The Blackman-Harris window coefficients are given by the following formula
$$a(k)=0.35875-0.48829 \, \cos(\frac{2\pi k}{N-1})+0.14128 \, \cos(\frac{4\pi k}{N-1})$$ $$-0.01168 \, \cos(\frac{6\pi k}{N-1})$$
where N is the length of the filter and k = 0, 1, …, N – 1.
The Blackman-Harris window is a generalized cosine window (see Hamming window).
An example Blackman-Harris window
Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Blackman-Harris window.
Given a sampling frequency of 2000 Hz and a filter cutoff frequency of 40 Hz, the impulse response of the filter with a rectangular window (with no window) and with the Blackman-Harris window is as follows.
The magnitude response of the same filter is shown on the graph below.
Measures for the Blackman-Harris window
The following graph compares the discrete Fourier transform of the Blackman-Harris window and the rectangular window.
The Blackman-Harris window measures are as follows.
| Coherent gain | 0.36 |
| Equivalent noise bandwidth | 2.01 |
| Processing gain | -3.03 dB |
| Scalloping loss | -0.82 dB |
| Worst case processing loss | -3.85 dB |
| Highest sidelobe level | -92.0 dB |
| Sidelobe falloff | -14.4 dB / octave, -48.0 dB / decade |
| Main lobe is -3 dB | 1.90 bins |
| Main lobe is -6 dB | 2.66 bins |
| Overlap correlation at 50% overlap | 0.037 |
| Amplitude flatness at 50% overlap | 0.435 |
| Overlap correlation at 75% overlap | 0.459 |
| Amplitude flatness at 75% overlap | 1.000 |
See also:
Window
Comments
Error?
Shouldn't all terms be positive in the cos sum?
(0.35875 + 0.48829 * cos(...) +0.14128 * cos(...) +0.01168 * cos(...)
error: reply
No, I was wrong! you should have the +- changes for the different terms in the sum. But then, I think, you should add the (-1)^k in the sum on the page about Hamming (generalized cosine summation)
Not necessarily
On that page, the generalized power of cosine window is written as the sum of terms in the form ak cos(X). We could put (-1)^k if that makes it clearer, but the coefficients ak could be positive or negative. You don't necessarily have to add (-1)^k. In some sense, not adding it is more "generic" / general.
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