Blackman-Harris Window FamilyThe 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 , and is the length zero-phase rectangular window (nonzero for ). The corresponding window transform is given by
where denotes the rectangular-window transform, and as usual. Note that for , we obtain the rectangular window, and for , the BH family specializes to the generalized Hamming family.
Blackman Window FamilyWhen in (3.26), we obtain the Blackman family:
Relative to the generalized Hamming family (§3.2), we have added one more cosine weighted by . We now therefore have three degrees of freedom to work with instead of two. In the Hamming family, we used one degree of freedom to normalize the window amplitude and the second was used either to maximize roll-off rate (Hann) or side-lobe rejection (Hamming). Now we can use two remaining degrees of freedom (after normalization) to optimize these objectives, or we can use one for each, resulting in three subtypes within the Blackman window family.
Blackman Window'' is the specific case for which , and . It has the following properties:
- 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
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.
side-lobe level, and the other is used to maximize roll-off rate. Harris [101, p. 64] defines the three-term Blackman-Harris window as the one which uses both degrees of freedom to minimize side-lobe level. An improved design is given in Nuttall [196, p. 89], and its properties are as follows:
- , 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
The Blackman-Harris window family can be very efficiently implemented
in the frequency domain as a
-point convolution with the
spectrum of the unwindowed data.
For example, to implement a zero-phase Hann window,
- Start with a length rectangular window
- Take an -point DFT
- Convolve the DFT data with the 3-point smoother
where is a nonnegative integer. Properties:
- 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)
Spectrum Analysis of an Oboe Tone
Generalized Hamming Window Family