Основы ЦОС: генерация сигналов¶
Рассмотрим основные техники генерации периодических, импульсных и шумовых сигналов для тестирования алгоритмов цифровой обработки сигналов (ЦОС), моделирования систем связи, радаров и радиотехнических систем в целом. Основные темы примера:
- задание периодического сигнала
- временная сетка дискретного сигнала
- генерация типичных периодических сигналов (синусоида, меандр, пила)
- генерация дискретного шума
- последовательности импульсов
- частотно-модулированные сигналы
Используемые в примере библиотеки:
Генерация периодических сигналов¶
Периодический сигнал - это повторяющаяся последовательность. Создадим один период значений сигнала, а затем повторим его пять раз функцией repeat
. Визуализируем во времени ступенчатой линией:
Теперь попробуем сгенерировать периодический сигнал при помощи математических функций. К примеру, функция sin
высчитывает значение синуса для входного значения угла в радианах. Для того, чтобы сгенерировать синусоиду с требуемой частотой повторения, стоит воспользоваться вектором времени - последовательностью равномерно нарастающих отсчётов времени с указанным периодом дискретизации. Зададим вектор времени для сигналов с частотой дискретизации 1000 Гц:
Теперь создадим периодический синусоидальный сигнал с основной частотой 10 Гц (форма синусоиды будет повторяться каждые 0.1 сек). Переменная sine_wave
- это вектор из 500 отсчётов. Визуализируем сигнал функцией plot
:
Мы можем одной строчкой кода сгенерировать несколько синусоид с разными амплитудами, частотами и начальными фазами. Для этого зададим эти параметры векторами. Результирующая переменная three_sines
уже будет матрицей размером 500х3. Мы можем передать её функции plot
для одновременного отображения трёх графиков на одних осях:
Если нам интересно наблюдать форму суммы трёх синусоид, то мы можем сложить столбцы матрицы при помощи функции sum
:
Дополнительные функции для генерации сигналов¶
Часто нам требуются прямоугольные, пилообразные и треугольные периодические сигналы, а также случайные последовательности для моделирования шума.
Библиотека Waveforms.jl
содержит функции для генерации перечисленных периодических сигналов, но в качестве "задающего" сигнала, определяющего основную частоту, используется равномерно нарастающий пилообразный сигнал, изменяющийся в пределах от 0 до 2pi. Создадим его из вектора времени и зациклим операцией деления mod
на 2pi:
Популярный сигнал в радиотехнике - меандр. Он же прямоугольный периодический сигнал со скважностью 2 (или коэффициентом заполнения в 50%). Мы можем сгенерировать его функцией squarewave
. Также эта функция может принимать дополнительный входной аргумент - коэффициент заполнения в пределах от 0 до 1. Сгенерируем второй прямоугольный сигнал с 20% заполнения:
Пилообразный сигнал, изменяющийся в пределах от -1 до +1 можно получить функцией sawtoothwave
:
Также рассмотрим функцию trianglewave
для генерации треугольного сигнала:
В Engee доступны функции для генерации случайных чисел с различными распределениями. Воспользуемся функцией rand
для создания шумового сигнала с равномерным распределением:
Теперь попробуем объединить шум и пилообразный сигнал, но сделаем это таким образом, чтобы уровень шума возрастал с уровнем сигнала. При помощи задающего сигнала сгенерируем пилообразный, изменяющийся в пределах от 0 до +1. Затем умножим шум на получившийся пилообразный сигнал и просуммируем нарастающую "пилу" с нарастающим шумом:
Генерация непериодических сигналов и пачки импульсов¶
В качестве примера непериодического сигнала рассмотрим распространённый sinc
-импульс. Мы получаем его при помощи тригонометрической функции и входного вектора значений угла в радианах. В примере мы генерируем импульс в пределах от -2pi до +2pi (с шагом pi/16).
Теперь создадим пачку sinc
-импульсов, следующих с определённым периодом. Для этого добавим к вектору одного импульса некоторое количество нулей в конце, и размножим результат функцией repeat
. Также добавим нашей пачке импульсов затухание - в этом нам поможет функция LinRange
:
Частотно-модулированные сигналы¶
В заключении рассмотрим типичные частотно-модулированные сигналы, такие как сигналы с линейной частотной модуляцией (ЛЧМ) и сигналы с квадратурной частотной модуляцией. Для создания подобных сигналов мы воспользуемся функцией chirp
:
А для отображения сигналов мы вызовем функцию DSP.spectrogram
, которая визуализирует изменения спектра сигнала (зависимости его мощности от частоты) во времени. Мы можем наблюдать линейный рост частоты для ЛЧМ-сигнала и квадратичный в случае опции method = "quadratic"
:
Заключение¶
Мы рассмотрели основные техники генерации периодических, апериодических и шумовых сигналов, помогающих нам разрабатывать алгоритмы обработки сигналов и моделировать радиотехнические системы.
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