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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:

$\displaystyle w_{BH}(n)= w_R(n)\sum_{l=0}^{L-1} \alpha_l \cos( l \Omega_M n), \protect$ (4.26)

where $ \Omega_M\isdef 2\pi/M$ , and $ w_R(n)$ is the length $ M$ zero-phase rectangular window (nonzero for $ n\in[-(M-1)/2,(M-1)/2]$ ). The corresponding window transform is given by

$\displaystyle W_{BH}(\omega) = \sum_{k=-(L-1)}^{L-1}\alpha_k W_R(\omega + k\Omega_M),$ (4.27)

where $ W_R(\omega) = M\cdot\hbox{asinc}_M(\omega)$ denotes the rectangular-window transform, and $ \Omega_M = 2\pi/M$ as usual.

Note that for $ L=1$ , we obtain the rectangular window, and for $ L=2$ , the BH family specializes to the generalized Hamming family.

Blackman Window Family

When $ L=3$ in (3.26), we obtain the Blackman family:

$\displaystyle w_{B}(n)= w_R(n)\left[\alpha_0 + \alpha_1 \cos(\Omega_M n) + \alpha_2 \cos(2\Omega_M n)\right]$ (4.28)

Relative to the generalized Hamming family (§3.2), we have added one more cosine weighted by $ \alpha_2$ . 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.

Classic Blackman

The so-called ``Blackman Window'' is the specific case for which $ \alpha_0 = 0.42$ $ \alpha_1 = 0.5$ , and $ \alpha_2 = 0.08$ . It has the following properties:

  • Side lobes roll off at about $ 18\dB $ per octave (like Hann)
  • Side-lobe level is about $ 58$ 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.

Figure 3.13: Classic Blackman window and transform.
\includegraphics[width=\twidth]{eps/blackmanWindow}

Three-Term Blackman-Harris Window

The classic Blackman window of the previous section is a three-term window in the Blackman-Harris family ($ L=2$ ), in which one degree of freedom is used to minimize 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:

  • $ \alpha_0 = 0.4243801$ $ \alpha_1 = 0.4973406$ , and $ \alpha_2 = 0.0782793$ .
  • Side-lobe level $ 71.48$ dB
  • Side lobes roll off $ \approx 6\dB $ 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)

Figure 3.14 plots the three-term Blackman-Harris Window and its transform. Figure 3.15 shows the same display for a much longer window of the same type, to illustrate its similarity to the rectangular window (and Hamming window) at high frequencies.

Figure 3.14: Three-term Blackman-Harris window and transform
\includegraphics[width=\twidth]{eps/blackmanHarris3}

Figure 3.15: Longer three-term Blackman-Harris window and transform
\includegraphics[width=\twidth]{eps/blackmanHarris3Long}

Frequency-Domain Implementation of the
Blackman-Harris Family

The Blackman-Harris window family can be very efficiently implemented in the frequency domain as a $ (2L-1)$ -point convolution with the spectrum of the unwindowed data.

For example, to implement a zero-phase Hann window,

  1. Start with a length $ M$ rectangular window
  2. Take an $ M$ -point DFT
  3. Convolve the DFT data with the 3-point smoother $ W=[1/4,1/2,1/4]$
Note that the frequency-domain implementation of the Hann window requires no multiplies in linear fixed-point data formats [188].

Similarly, any Blackman window may be implemented as a 5-point smoother in the frequency domain. More generally, any $ L$ -term Blackman-Harris window requires convolution of the critically sampled spectrum with a smoother of length $ 2L-1$ .

Power-of-Cosine Window Family

Definition:

$\displaystyle w(n)=w_R(n) \cos^P\left( \frac{ \pi n}{M} \right)$ (4.29)

where $ P$ is a nonnegative integer.

Properties:

  • The first $ P$ terms of the window's Taylor expansion, evaluated at the endpoints are identically 0 .
  • Roll-off rate $ \approx 6(P+1)$ dB/octave.

Special Cases:

  • $ P=0 \Rightarrow$ Rectangular window
  • $ P=1 \Rightarrow$ MLT sine window
  • $ P=2 \Rightarrow$ Hann window (``raised cosine'' = ``$ \cos^2$ '')
  • $ P=4 \Rightarrow$ Alternative Blackman (maximized roll-off rate)

Thus, $ \cos^P$ windows parametrize $ L$ -term Blackman-Harris windows (for $ L=P/2+1$ ) which are configured to use all degrees-of-freedom to maximize roll-off rate.

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Spectrum Analysis of an Oboe Tone
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Generalized Hamming Window Family