Blackman-Harris Window Family
The Blackman-Harris (BH) window family is a straightforward generalization of the Hamming family introduced in §3.2. Recall from that discussion that the generalized Hamming family was constructed using a summation of three shifted and scaled aliased-sinc-functions (shown in Fig.3.8). The Blackman-Harris family is obtained by adding still more shifted sinc functions:where



![$ n\in[-(M-1)/2,(M-1)/2]$](http://www.dsprelated.com/josimages_new/sasp2/img403.png)
![]() |
(4.27) |
where




Blackman Window Family
When
![]() |
(4.28) |
Relative to the generalized Hamming family (§3.2), we have added one more cosine weighted by

Classic Blackman
The so-called ``Blackman Window'' is the specific case for which


- Side lobes roll off at about
per octave (like Hann)
- Side-lobe level is about
dB (worst case)
- One degree of freedom used to increase the roll-off rate from 6dB/octave (like rectangular) to 18 dB per octave by matching amplitude and slope to 0 at the window endpoints
- One degree of freedom is used to minimize side lobes (like Hamming)
- One degree of freedom is used to scale the window
Matlab for the Classic Blackman Window
N = 101; L = 3; No2 = (N-1)/2; n=-No2:No2; ws = zeros(L,3*N); z = zeros(1,N); for l=0:L-1 ws(l+1,:) = [z,cos(l*2*pi*n/N),z]; end alpha = [0.42,0.5,0.08]; % Classic Blackman w = alpha * ws;Figure 3.13 plots the classic Blackman Window and its transform.
Three-Term Blackman-Harris Window
The classic Blackman window of the previous section is a three-term window in the Blackman-Harris family (
-
, and
.
- Side-lobe level
dB
- Side lobes roll off
per octave in the absence of aliasing (like rectangular and Hamming)
- All degrees of freedom (scaling aside) are used to minimize side lobes (like Hamming)
Frequency-Domain Implementation of the
Blackman-Harris Family
The Blackman-Harris window family can be very efficiently implemented
in the frequency domain as a 
- Start with a length
rectangular window
- Take an
-point DFT
- Convolve the DFT data with the 3-point smoother


Power-of-Cosine Window Family
Definition:![]() |
(4.29) |
where

- The first
terms of the window's Taylor expansion, evaluated at the endpoints are identically 0 .
- Roll-off rate
dB/octave.
-
Rectangular window
-
MLT sine window
-
Hann window (``raised cosine'' = ``
'')
-
Alternative Blackman (maximized roll-off rate)



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Spectrum Analysis of an Oboe Tone
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Generalized Hamming Window Family