def plot_ratios(in_wave, out_wave):
# compare filters for cumsum and integration
diff_window = np.array([1.0, -1.0])
padded = thinkdsp.zero_pad(diff_window, len(in_wave))
diff_wave = thinkdsp.Wave(padded, framerate=in_wave.framerate)
diff_filter = diff_wave.make_spectrum()
cumsum_filter = diff_filter.copy()
cumsum_filter.hs = 1 / cumsum_filter.hs
cumsum_filter.plot(label='cumsum filter', color=GRAY, linewidth=7)
integ_filter = cumsum_filter.copy()
integ_filter.hs = integ_filter.framerate / (PI2 * 1j * integ_filter.fs)
integ_filter.plot(label='integral filter')
thinkplot.config(xlim=[0, integ_filter.max_freq],
yscale='log', legend=True)
thinkplot.save('diff_int8')
# compare cumsum filter to actual ratios
cumsum_filter.plot(label='cumsum filter', color=GRAY, linewidth=7)
in_spectrum = in_wave.make_spectrum()
out_spectrum = out_wave.make_spectrum()
ratio_spectrum = out_spectrum.ratio(in_spectrum, thresh=1)
ratio_spectrum.plot(label='ratio', style='.', markersize=4)
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, integ_filter.max_freq],
yscale='log', legend=True)
thinkplot.save('diff_int9')
def plot_ratios(in_wave, out_wave):
# compare filters for cumsum and integration
diff_window = np.array([1.0, -1.0])
padded = thinkdsp.zero_pad(diff_window, len(in_wave))
diff_wave = thinkdsp.Wave(padded, framerate=in_wave.framerate)
diff_filter = diff_wave.make_spectrum()
cumsum_filter = diff_filter.copy()
cumsum_filter.hs = 1 / cumsum_filter.hs
cumsum_filter.plot(label='cumsum filter', color=GRAY, linewidth=7)
integ_filter = cumsum_filter.copy()
integ_filter.hs = integ_filter.framerate / (PI2 * 1j * integ_filter.fs)
integ_filter.plot(label='integral filter')
thinkplot.config(xlim=[0, integ_filter.max_freq],
yscale='log',
legend=True)
thinkplot.save('diff_int8')
# compare cumsum filter to actual ratios
cumsum_filter.plot(label='cumsum filter', color=GRAY, linewidth=7)
in_spectrum = in_wave.make_spectrum()
out_spectrum = out_wave.make_spectrum()
ratio_spectrum = out_spectrum.ratio(in_spectrum, thresh=1)
ratio_spectrum.plot(label='ratio', style='.', markersize=4)
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, integ_filter.max_freq],
yscale='log',
legend=True)
thinkplot.save('diff_int9')
def fft_autocorr(signal):
N = len(signal)
signal = thinkdsp.zero_pad(signal, 2*N)
window = np.flipud(signal)
corrs = fft_convolve(signal, window)
corrs = np.roll(corrs, N//2+1)[:N]
return corrs
def fft_autocorr(signal):
"""Computes the autocorrelation function using FFT.
"""
N = len(signal)
signal = thinkdsp.zero_pad(signal, 2*N)
window = np.flipud(signal)
corrs = fft_convolve(signal, window)
corrs = np.roll(corrs, N//2+1)[:N]
return corrs
def fft_autocorr(signal):
"""Computes the autocorrelation function using FFT.
"""
N = len(signal)
signal = thinkdsp.zero_pad(signal, 2*N)
window = np.flipud(signal)
corrs = fft_convolve(signal, window)
corrs = np.roll(corrs, N//2+1)[:N]
return corrs
def make_filter(window, wave):
"""Computes the filter that corresponds to a window.
window: NumPy array
wave: wave used to choose the length and framerate
returns: new Spectrum
"""
padded = thinkdsp.zero_pad(window, len(wave))
window_wave = thinkdsp.Wave(padded, framerate=wave.framerate)
window_spectrum = window_wave.make_spectrum()
return window_spectrum
def make_filter(window, wave):
"""Computes the filter that corresponds to a window.
window: NumPy array
wave: wave used to choose the length and framerate
returns: new Spectrum
"""
padded = thinkdsp.zero_pad(window, len(wave))
window_wave = thinkdsp.Wave(padded, framerate=wave.framerate)
window_spectrum = window_wave.make_spectrum()
return window_spectrum
def plot_diff_filters(wave):
window1 = np.array([1, -1])
window2 = np.array([-1, 4, -3]) / 2.0
window3 = np.array([2, -9, 18, -11]) / 6.0
window4 = np.array([-3, 16, -36, 48, -25]) / 12.0
window5 = np.array([12, -75, 200, -300, 300, -137]) / 60.0
thinkplot.preplot(5)
for i, window in enumerate([window1, window2, window3, window4, window5]):
padded = thinkdsp.zero_pad(window, len(wave))
fft_window = np.fft.rfft(padded)
n = len(fft_window)
thinkplot.plot(abs(fft_window)[:], label=i + 1)
thinkplot.show()
def plot_diff_filters(wave):
window1 = np.array([1, -1])
window2 = np.array([-1, 4, -3]) / 2.0
window3 = np.array([2, -9, 18, -11]) / 6.0
window4 = np.array([-3, 16, -36, 48, -25]) / 12.0
window5 = np.array([12, -75, 200, -300, 300, -137]) / 60.0
thinkplot.preplot(5)
for i, window in enumerate([window1, window2, window3, window4, window5]):
padded = thinkdsp.zero_pad(window, len(wave))
fft_window = np.fft.rfft(padded)
n = len(fft_window)
thinkplot.plot(abs(fft_window)[:], label=i+1)
thinkplot.show()
def plot_gaussian():
"""Makes a plot showing the effect of convolution with a boxcar window.
"""
# start with a square signal
signal = thinkdsp.SquareSignal(freq=440)
wave = signal.make_wave(duration=1, framerate=44100)
spectrum = wave.make_spectrum()
# and a boxcar window
boxcar = np.ones(11)
boxcar /= sum(boxcar)
# and a gaussian window
gaussian = scipy.signal.gaussian(M=11, std=2)
gaussian /= sum(gaussian)
thinkplot.preplot(2)
thinkplot.plot(boxcar, label='boxcar')
thinkplot.plot(gaussian, label='Gaussian')
thinkplot.config(xlabel='Index', legend=True)
thinkplot.save(root='convolution7')
ys = np.convolve(wave.ys, gaussian, mode='same')
smooth = thinkdsp.Wave(ys, framerate=wave.framerate)
spectrum2 = smooth.make_spectrum()
# plot the ratio of the original and smoothed spectrum
amps = spectrum.amps
amps2 = spectrum2.amps
ratio = amps2 / amps
ratio[amps<560] = 0
# plot the same ratio along with the FFT of the window
padded = thinkdsp.zero_pad(gaussian, len(wave))
dft_gaussian = np.fft.rfft(padded)
thinkplot.plot(abs(dft_gaussian), color=GRAY, label='Gaussian filter')
thinkplot.plot(ratio, label='amplitude ratio')
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050])
thinkplot.save(root='convolution8')
def plot_gaussian():
"""Makes a plot showing the effect of convolution with a boxcar window.
"""
# start with a square signal
signal = thinkdsp.SquareSignal(freq=440)
wave = signal.make_wave(duration=1, framerate=44100)
spectrum = wave.make_spectrum()
# and a boxcar window
boxcar = np.ones(11)
boxcar /= sum(boxcar)
# and a gaussian window
gaussian = scipy.signal.gaussian(M=11, std=2)
gaussian /= sum(gaussian)
thinkplot.preplot(2)
thinkplot.plot(boxcar, label='boxcar')
thinkplot.plot(gaussian, label='Gaussian')
thinkplot.config(xlabel='Index', legend=True)
thinkplot.save(root='convolution7')
ys = np.convolve(wave.ys, gaussian, mode='same')
smooth = thinkdsp.Wave(ys, framerate=wave.framerate)
spectrum2 = smooth.make_spectrum()
# plot the ratio of the original and smoothed spectrum
amps = spectrum.amps
amps2 = spectrum2.amps
ratio = amps2 / amps
ratio[amps<560] = 0
# plot the same ratio along with the FFT of the window
padded = thinkdsp.zero_pad(gaussian, len(wave))
dft_gaussian = np.fft.rfft(padded)
thinkplot.plot(abs(dft_gaussian), color=GRAY, label='Gaussian filter')
thinkplot.plot(ratio, label='amplitude ratio')
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050])
thinkplot.save(root='convolution8')
def plot_fft_convolve():
"""Makes a plot showing that FFT-based convolution works.
"""
names = ['date', 'open', 'high', 'low', 'close', 'volume']
df = pd.read_csv('fb.csv',
header=0, names=names, parse_dates=[0])
close = df.close.values[::-1]
# compute a 30-day average using np.convolve
window = scipy.signal.gaussian(M=30, std=6)
window /= window.sum()
smoothed = np.convolve(close, window, mode='valid')
# compute the same thing using fft_convolve
N = len(close)
padded = thinkdsp.zero_pad(window, N)
M = len(window)
smoothed4 = fft_convolve(close, padded)[M-1:]
# check for the biggest difference
diff = smoothed - smoothed4
print(max(abs(diff)))
# compute autocorrelation using np.correlate
corrs = np.correlate(close, close, mode='same')
corrs2 = fft_autocorr(close)
# check for the biggest difference
diff = corrs - corrs2
print(max(abs(diff)))
# plot the results
lags = np.arange(N) - N//2
thinkplot.plot(lags, corrs, color=GRAY, linewidth=7, label='np.convolve')
thinkplot.plot(lags, corrs2.real, linewidth=2, label='fft_convolve')
thinkplot.config(xlabel='Lag',
ylabel='Correlation',
xlim=[-N//2, N//2])
thinkplot.save(root='convolution9')
def plot_fft_convolve():
"""Makes a plot showing that FFT-based convolution works.
"""
names = ['date', 'open', 'high', 'low', 'close', 'volume']
df = pd.read_csv('fb.csv',
header=0, names=names, parse_dates=[0])
close = df.close.values[::-1]
# compute a 30-day average using np.convolve
window = scipy.signal.gaussian(M=30, std=6)
window /= window.sum()
smoothed = np.convolve(close, window, mode='valid')
# compute the same thing using fft_convolve
N = len(close)
padded = thinkdsp.zero_pad(window, N)
M = len(window)
smoothed4 = fft_convolve(close, padded)[M-1:]
# check for the biggest difference
diff = smoothed - smoothed4
print(max(abs(diff)))
# compute autocorrelation using np.correlate
corrs = np.correlate(close, close, mode='same')
corrs2 = fft_autocorr(close)
# check for the biggest difference
diff = corrs - corrs2
print(max(abs(diff)))
# plot the results
lags = np.arange(N) - N//2
thinkplot.plot(lags, corrs, color=GRAY, linewidth=7, label='np.convolve')
thinkplot.plot(lags, corrs2.real, linewidth=2, label='fft_convolve')
thinkplot.config(xlabel='Lag',
ylabel='Correlation',
xlim=[-N//2, N//2])
thinkplot.save(root='convolution9')
def plot_filter(M=11, std=2):
signal = thinkdsp.SquareSignal(freq=440)
wave = signal.make_wave(duration=1, framerate=44100)
spectrum = wave.make_spectrum()
gaussian = scipy.signal.gaussian(M=M, std=std)
gaussian /= sum(gaussian)
high = gaussian.max()
thinkplot.preplot(cols=2)
thinkplot.plot(gaussian)
thinkplot.config(xlabel='Index',
ylabel='Window',
xlim=[0, len(gaussian) - 1],
ylim=[0, 1.1 * high])
ys = np.convolve(wave.ys, gaussian, mode='same')
smooth = thinkdsp.Wave(ys, framerate=wave.framerate)
spectrum2 = smooth.make_spectrum()
# plot the ratio of the original and smoothed spectrum
amps = spectrum.amps
amps2 = spectrum2.amps
ratio = amps2 / amps
ratio[amps < 560] = 0
# plot the same ratio along with the FFT of the window
padded = thinkdsp.zero_pad(gaussian, len(wave))
dft_gaussian = np.fft.rfft(padded)
thinkplot.subplot(2)
thinkplot.plot(abs(dft_gaussian), color=GRAY, label='Gaussian filter')
thinkplot.plot(ratio, label='amplitude ratio')
thinkplot.show(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050],
ylim=[0, 1.05])
def plot_derivative(wave, wave2):
# compute the derivative by spectral decomposition
spectrum = wave.make_spectrum()
spectrum3 = wave.make_spectrum()
spectrum3.differentiate()
# plot the derivative computed by diff and differentiate
wave3 = spectrum3.make_wave()
wave2.plot(color='0.7', label='diff')
wave3.plot(label='derivative')
thinkplot.config(xlabel='days',
xlim=[0, 1650],
ylabel='dollars',
loc='upper left')
thinkplot.save(root='diff_int4')
# plot the amplitude ratio compared to the diff filter
amps = spectrum.amps
amps3 = spectrum3.amps
ratio3 = amps3 / amps
thinkplot.preplot(1)
thinkplot.plot(ratio3, label='ratio')
window = np.array([1.0, -1.0])
padded = thinkdsp.zero_pad(window, len(wave))
fft_window = np.fft.rfft(padded)
thinkplot.plot(abs(fft_window), color='0.7', label='filter')
thinkplot.config(xlabel='frequency (1/days)',
xlim=[0, 1650/2],
ylabel='amplitude ratio',
#ylim=[0, 4],
loc='upper left')
thinkplot.save(root='diff_int5')
def plot_ratios(wave, wave2):
spectrum = wave.make_spectrum()
spectrum2 = wave2.make_spectrum()
amps = spectrum.amps
amps2 = spectrum2.amps
n = min(len(amps), len(amps2))
ratio = amps2[:n] / amps[:n]
thinkplot.preplot(1)
thinkplot.plot(ratio, label='ratio')
window = np.array([1.0, -1.0])
padded = thinkdsp.zero_pad(window, len(wave))
fft_window = np.fft.rfft(padded)
thinkplot.plot(abs(fft_window), color='0.7', label='filter')
thinkplot.config(xlabel='Frequency (1/days)',
#xlim=[0, spectrum.fs[-1]],
ylabel='Amplitude ratio',
#ylim=[0, 4],
loc='upper left')
thinkplot.save(root='diff_int3')
# In[3]: signal = thinkdsp.SquareSignal(freq=440) wave = signal.make_wave(duration=1, framerate=4400) segment = wave.segment(duration=0.01) # In[4]: window = np.ones(11) window /= sum(window) # In[5]: ys = segment.ys N = len(ys) padded = thinkdsp.zero_pad(window, N) # In[6]: prod = padded * ys # In[7]: sum(ys) # In[8]: convolved = np.convolve(ys, window, mode='valid') smooth2 = thinkdsp.Wave(convolved, framerate=wave.framerate) smooth2
def plot_boxcar():
"""Makes a plot showing the effect of convolution with a boxcar window.
"""
# start with a square signal
signal = thinkdsp.SquareSignal(freq=440)
wave = signal.make_wave(duration=1, framerate=44100)
# and a boxcar window
window = np.ones(11)
window /= sum(window)
# select a short segment of the wave
segment = wave.segment(duration=0.01)
# and pad with window out to the length of the array
N = len(segment)
padded = thinkdsp.zero_pad(window, N)
# compute the first element of the smoothed signal
prod = padded * segment.ys
print(sum(prod))
# compute the rest of the smoothed signal
smoothed = np.zeros(N)
rolled = padded
for i in range(N):
smoothed[i] = sum(rolled * segment.ys)
rolled = np.roll(rolled, 1)
# plot the results
segment.plot(color=GRAY)
smooth = thinkdsp.Wave(smoothed, framerate=wave.framerate)
smooth.plot()
thinkplot.config(xlabel='Time(s)', ylim=[-1.05, 1.05])
thinkplot.save(root='convolution2')
# compute the same thing using np.convolve
segment.plot(color=GRAY)
ys = np.convolve(segment.ys, window, mode='valid')
smooth2 = thinkdsp.Wave(ys, framerate=wave.framerate)
smooth2.plot()
thinkplot.config(xlabel='Time(s)', ylim=[-1.05, 1.05])
thinkplot.save(root='convolution3')
# plot the spectrum before and after smoothing
spectrum = wave.make_spectrum()
spectrum.plot(color=GRAY)
ys = np.convolve(wave.ys, window, mode='same')
smooth = thinkdsp.Wave(ys, framerate=wave.framerate)
spectrum2 = smooth.make_spectrum()
spectrum2.plot()
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude',
xlim=[0, 22050])
thinkplot.save(root='convolution4')
# plot the ratio of the original and smoothed spectrum
amps = spectrum.amps
amps2 = spectrum2.amps
ratio = amps2 / amps
ratio[amps<560] = 0
thinkplot.plot(ratio)
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050])
thinkplot.save(root='convolution5')
# plot the same ratio along with the FFT of the window
padded = thinkdsp.zero_pad(window, len(wave))
dft_window = np.fft.rfft(padded)
thinkplot.plot(abs(dft_window), color=GRAY, label='DFT(window)')
thinkplot.plot(ratio, label='amplitude ratio')
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050])
thinkplot.save(root='convolution6')
def plot_boxcar():
"""Makes a plot showing the effect of convolution with a boxcar window.
"""
# start with a square signal
signal = thinkdsp.SquareSignal(freq=440)
wave = signal.make_wave(duration=1, framerate=44100)
# and a boxcar window
window = np.ones(11)
window /= sum(window)
# select a short segment of the wave
segment = wave.segment(duration=0.01)
# and pad with window out to the length of the array
N = len(segment)
padded = thinkdsp.zero_pad(window, N)
# compute the first element of the smoothed signal
prod = padded * segment.ys
print(sum(prod))
# compute the rest of the smoothed signal
smoothed = np.zeros(N)
rolled = padded
for i in range(N):
smoothed[i] = sum(rolled * segment.ys)
rolled = np.roll(rolled, 1)
# plot the results
segment.plot(color=GRAY)
smooth = thinkdsp.Wave(smoothed, framerate=wave.framerate)
smooth.plot()
thinkplot.config(xlabel='Time(s)', ylim=[-1.05, 1.05])
thinkplot.save(root='convolution2')
# compute the same thing using np.convolve
segment.plot(color=GRAY)
ys = np.convolve(segment.ys, window, mode='valid')
smooth2 = thinkdsp.Wave(ys, framerate=wave.framerate)
smooth2.plot()
thinkplot.config(xlabel='Time(s)', ylim=[-1.05, 1.05])
thinkplot.save(root='convolution3')
# plot the spectrum before and after smoothing
spectrum = wave.make_spectrum()
spectrum.plot(color=GRAY)
ys = np.convolve(wave.ys, window, mode='same')
smooth = thinkdsp.Wave(ys, framerate=wave.framerate)
spectrum2 = smooth.make_spectrum()
spectrum2.plot()
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude',
xlim=[0, 22050])
thinkplot.save(root='convolution4')
# plot the ratio of the original and smoothed spectrum
amps = spectrum.amps
amps2 = spectrum2.amps
ratio = amps2 / amps
ratio[amps<560] = 0
thinkplot.plot(ratio)
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050])
thinkplot.save(root='convolution5')
# plot the same ratio along with the FFT of the window
padded = thinkdsp.zero_pad(window, len(wave))
dft_window = np.fft.rfft(padded)
thinkplot.plot(abs(dft_window), color=GRAY, label='DFT(window)')
thinkplot.plot(ratio, label='amplitude ratio')
thinkplot.config(xlabel='Frequency (Hz)',
ylabel='Amplitude ratio',
xlim=[0, 22050])
thinkplot.save(root='convolution6')
