Дискретная свёртка с промежуточными точками
Я интерполирую сигнал с помощью свёртки с функцией кардинального синуса. Формула интерполяции:
Здесь x(t) - исходный сигнал, hs(t) - функция кардинального синуса, x^(t) - интерполированный сигнал. Формула, в принципе, понятна, и я набросал её наивную реализацию, которая между каждыми двумя соседними точками сигнала вставляет numValuesToInsert новых точек:
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
def interpolate_hard(time, samples, numValuesToInsert = 3):
samplingPeriod = time[1] - time[0]
N = samples.size
iTime = np.linspace(time[0], time[-1], time.size * (numValuesToInsert + 1) - numValuesToInsert)
iSamples = np.empty(samples.size * (numValuesToInsert + 1) - numValuesToInsert)
iSamplingPeriod = iTime[1] - iTime[0]
for j in range(iSamples.size):
summ = 0
for i in range(0, N):
summ += samples[i] * np.sinc((j * iSamplingPeriod - i*samplingPeriod) / samplingPeriod)
iSamples[j] = summ
return iTime, iSamples
samples = np.array([0.02463, 0.11107, 0.22659, 0.27451, 0.17808, -0.07051, -0.41178, -0.75116, -0.98205, -1.0257, -0.8773, -0.6021, -0.30154, -0.06442, 0.08611, 0.1938, 0.33872, 0.59051, 0.98102, 1.51593, 2.15984, 2.78196, 3.15713, 3.07422, 2.42536, 1.24996, -0.27762, -1.91811, -3.39176, -4.44633, -4.94513, -4.8844, -4.39088, -3.63353, -2.74111, -1.80521, -0.83655, 0.229, 1.44554, 2.80646, 4.21961, 5.50067, 6.40278, 6.69, 6.20554, 4.90547, 2.9173, 0.55879, -1.79256, -3.80937, -5.25163, -6.02999, -6.18016, -5.78575, -4.97589, -3.89464, -2.65637, -1.29684, 0.22418, 1.91521, 3.71543, 5.4786, 6.94698, 7.82051, 7.87678, 7.03643, 5.36419, 3.05524, 0.41419, -2.21788, -4.53591, -6.28681, -7.34791, -7.73723, -7.50724, -6.72628, -5.49016, -3.87737, -1.94811, 0.2052, 2.47964, 4.72682, 6.7114, 8.16833, 8.84068, 8.57557, 7.39706, 5.45901, 3.00372, 0.35264, -2.18094, -4.39346, -6.16812, -7.42778, -8.12701, -8.21951, -7.66299, -6.47108, -4.69, -2.40592, 0.2053, 2.93061, 5.53068, 7.72455, 9.23357, 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sampleRate = 1e6
sampleTime = np.arange(samples.size) / sampleRate
iTime, iSamples = interpolate_hard(sampleTime, samples)
plt.plot(sampleTime, samples, label = 'Samples')
plt.plot(iTime, iSamples, label = 'Interpolated')
plt.legend()
plt.show()
Функция работает невероятно долго. Я даже не стал дожидаться, когда эта программа завершится. Если ограничить размер сигнала 500-ми точками, то обработка у меня выполняется примерно 5 секунд.
Какие проблемы я здесь вижу?
Во-первых, это вычисление функции кардинального синуса на каждом шаге алгоритма, в то время как значения часто будут повторяться. Во-вторых, сама свёртка - крайне медленная операция. А с учётом того, что это потом нужно будет переносить в контроллер, требуется найти более быстрое решение.
Как я дальше решал
Я подумал, как можно здесь использовать стандартные функции convolve. На схеме ниже в верхней строке исходный сигнал, в нижней строке значения функции кардинального синуса для первых двух точек интерполированного сигнала, внутри квадратов - индексы.
Из схемы следует, что необходимо рассчитать numValuesToInsert+1 разных ядер свёртки и выполнить операцию свёртки исходного сигнала с каждым из них. С учётом того, что свёртка сигнала с 0-м ядром даст тот же сигнал, то эту операцию выполнять бессмысленно: можно просто взять значения из исходного сигнала. Так появилась новая реализация:
def interpolate(time, samples, numValuesToInsert = 3):
samplingPeriod = time[1] - time[0]
N = samples.size
iSamplingPeriod = samplingPeriod / (numValuesToInsert + 1)
iSamples = np.empty(N * (numValuesToInsert + 1) - numValuesToInsert)
# На позиции, кратные numValuesToInsert+1, копируем исходный сигнал
iSamples[np.arange(0, iSamples.size) % (numValuesToInsert + 1) == 0] = samples
for of in range(1, numValuesToInsert+1):
# Расчёт ядра свёртки
sinc = np.sinc(np.linspace((of * iSamplingPeriod - (N - 1) * samplingPeriod) / samplingPeriod,
((N - 1) * samplingPeriod + of * iSamplingPeriod) / samplingPeriod,
N * 2 + 1))
# Сама свёртка
iS = signal.convolve(samples, sinc, 'full')[N:2*N]
r = np.arange(0, iSamples.size) % (numValuesToInsert + 1) == of
iS = iS[:r[r==True].size]
# Копируем результат на нужные позиции
iSamples[r] = iS
iTime = np.linspace(time[0], time[-1], time.size * (numValuesToInsert + 1) - numValuesToInsert)
return iTime, iSamples
Эта функция уже работает нормально.
Вопросы
Есть ли способы решить это с помощью одной свёртки? Какие есть другие алгоритмы (например, в частотной области), с помощью которых можно достичь такого же результата?
Источник: Stack Overflow на русском