Амплітудно-фазова маніпуляція

Збільшення числа позицій маніпуляції m при ФМ, як це видно з (3.16) зменшує займану каналом передачі смугу частот. Однак при m > 8 істотно збільшується імовірність бітової помилки прийому P_пом.

Подальше зменшення займаної каналом передачі смуги частот можливе при використанні амплітудно-фазової маніпуляції (АФМ) (Amplitude-Phase Shift Keying–APSK), коли одночасно зі зміною фази сигналу змінюється і його амплітуда. Аналіз показує, що при m > 8 АФМ поряд з високою спектральною ефективністю характеризуються і більш високою в порівнянні з ФМ завадостійкістю. Так, при m = 16 енергетичний програш ФМ-16 у порівнянні з АФМ-16 може досягати 4,27 дБ, при m = 32 – 7,1 дБ, при m = 64 – 10,1 дБ, при m = 128 – 13,1 дБ.

При АФМ інформативними параметрами сигналу є амплітуда та фаза (комплексна амплітуда). Сигнал з m - позиційною АФМ можна записати в вигляді:

,

(3.19)

де S_i – амплітуда та φ _i – фаза i- го сигналу задаються відповідно до символів інформаційної послідовності.

Сигнал з АФМ як і ФМ сигнал можна також представити в квадратурному вигляді

,

(3.20)

де коефіцієнти d_I, d_Q визначаються, як і S_i та φ _i, символами повідомлення;

– синфазна та квадратурна складові сигналу.

Таким чином, АФМ сигнал може розглядатись як вектор з координатами d_I, d_Q в декартовій (3.20), або як вектор з відповідною амплітудою S_i і фазою φ _i в полярній системах координат:

.

(3.21)

В літературі представлення АФМ сигналу іноді дається через енергію сигналу Е_і та його тривалість Т

.

(3.22)

Представлення сигналу в вигляді (3.22) еквівалентне (3.19), так як

; ; ,

(3.23)

де P – середня потужність сигналу (нормирована на 1 Ом), – середньоквадратичне значення сигналу, яке менше амплітудного S ₀ в раз.

Представлення амплітуди сигналу у вигляді може бути використане і при представлені сигналів при інших видах маніпуляції.

Форма спектра АФМ сигналу не відрізняється від форми спектру ФМ сигналу. Як приклад на рис. 3.19 приведений графік енергетичного спектру
АФМ-16 сигналу.

2 МГц

f 0 = 100 МГц V = 4 Мбіт

Рис. 3.19. АЧС АФМ-16 сигналу

При дослідженні АФМ сигналів широко використовується їх геометрична теорія. Сигнали зображуються точками, що є кінцями векторів на площині. У термінах геометричної теорії оптимізація АФМ сигналу зводиться до знаходження такого розташування сигнальних точок, при якому області сигналів мають максимальну величину, найбільш близькі одна до іншої по розмірах і наближаються по формі до окружності.

У принципі для кожного числа m можна побудувати нескінченно велику кількість ансамблів АФМ сигналів. Тому важливим є вирішення завдання знаходження оптимальних ансамблів.

Користуючись геометричною теорією, кожному сигналу s_i (t), i = 1,..., m, можна поставити у відповідність деяку область простору сигналів А_i, яку називають власною областю або областю правильного прийому. Для дискретних повідомлень імовірність правильного прийому є ймовірність попадання кінця вектора прийнятого сигналу y (t) у власну область А_i сигналу s_i (t).

При рівно ймовірній передачі повідомлень оптимізація ансамблю полягає в такому розміщенні сигнальних точок, при якому власні області А_i, i = 1,..., m, приблизно однакові та мають максимальний об’єм (площу). Це завдання зводиться в загальному випадку до найщільнішого вкладання сфер однакового радіуса в m -мірному просторі сигналів або, як у розглянутому випадку, до найщільнішого вкладання окружностей на площині. При цьому центри сфер або окружностей відповідають сигнальним точкам.

Більшість відомих ансамблів АФМ сигналів знайдені евристичним методом. На рис. Ошибка! Неизвестный аргумент ключа. показані ансамблі сигналів на основі трикутної, квадратної та круглої мереж.

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М = 3

М = 7

М = 19

М = 4

М = 16

М = 8

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