В данном примере мы разберём возможности задания сигналов, способы их фильтрации и методы изменения частоты сигнала, а также проведём анализ оценки спектральной плотности мощности сигналов на каждом из этапов взаимодействия с сигналом.
Resolving package versions...
Updating `~/.project/Project.toml`
[a7b11256] + DigitalComm v1.2.0
[81d43f40] + Noise v0.3.3
Updating `~/.project/Manifest.toml`
[a7b11256] + DigitalComm v1.2.0
[81d43f40] + Noise v0.3.3
[d96e819e] + Parameters v0.12.3
[e409e4f3] + PoissonRandom v0.4.4
[3a884ed6] + UnPack v1.0.2
Precompiling project...
✓ Noise
1 dependency successfully precompiled in 4 seconds. 23 already precompiled.
Над данной синусоидой будут проведены все преобразования в примере.
┌ Warning: attempting to remove probably stale pidfile
│ path = "/user/.jlassetregistry.lock"
└ @ Pidfile /usr/local/ijulia-core/packages/Pidfile/DDu3M/src/Pidfile.jl:260
Проведём оценку спектральной плотности мощности сигнала через periodogram
— это функция построения периодограммы, основанная на вычислении квадрата модуля преобразования Фурье для последовательности данных.
Спектральная плотность мощности (СПМ) в физике и обработке сигналов — функция, описывающая распределение мощности сигнала в зависимости от частоты, то есть мощность, приходящаяся на единичный интервал частоты.
Следующим этапом преобразования нашего сигнала станет его квантование — разбиение диапазона отсчётных значений сигнала на конечное число уровней и округление этих значений до одного из двух ближайших к ним уровней.
Как мы видим, данное преобразование не влияет на спектральную плотность мощности сигнала и не смещает его частоту.
Следующее действие, которое мы проведём с исходной синусоидой, — это прореживание данных. Мы возьмём каждый второй отсчёт из исходного сигнала.
Периодограмма показывает, что прореживание данных увеличивает частоту сигнала. Это связано с тем, что из-за уменьшения количества отсчётов в два раза частота колебаний соответственно увеличилась в два раза.
Шум — это нежелательные явления, мешающие нам получать информацию из полезного сигнала.
Шум случаен по своей природе.
В цифровой обработке сигналов чаще всего используется аддитивный белый гауссовский шум (АБГШ).
Его характеристики:
- равномерная спектральная плотность мощности (поэтому он называется белым);
- нормальное распределение временных значений (поэтому он называется гауссовским);
- суммируется с полезным сигналом (поэтому он называется аддитивным);
- АБГШ статистически независим от полезного сигнала.
Важно отметить, что функция addNoise
задаёт кортеж, из которого нам нужен только первый вектор.
Далее в коде показаны типы данных входа addNoise
, его выхода и результата извлечения из кортежа первого вектора.
Out[0]:
Vector{Float64} (alias for Array{Float64, 1})
Out[0]:
Tuple{Vector{Float64}, Vector{Float64}}
Out[0]:
Vector{Float64} (alias for Array{Float64, 1})
Фильтрация сигнала в Engee¶
В данном примере мы разберём три фильтра, но в документации Engee вы можете найти любой необходимый вам фильтр.
Первый способ, который мы рассмотрим, — это задание фильтра через коэффициенты фильтра, в виде двух векторов a
и b
, затем реализуем представление фильтра через коэффициенты числителя
b
и знаменатель a
передаточной функции с помощью функции PolynomialRatio
, содержащейся в библиотеке DSP
.
Примечание:
b
и a
могут быть указаны как объекты Polynomial
, или
векторы, упорядоченные от наибольшей степени к наименьшей.
Данный фильтр не является децемирующим. В этом можно убедиться, сравнив размеры входа и выхода фильтра.
Второй вариант задания такого же фильтра — через импульсную характеристику при помощи функции impresp
, а затем вычисление выхода фильтра с помощью функции conv
.
Сравним выход функции filt
и выход функции conv
.
Как мы видим из графика, выходы функции filt
и выход функции conv
практически полностью совпадают.
Третий вариант фильтрации, который мы разберём, — это задание оконного КИХ-фильтра. Он будет отличаться от предыдущих, так как имеет задержку относительно входного сигнала, равную размеру окна.
Для сравнения выходов фильтров воспользуемся периодограммой Уэлча.
Как мы видим, основные различия между выходами фильтров кроются в их спектральной плотности мощности.
Изменение частоты сигнала¶
Для того, чтобы увеличить или уменьшить количество отсчётов, в Engee используется команда resample
, которая позволяет выполнить upsampling
и downsampling
сигнала.
Примечание: upsampling
и downsampling
задаются через отношение n//k
, где:
n
: количество точек, которые необходимо создать;
k
: количество ближайших соседей, которых следует учитывать для каждой точки.
Как мы видим, частота отфильтрованного сигнала увеличилась вдвое.
Как мы видим, частота отфильтрованного сигнала уменьшилась вдвое.
Размер входа: 2501
Размер при downsampling: 1251
Размер при upsampling: 5001
Те же эффекты мы можем наблюдать при сравнении сигналов и их размеров. При downsampling
уменьшается количество отсчётов сигнала, а при upsampling
, наоборот, увеличивается относительно исходного сигнала.
Вывод¶
В данном примере мы продемонстрировали возможности взаимодействия с сигналами и показали, что Engee может быть в полной мере применим для цифровой обработки сигналов.
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