窓関数の種類について
DFTを行う場合、入力信号がDFT領域に整数周期で入っていないとリーケージ誤差(漏れ誤差)が発生する。 この誤差の影響を抑えるために窓関数を掛けるが、種類が色々ありどのような特性を持っているのか把握しきれていない。 とりあえず、scipy.signalに含まれる窓関数とよく使っている7次ブラックマンハリスについて特性の一覧を出してみた。
import numpy as np from scipy import signal from scipy.fftpack import fft, fftfreq, fftshift import matplotlib as mpl import matplotlib.pyplot as plt #import subprocess #print("subprocess imported") N = 64 FFTN = 4096 filename = "window_function.png" n = np.linspace(0,N,N,False) w = 2*np.pi*n/N # generate window w_boxcar = signal.boxcar(N, sym=False) w_barthann = signal.barthann(N, sym=False) w_bartlett = signal.bartlett(N, sym=False) w_blackman = signal.blackman(N, sym=False) w_4_blackmanharris = signal.blackmanharris(N, sym=False) w_7_blackmanharris = ( 0.27105140069342 -0.43329793923448*np.cos(w) +0.21812299954311*np.cos(2*w) -0.06592544638803*np.cos(3*w) +0.01081174209837*np.cos(4*w) -0.00077658482522*np.cos(5*w) +0.00001388721735*np.cos(6*w) ) w_bohman = signal.bohman(N, sym=False) w_chebwin = signal.chebwin(N, at=100, sym=False) w_cosine = signal.cosine(N, sym=False) w_exponential = signal.exponential(N, tau=30, sym=False) w_flattop = signal.flattop(N, sym=False) w_gaussian = signal.gaussian(N, std=N/8, sym=False) w_general_gaussian = signal.general_gaussian(N, p=1.5, sig=N/6, sym=False) w_hamming = signal.hamming(N, sym=False) w_hann = signal.hann(N, sym=False) w_kaiser = signal.kaiser(N, beta=np.pi*4, sym=False) w_nuttall = signal.nuttall(N, sym=False) w_parzen = signal.parzen(N, sym=False) w_slepian = signal.slepian(N, width=0.1, sym=False) w_triang = signal.triang(N, sym=False) w_tukey = signal.tukey(N, sym=False) #描画設定 #mpl.rcParams['font.family'] = 'sans-serif' mpl.rcParams['font.family'] = 'Times New Roman' mpl.rcParams['mathtext.default'] = 'regular' mpl.rcParams['xtick.top'] = 'True' mpl.rcParams['ytick.right'] = 'True' mpl.rcParams['xtick.direction'] = 'in' mpl.rcParams['ytick.direction'] = 'in' mpl.rcParams['axes.grid'] = 'True' mpl.rcParams['axes.xmargin'] = '0' mpl.rcParams['axes.ymargin'] = '.05' mpl.rcParams['savefig.facecolor'] = '#ffffff' mpl.rcParams['savefig.edgecolor'] = 'None' mpl.rcParams['savefig.bbox'] = 'tight' mpl.rcParams['axes.titlesize'] = 12 mpl.rcParams['axes.labelsize'] = 12 mpl.rcParams['lines.linewidth'] = 1.0 mpl.rcParams['lines.markersize'] = 10 mpl.rcParams['xtick.labelsize'] = 10 mpl.rcParams['ytick.labelsize'] = 10 fig, axs = plt.subplots(8,3,figsize=(14,28)) for ax in axs[:,0]: ax.set_xlim((0,2*np.pi)) ax.set_xticks(np.linspace(0, 2*np.pi, 11)) ax.set_xticklabels(["{0:.1f}$\pi$".format(x/np.pi) for x in ax.get_xticks()]) ax.set_xlabel("Phase [rad]") ax.set_ylim((-0.2,1.2)) ax.set_ylabel("Amplitude") ax.minorticks_on() ax.tick_params(which='both', direction='in', zorder=1.8) ax.grid(True,which='major') for ax in axs[:,1]: ax.set_xlim((-10,10)) ax.set_xlabel("bins") ax.set_ylim((-200,10)) ax.set_ylabel("Magnitude [dB]") ax.grid(True,which='both') for ax in axs[:,2]: ax.set_xlim((-N/2,N/2)) ax.set_xlabel("bins") ax.set_ylim((-200,10)) ax.set_ylabel("Magnitude [dB]") ax.grid(True,which='major') #freq = np.linspace(-N/2, N/2, FFTN,False) freq = fftshift(fftfreq(FFTN,1/N)) def mag(w): w_normalize = w/(np.sum(w)/len(w)) A = fft(w_normalize, FFTN) / (len(w)) responce = 20 * np.log10(np.abs(fftshift(A)+1E-32)) #responce[responce<-200] = None return responce def mags(ws): return np.array([mag(w) for w in ws]) g0 = np.array([ w_boxcar, w_hann, w_hamming, ]) g0_legend = [ "boxcar", "hann", "hamming", ] g1 = np.array([ w_bohman, w_chebwin, ]) g1_legend = [ "bohman", "chebwin", ] g2 = np.array([ w_blackman, w_nuttall, w_4_blackmanharris, w_kaiser, w_7_blackmanharris, ]) g2_legend = [ "blackman", "nuttall", "4 term blackman-harris", "kaiser", "7 term blackman-harris", ] g3 = np.array([ w_flattop, ]) g3_legend = [ "flattop", ] g4 = np.array([ w_tukey, w_parzen, ]) g4_legend = [ "tukey", "parzen", ] g5 = np.array([ w_triang, w_bartlett, w_barthann, w_hann, ]) g5_legend = [ "triang", "bartlett", "barthann", "hann", ] g6 = np.array([ w_gaussian, w_general_gaussian, ]) g6_legend = [ "gaussian", "general_gaussian", ] g7 = np.array([ w_cosine, w_slepian, ]) g7_legend = [ "cosine", "slepian", ] axs[0,1].set_ylim((-150,10)) axs[0,2].set_ylim((-150,10)) axs[0,0].plot(w, g0.T) axs[0,1].plot(freq, mags(g0).T) axs[0,2].plot(freq, mags(g0).T) axs[0,2].legend( g0_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[1,1].set_ylim((-150,10)) axs[1,2].set_ylim((-150,10)) axs[1,0].plot(w, g1.T) axs[1,1].plot(freq, mags(g1).T) axs[1,2].plot(freq, mags(g1).T) axs[1,2].legend( g1_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[2,1].set_ylim((-200,10)) axs[2,2].set_ylim((-200,10)) axs[2,0].plot(w, g2.T) axs[2,1].plot(freq, mags(g2).T) axs[2,2].plot(freq, mags(g2).T) axs[2,2].legend( g2_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[3,1].set_ylim((-0.5,0.5)) axs[3,2].set_ylim((-100,10)) axs[3,1].set_xlim((-2,2)) axs[3,0].plot(w, g3.T) axs[3,1].plot(freq, mags(g3).T) axs[3,2].plot(freq, mags(g3).T) axs[3,2].legend( g3_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[4,1].set_ylim((-150,10)) axs[4,2].set_ylim((-150,10)) axs[4,0].plot(w, g4.T) axs[4,1].plot(freq, mags(g4).T) axs[4,2].plot(freq, mags(g4).T) axs[4,2].legend( g4_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[5,1].set_ylim((-120,10)) axs[5,2].set_ylim((-120,10)) axs[5,0].plot(w, g5.T) axs[5,1].plot(freq, mags(g5).T) axs[5,2].plot(freq, mags(g5).T) axs[5,2].legend( g5_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[6,1].set_ylim((-100,10)) axs[6,2].set_ylim((-100,10)) axs[6,0].plot(w, g6.T) axs[6,1].plot(freq, mags(g6).T) axs[6,2].plot(freq, mags(g6).T) axs[6,2].legend( g6_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) axs[7,1].set_ylim((-80,10)) axs[7,2].set_ylim((-80,10)) axs[7,0].plot(w, g7.T) axs[7,1].plot(freq, mags(g7).T) axs[7,2].plot(freq, mags(g7).T) axs[7,2].legend( g7_legend, bbox_to_anchor = (1.05, 1), loc = 'upper left', borderaxespad = 0 ) fig.savefig(filename)
