FFT of a Real Periodic Signal: Naive versus Windowed¶
FFT demo of a real, periodic signal¶
a) naive way b) windowing with DC treatment
FFT of a Real Periodic Signal: Naive versus Windowed¶
from matplotlib.pyplot import annotate, figure, grid, legend, plot, show, title, xlabel, xlim, ylabel
from numpy import arange, ceil, cos, fix, loadtxt, log2, mean, pi, sin, sqrt, zeros
from numpy.fft import fft
# redefine default figure size and fonts
import matplotlib as mpl
# mpl.rc('text', usetex = True)
mpl.rc('font', family = 'sans serif',size=16)
mpl.rc('figure',figsize=(12,8))
mpl.rc('lines', linewidth=1, color='lightblue',linestyle=':',marker='o')
Given periodic signal, sampling frequency and total time¶
f_s = 100.0 # sampling frequency (Hz)
T = 3.0 # total actual sample time (s)
g = loadtxt('data/FFT_Example_data_with_window.txt')
# first few values
g[:5]
array([-1.7527, -0.565 , 0.8801, 3.1924, 1.2804])
# first, visualize the signal
del_t = 1/f_s # time resolution [s]
t = arange(0.0,T+del_t,del_t) # time, t (s)
# plotSignal(t,g,f_s)
plot(t,g,marker='o',markerfacecolor='b',linestyle=':',color='lightgrey')
xlabel('$t$ (s)')
ylabel('$g_i$ (V)')
title('Periodic time signal sample')
show()
DC = 0.452950 [V]
Calculate the necessary parameters: \(N, \Delta t, \Delta f, f_{fold},N_{freq}\)¶
N = f_s * T # number of data points
del_t_1 = 1.0 / f_s # time resolution (s)
del_f = 1.0 / T # frequency resolution(Hz)
f_fold = f_s / 2.0 # folding frequency = Nyquist frequency of FFT (Hz)
N_freq = int(N / 2.0) # number of useful frequency points
Frequency analysis using naive FFT¶
[136.3379 +0.j 0.85689194+0.01756347j 0.7956033 +0.03999151j 0.83065943+0.02317239j 0.81648114+0.01029308j 0.86012694-0.03048062j 0.88366205-0.03279026j 0.91944369-0.03388283j 0.82537342-0.0616697j 0.78619988+0.03823325j] (301,)
Magnitude = abs(G)/(N_freq) # complex -> amplitude: |F|/(N/2)
figure()
plot(frequency,Magnitude[:N_freq])
grid('on')
xlim([-.5,f_fold])
xlabel('$f$ [Hz]')
ylabel('$|F|$ [V]')
annotate('DC', xy=(0,.9), xycoords='data',
xytext=(60, 60), textcoords='offset points',
size=20,
arrowprops=dict(arrowstyle="fancy",
fc="0.6", ec="none",
connectionstyle="angle3,angleA=0,angleB=90"),
)
show()
Notes¶
- Note the value at 0 Hz, what can we learn from it (we saw it's about 0.45 Volt?
- What do we learn from about the frequencies? about 2.1V at 16 Hz and 0.7 Volt at 47 Hz?
- Let's remove DC first and see the result
figure()
plot(t,g,t,g-DC,'ms')
xlabel('$t$ [sec]')
ylabel('$g$ [V]')
legend(('original','DC removed'))
# repeat the frequency analysis
G_1 = fft(g - DC) # FFT of the signal without DC
Magnitude1 = abs(G_1) / N_freq # complex -> amplitude: |F|/(N/2)
figure()
plot(frequency, Magnitude[:N_freq], frequency, Magnitude1[:N_freq], '--ms')
annotate('DC', xy=(0, 0), xycoords='data', xytext=(60, 60), textcoords='offset points', size=20, arrowprops=dict(arrowstyle='fancy', fc='0.6', ec='none', connectionstyle='angle3,angleA=0,angleB=45'))
grid('on')
xlim([-0.5, f_fold])
xlabel('$f$ [Hz]')
ylabel('$|F|$ [V]')
legend(('original', 'DC removed'))
show()
Let's use do it the right way:¶
- use 2^k number of points for faster FFT
- multiply the signal by a low-pass filter:
- assure there is no aliasing
- get read of the edges and make it less leaking
# let's check how much we gain if we do it right size:
fft(g)
fft(g[:256]) # 256 points instead of 301, not waisting much data
_g1 = zeros((512,))
_g1[:301] = g.copy()
# even if it's longer, but the right size with zeros at the end
fft(_g1)
array([ 1.36337900e+02+0.00000000e+00j, -2.15037892e+01-6.70899063e+01j,
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2.38009866e+00+4.40250032e+00j, -4.38903862e+00+4.01587044e+00j,
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-2.30327105e+00-3.04246134e-02j, 2.96478277e+00+7.05799808e+00j,
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-3.49126610e+00+1.22162369e+01j, -2.06970867e+00-9.59591827e-02j,
5.67850309e+00+1.61176004e+01j, -1.30017620e+01+9.62075975e+00j,
1.63920150e+01+1.16122483e+01j, -2.15037892e+01+6.70899063e+01j])
N_2 = 2 ** fix(log2(N)).astype(int)
T_2 = N_2 / f_s
del_f_2 = 1 / T_2
N_freq_2 = N_2 / 2
t_2 = arange(0.0, T_2 + del_t_1, del_t_1)
frequency_2 = arange(0, f_fold, del_f_2)
len2, = t_2.shape
_g_uncoupled_2 = g - DC
np.int64(256)
# create the low pass filter, called Hanning
_u_Hann_2 = 0.5 * (1 - cos(2 * pi * t_2[:-1] / T_2)) #u_Hanning(t)
_DC_2 = mean(g[:len2 - 1]) #DC = mean value of input signal (V) (average of all the useful data)
_g_uncoupled_2 = g[:len2 - 1] - _DC_2 # uncoupled
g_Hann_2 = _g_uncoupled_2 * _u_Hann_2
figure()
plot(_u_Hann_2, 'r--')
plot(g_Hann_2, 'b-')
plot(g, 'c:')
# take the FFT of the filtered, shorter signal
G_Hann_2 = fft(g_Hann_2,N_2) #G(omega) with Hanning window
Magnitude_Hann_2 = abs(G_Hann_2)*sqrt(8./3.)/(N_2/2) #|F|*sqrt(8/3)/(N/2)
# Magnitude_Hann_2[0] = Magnitude_Hann_2[0]/2 + DC_2 #(also divide the first one by 2, and add back the DC value)
# len_loc, = Magnitude_Hann_2.shape
# A_2 = Magnitude_Hann_2[0:round(len_loc/2)]
# Freq_2 = frequency_2[0:round(len_loc/2)]
figure()
plot(frequency,Magnitude[:frequency.shape[0]],'b:x')
plot(frequency_2,Magnitude_Hann_2[:frequency_2.shape[0]], 'r-s')
xlabel('frequency, (Hz)')
ylabel('abs(F)')
title('FFT Frequency Spectrum')
legend(('Simple','Window + DC'))
- 0: 0.3333333333333333
- 1: 0.390625
N_3 = 2 ** ceil(log2(N)).astype(int)
T_3 = N_3 / f_s
del_f_3 = 1 / T_3
N_freq_3 = int(N_3 / 2)
t_3 = arange(0.0, T_3 + del_t_1, del_t_1)
frequency_3 = arange(0, f_fold, del_f_3)
len3, = t_3.shape
_g_uncoupled_2 = g - DC
_u_Hann_2 = 0.5 * (1 - cos(2 * pi * t_2[:-1] / T_2))
_DC_2 = mean(g[:int(len2 - 1)])
_g_uncoupled_2 = g[:int(len2 - 1)] - _DC_2
g_Hann_2_1 = _g_uncoupled_2 * _u_Hann_2
g_Hann_3 = zeros((N_3,))
g_Hann_3[:g_Hann_2_1.shape[0]] = g_Hann_2_1.copy()
figure()
plot(g_Hann_3, 'b-')
G_Hann_3 = fft(g_Hann_3, N_3)
Magnitude_Hann_3 = abs(G_Hann_3) * sqrt(8.0 / 3.0) / (N_2 / 2)
len_loc, = Magnitude_Hann_3.shape
A_3 = Magnitude_Hann_3[0:int(len_loc / 2)]
Freq_3 = frequency_3[0:int(len_loc / 2)]
np.int64(512)
figure()
plot(frequency,Magnitude[:frequency.shape[0]],'b:x')
plot(frequency_2,Magnitude_Hann_2[:frequency_2.shape[0]], 'g--o')
xlabel('frequency, (Hz)')
plot(frequency_3,Magnitude_Hann_3[:frequency_3.shape[0]], 'r-s')
xlabel('frequency, (Hz)')
ylabel('abs(F)')
title('FFT Frequency Spectrum')
legend(('Simple','256', 'Window + DC'))
1.7914015486916908 15.625
b = a.copy()
b[:a.argmax()+10] = 0 # remove the first peak
a2,f2 = b.max(),f[b.argmax()]
print( a2,f2)
0.7161739235195177 46.875
t_1 = arange(0.0, T + del_t_1, del_t_1)
_g1 = DC + a1 * sin(2 * pi * f1 * t_1) + a2 * sin(2 * pi * f2 * t_1)
plot(t_1, g, 'b--', t_1, _g1, 'r-')
