Hamming, Blackman, and Blackman-Harris

Python. This is my implementation for some of windows used in audio applications. I chose Hamming, Blackman, and Blackman-Harris window functions. Instead of using FFT pack, I directly applied DFT since the DFT size is not too big. I scaled the amplitude for a better view. In the figure, we see their main-lobe width and side-lobes. The main lobe and the highest side lobe are important for us. I chose the small window and DFT sizes (64 and 512).

import numpy as np
import matplotlib.pyplot as plt
 
def WindowFreqResponses(M = 64, N = 512):
 
	#Window and DFT sizes have to be even
	
	#HAMMING WINDOW 	
	nh = np.arange(-M/2, M/2)
	w1 = 0.54 + 0.46 * np.cos(2 * np.pi * nh / M)   
	
	#BLACKMAN WINDOW 	
	nb = np.arange(0, M)     	       
	w2 = 0.42 - 0.5 *np.cos(2 * np.pi * nb / M) + 0.08 * np.cos(4 * np.pi * nb / M)
	
	#BLACKMAN-HARRIS 	
	nbh = np.arange(-M/2, M/2)
	w3 = np.zeros(M)
	coefficient = [0.35875, 0.48829, 0.14128, 0.01168]
 
	for i in range(0, 4):
		w3 += coefficient[i] * np.cos(2 * np.pi * i * nbh / M)
	
	w3 = w3 / M	
	
	#Taking DFT and plotting for each window 
	
	plt.subplot(131)
	DFT(w1, M, N)
	plt.title('Hamming')
	plt.ylabel('amplitude (dB)')
	plt.subplot(132)
	DFT(w2, M, N)
	plt.title('Blackman')
	plt.xlabel('bins')
	plt.subplot(133)
	DFT(w3, M, N)
	plt.title('Blackman-Harris')
	
	plt.show()
	
	return
	
def DFT(w, M, N):
    
    hWindow = int(M/2)                                    #Half of window size    		    
    fftbuffer = np.zeros(N)                               #Zero-padding
    fftbuffer[:hWindow] = w[hWindow:]
    fftbuffer[N-hWindow:] = w[:hWindow]
    
    #DFT Calculation
	
    X = np.array([])                               
                                                     	      
    for k in range(N):                              
        s = np.exp(-1j * 2 * np.pi * k * np.arange(N)/ N)
        X = np.append(X, sum(fftbuffer*s))
        
    mX = abs(X)                                           #Amplitude
    mX[mX<np.finfo(float).eps] = np.finfo(float).eps      #For machine epsilon
    mXdb = 20*np.log10(mX)                                #Amplitude in dB
    
    hDFT = int(N/2)                                       #Half of DFT size
    scaledmX = np.zeros(N)                                #For a better view
    scaledmX[:hDFT] = mXdb[hDFT:]   	
    scaledmX[hDFT:] = mXdb[:hDFT]
    plt.plot(np.arange(-N/2,N/2)/float(N)*M, scaledmX - max(scaledmX), 'b')
    plt.axis([-15, 15, -100, 0])

    return

    ###
    
WindowFreqResponses()

import numpy as np

import matplotlib.pyplot as plt

def WindowFreqResponses(M = 64, N = 512):

#Window and DFT sizes have to be even

#HAMMING WINDOW

nh = np.arange(-M/2, M/2)

w1 = 0.54 + 0.46 * np.cos(2 * np.pi * nh / M)

#BLACKMAN WINDOW

nb = np.arange(0, M)

w2 = 0.42 - 0.5 *np.cos(2 * np.pi * nb / M) + 0.08 * np.cos(4 * np.pi * nb / M)

#BLACKMAN-HARRIS

nbh = np.arange(-M/2, M/2)

w3 = np.zeros(M)

coefficient = [0.35875, 0.48829, 0.14128, 0.01168]

for i in range(0, 4):

w3 += coefficient[i] * np.cos(2 * np.pi * i * nbh / M)

w3 = w3 / M

#Taking DFT and plotting for each window

plt.subplot(131)

DFT(w1, M, N)

plt.title('Hamming')

plt.ylabel('amplitude (dB)')

plt.subplot(132)

DFT(w2, M, N)

plt.title('Blackman')

plt.xlabel('bins')

plt.subplot(133)

DFT(w3, M, N)

plt.title('Blackman-Harris')

plt.show()

return

def DFT(w, M, N):

hWindow = int(M/2) #Half of window size

fftbuffer = np.zeros(N) #Zero-padding

fftbuffer[:hWindow] = w[hWindow:]

fftbuffer[N-hWindow:] = w[:hWindow]

#DFT Calculation

X = np.array([])

for k in range(N):

s = np.exp(-1j * 2 * np.pi * k * np.arange(N)/ N)

X = np.append(X, sum(fftbuffer*s))

mX = abs(X) #Amplitude

mX[mX<np.finfo(float).eps] = np.finfo(float).eps #For machine epsilon

mXdb = 20*np.log10(mX) #Amplitude in dB

hDFT = int(N/2) #Half of DFT size

scaledmX = np.zeros(N) #For a better view

scaledmX[:hDFT] = mXdb[hDFT:]

scaledmX[hDFT:] = mXdb[:hDFT]

plt.plot(np.arange(-N/2,N/2)/float(N)*M, scaledmX - max(scaledmX), 'b')

plt.axis([-15, 15, -100, 0])

return

###

WindowFreqResponses()

The results are (approximately):

Hamming: Main lobe width is 4 bins and side-lobe level is -42.7 dB.

Blackman: Main lobe width is 6 bins and side-lobe level is -58.2 dB.

Blackman-Harris: Main lobe width is 8 bins and side-lobe level is -92 dB.

References:

[1] Mathematics of the Discrete Fourier Transform, Julius O. Smith III

[2] FFT Bin Interpolation, Ted Knowlton

[3] Windows, Matlab Documentation