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Filtering_Part2.txt
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Filtering_Part2.txt
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# omega_c = pi/4
from __future__ import division
from scipy import signal
wc = pi/4
M=20
N = 512 # DFT size
n = arange(-M,M)
h = wc/pi * sinc(wc*(n)/pi) # see definition of np.sinc()
w,Hh = signal.freqz(h,1,whole=True, worN=N) # get entire frequency domain
wx = fft.fftfreq(len(w)) # shift to center for plotting
fig,axs = subplots(3,1)
fig.set_size_inches((8,8))
subplots_adjust(hspace=0.3)
ax=axs[0]
ax.stem(n+M,h,basefmt='b-')
ax.set_xlabel("n",fontsize=16)
ax.set_ylabel("amplitude",fontsize=16)
ax=axs[1]
ax.plot(w-pi,abs(fft.fftshift(Hh)))
ax.axis(xmax=pi/2,xmin=-pi/2)
ax.vlines([-wc,wc],0,1.2,color='g',lw=2.,linestyle='--',)
ax.hlines(1,-pi,pi,color='g',lw=2.,linestyle='--',)
ax.set_xlabel(r"$\omega$",fontsize=22)
ax.set_ylabel(r"$|H(\omega)| $",fontsize=22)
ax.grid()
ax=axs[2]
ax.plot(w-pi,20*log10(abs(fft.fftshift(Hh))))
ax.axis(ymin=-40,xmax=pi/2,xmin=-pi/2)
ax.vlines([-wc,wc],10,-40,color='g',lw=2.,linestyle='--',)
ax.hlines(0,-pi,pi,color='g',lw=2.,linestyle='--',)
ax.grid()
ax.set_xlabel(r"$\omega$",fontsize=22)
ax.set_ylabel(r"$20\log_{10}|H(\omega)| $",fontsize=18)
#ax.axis( ymin=-40,xmax=pi/2)
# fig.savefig('[email protected]', bbox_inches='tight', dpi=300)
fig,ax = subplots()
fig.set_size_inches(6,3)
k=arange(M)
omega = linspace(0,pi,100)
ax.plot(omega,(sin(k*omega[:,None]+k*wc)-sin(k*omega[:,None]-k*wc)).sum(axis=1))
ax.set_ylabel(r"$Y_{re}(\omega)$",fontsize=18)
ax.grid()
ax.set_title(r"$\omega_c = \pi/4$",fontsize=22)
ax.set_xlabel(r" $\omega $",fontsize=22)
ax.set_xticks([0, pi/4,pi/2.,3*pi/4, pi,])
ax.set_xticklabels(['$0$',r'$\frac{\pi}{4}$',r'$\frac{\pi}{2}$',r'$\frac{3\pi}{4}$', r'$\pi$'],fontsize=18)
ax.set_xlim(xmax=pi)
ax.annotate("Gibbs phenomenon",xy=(pi/4,10),fontsize=14,
xytext=(20,0),
textcoords='offset points',
arrowprops={'facecolor':'b','arrowstyle':'->'})
# fig.savefig('[email protected]', bbox_inches='tight', dpi=300)
wc = pi/4
M=20
N = 512 # DFT size
n = arange(-M,M)
win = signal.hamming(len(n))
h = wc/pi * sinc(wc*(n)/pi)*win # see definition of np.sinc()
w,Hh = signal.freqz(h,1,whole=True, worN=N) # get entire frequency domain
wx = fft.fftfreq(len(w)) # shift to center for plotting
fig,axs = subplots(3,1)
fig.set_size_inches((8,8))
subplots_adjust(hspace=0.3)
ax=axs[0]
ax.stem(n+M,h,basefmt='b-')
ax.set_xlabel("n",fontsize=16)
ax.set_ylabel("amplitude",fontsize=16)
ax=axs[1]
ax.plot(w-pi,abs(fft.fftshift(Hh)))
ax.axis(xmax=pi/2,xmin=-pi/2)
ax.vlines([-wc,wc],0,1.2,color='g',lw=2.,linestyle='--',)
ax.hlines(1,-pi,pi,color='g',lw=2.,linestyle='--',)
ax.set_xlabel(r"$\omega$",fontsize=22)
ax.set_ylabel(r"$|H(\omega)| $",fontsize=22)
ax.grid()
ax=axs[2]
ax.plot(w-pi,20*log10(abs(fft.fftshift(Hh))))
ax.axis(ymin=-80,xmax=pi/2,xmin=-pi/2)
ax.vlines([-wc,wc],10,-80,color='g',lw=2.,linestyle='--',)
ax.hlines(0,-pi,pi,color='g',lw=2.,linestyle='--',)
ax.grid()
ax.set_xlabel(r"$\omega$",fontsize=22)
ax.set_ylabel(r"$20\log_{10}|H(\omega)| $",fontsize=18)
#ax.axis( ymin=-40,xmax=pi/2)
# fig.savefig('[email protected]', bbox_inches='tight', dpi=300)
Ns =300 # number of samples
N = 1024 # DFT size
fs = 1e3 # sample rate in Hz
fpass = 100 # in Hz
fstop = 150 # in Hz
delta = 60 # in dB, desired attenuation in stopband
from matplotlib.patches import Rectangle
M,beta= signal.fir_filter_design.kaiserord(delta, (fstop-fpass)/(fs/2.))
hn = signal.firwin(M,(fstop+fpass)/2.,window=('kaiser',beta),nyq=fs/2.)
w,H = signal.freqz(hn) # frequency response
fig,ax = subplots()
fig.set_size_inches((8,3))
ax.plot(w/pi*fs/2.,20*log10(abs(H)))
ax.set_xlabel("frequency( Hz )",fontsize=16)
ax.set_ylabel(r"$20\log_{10} |H(f)| $",fontsize=22)
ymin,ymax = -80,5
ax.axis(ymin = ymin,ymax=ymax)
ax.add_patch(Rectangle((0,ymin),width=fpass,height=ymax-ymin,color='g',alpha=0.3))
ax.add_patch(Rectangle((fpass,ymin),width=fstop-fpass,height=ymax-ymin,color='r',alpha=0.3))
ax.add_patch(Rectangle((fstop,ymin),width=fs/2-fstop,height=ymax-ymin,color='y',alpha=0.3))
ax.set_title("Number of taps=%d"%M)
ax.text(10,-15,'passband',fontsize=14,bbox=dict(color='white'))
ax.text(200,-15,'stopband',fontsize=16,bbox=dict(color='white'))
ax.grid()
# fig.savefig('[email protected]', bbox_inches='tight', dpi=300)
# two-tone example using filter
t = arange(0,Ns)/fs
x = cos(2*pi*30*t)+cos(2*pi*200*t)
X = fft.fft(x,N)
y=signal.lfilter(hn,1,x)
Y = fft.fft(y,N)
fig,ax = subplots()
fig.set_size_inches((10,4))
ax.plot(arange(N)/N*fs,20*log10(abs(X)),'r-',label='filter input')
ax.plot(arange(N)/N*fs,20*log10(abs(Y)),'g-',label='filter output')
ax.set_xlim(xmax = fs/2)
ax.set_ylim(ymin=-20)
ax.set_ylabel(r'dB',fontsize=22)
ax.set_xlabel("frequency (Hz)",fontsize=18)
ax.grid()
ax.annotate('attenuated in\nstopband',fontsize=16,xy=(200,32),
xytext=(50,3),textcoords='offset points',
arrowprops=dict(arrowstyle='->',lw=3),
)
ax.legend(loc=0);
# fig.savefig('[email protected]', bbox_inches='tight', dpi=300)