Window Functions and Spectral Leakage - Signal Processing of Random Physiological Signals - page 192

1 dB

-20

1.2

1.0

-40

-n bins

0 T 0 n bins

(a) (b)

FIGURE 16.15 : The Exact Blackman Window. Trace (a) is the time domain weighted coeffi-
cients. Trace (b) shows the main frequency lobe and the resulting side lobes. The first sidelobe is
− 51 dB from the peak of the frequency main lobe

Notice that the third term deals with the second harmonic. If we add the terms

at n 0, something that we want to see. The side-lobe level of the

Exact Blackman Window is

= 0 then w ( n ) =

51 dB almost 3.5 times lower than the normal rectan-
gular window. The roll-off for the Exact Blackman Window is

−

18 dB/octave. These
characteristics are impressive, since the second side-lobe is at a level of

−

69 dB. What
these numbers indicate is that the second harmonic leakage is about 1200 times less
than the maximum value at the center frequency f 0 . For a normal power spectrum, most
analysts are not interested below

−

69 dB is exceptionally good
(the decibel values are referred to normalized spectra). The disadvantage of the Exact
Blackman Window is the coherent gain, which unfortunately is less than 0.5, closer to
0.47. However, this error can be compensated in the frequency domain by multiplying
with the inverse of the coherent gain.
The Exact Blackman Window time domain weighted coefficients are shown
in Fig. 16.15(a) with its resulting Log-magnitude of the Fourier transform shown in
Fig. 16.15(b).

− 60 dB; hence, a value of −

16.4 SUMMARY
Leakage is the result of making an infinitely long function finite in length. A smearing
effect is the result of convolving the infinite period of the signal, with the SINC function.