Blackman-Harris window

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.

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.

Impulse response of a low pass filter with and without the Blackman-Harris window

The magnitude response of the same filter is shown on the graph below.

Magnitude response of a low pass filter with and without the Blackman-Harris window

Measures for the Blackman-Harris window

The following graph compares the discrete Fourier transform of the Blackman-Harris window and the rectangular window.

Discrete Fourier transform of the Blackman-Harris 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

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