f6465506167657300110554696d6573204e65
7720526f6d616e00110353796d626f6c001105436f7572696572204e65770011044d5420
457874726100120008212f458f442f4150f4100f475f4150f21f1e4150f4150f4100f445
f425f48f425f4100f4100f435f4100f48f45f42a5f48f48f4100f4100f40f48f417f48f4
100f412a5f445f45f45f45f45f410f0c0100010001020202020002000101010003000100
0400000a0100020484bd036e02008228000200836e0002008229000000030b0000002606
0f000c00ffffffff01000000000000001c000000fb021000070000000000bc02000000cc
0102022253797374656d00c7030a66c700000a0021008a010000000001000000d4f31200
040000002d01010004000000f0010000030000000000 и импульсную характеристику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Короткий сегмент невокализованной речи можно представить в виде
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 (14)

где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– сигнал
возбуждения в виде случайного шума;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 – импульсная реакция системы, объединяющая воздействие
речевого тракта и излучения.

Для случая вокализованной речи передаточная функция линейной системы
имеет вид
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 (15)

Для невокализованной речи получаем

picscalex100010009000003e00100000200970000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c024002
400b1300000026060f001c00ffffffff00004e0010000000c0ffffffa6ffffff000b0000
e60100000b00000026060f000c004d617468547970650000600005000000090200000002
050000001402800116021c000000fb0280fe000000000000900100000000000200105469
6d6573204e657720526f6d616e00d89ff377e19ff3772020f577d20c660a040000002d01
000010000000320a00000000060000002829282928294401060344018c01440100030500
00001402e30171011c000000fb0222ff0000000000009001010000000002001054696d65
73204e657720526f6d616e00d89ff377e19ff3772020f577d20c660a040000002d010100
04000000f001000009000000320a00000000010000007579bc0105000000140280014600
1c000000fb0280fe0000000000009001010000000002001054696d6573204e657720526f
6d616e00d89ff377e19ff3772020f577d20c660a040000002d01000004000000f0010100
10000000320a0000000006000000487a567a527a6c028802c2013e019201000305000000
1402800132041c000000fb0280fe0000000000009001000000010002001053796d626f6c
0000350a0ae298d91200d89ff377e19ff3772020f577d20c660a040000002d0101000400
0000f001000009000000320a00000000010000003d7900039700000026060f0023014d61
74685479706555551701050100050244534d543500001357696e416c6c4261736963436f
6465506167657300110554696d6573204e657720526f6d616e00110353796d626f6c0011
05436f75726965724d43590011044d5420457874726100120008212f458f442f4150f410
0f475f4150f21f1e4150f4150f4100f445f425f48f425f4100f4100f435f4100f48f45f4
2a5f48f48f4100f4100f40f48f417f48f4100f412a5f445f45f45f45f45f410f0c010001
00010202020200020001010100030001000400000a0100020083480003001b00000b0100
0200837500000101000a02008228000200837a0002008229000204863d003d0200835600
02008228000200837a000200822900020083520002008228000200837a00020082290000
008c0b00000026060f000c00ffffffff01000000000000001c000000fb02100007000000
0000bc02000000cc0102022253797374656d000ad20c660a00000a0021008a0100000000
00000000e4e31200040000002d01000004000000f0010100030000000000 (16)

Комплексный кепстр быстро затухает с ростом п. Кроме того, отметим, что
вклад в комплексный кепстр от периодического возбуждения проявится в
наличии импульсов в точках, кратных периоду возбуждения. Пример анализа
(рис.6) иллюстрирует основные особенности вокализованного речевого
сигнала.
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фазовая кривая, лишенная разрывов. Результат преобразования Фурье в
комплексный кепстр кривых на рис.6,б и 6,г представлен на рис.6,д.

Отметим наличие пиков в положительном и отрицательном времени и быстрое
затухание компонент в области малых времен, что обусловлено совместным
воздействием речевого тракта, источника возбуждения и излучением.
Кепстр, являющийся обратным преобразованием Фурье логарифма амплитуды
модуля спектра, показан на рис.6,е. В данном случае сохранены все
основные особенности комплексного кепстра, поскольку он является четной
частью комплексного кепстра.

Последовательность графиков на рис.6 показывает, как можно использовать
гомоморфную фильтрацию для анализа речевого сигнала. Прежде всего
отметим, что импульс в кепстре, обусловленный квазипериодическим
возбуждением, оказывается отделенным от остальных компонент. Это
приводит к соответствующей системе гомоморфной фильтрации речевого
сигнала, представленной на рис. 7.

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Сегмент речевого сигнала взвешивается с некоторым окном и требуемые
компоненты кепстра выделяются с использованием «окна по кепстру»
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00000a000000320a000000000200000028294f01000305000000140260012f001c000000
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