SIGNAL CONDITIONING, DATA ACQUISITION, AND SPECTRAL ANALYSIS - Design and Development of Medical Electronic Instrumentation - page 237

TABLE 5.6 Typical Window Functions for Use with the FFT in Spectral Estimation

3-dB Bandwidth Scallop Loss Highest

Window a (bins) (dB) Sidelobe (dB)

Rectangular

N
2 n

N
2

1 for 1

w ( n ) 0.89 3.92 13

0 for all other

Triangular

1 2 n

N 1

N
2

N
2 1

n

for

w ( n ) 1.28 1.82 27

0 for all other

Hamming

2

π (

(
N

2 n ) 1 1

)

N
2 n

N
2 1

0.54 0.46 cos for

1.30 1.78 43

w ( n )

0 for all other

Hanning

2

π (

(
N

2 n ) 1 1

)

N
2 n

N
2

0.5 0.5 cos for 1

w ( n ) 1.44 1.42 32

0 for all other

a Windows are N -point long and are assumed here to be symmetric around n

0.

sinusoidal with a frequency of 10 Hz. In a perfect system, this signal would not appear in
the PSD because it falls between two discrete frequency channels, much as a picket fence
allows us to see details in the scene behind it only if they happen to fall within a slot
between the boards. In reality, however, because the FFT produces slightly overlapping
“bins” of

finite bandwidth, components with frequencies that fall between the theoretical
discrete lines are distributed among adjacent bins, but at reduced magnitudes. This atten-
uation is the actual picket-fence or scalloping error.
Both of these problems are somewhat corrected by the use of an appropriate window. So
far, all samples presented to the FFT have been considered equal, which means that a weight
of 1 has been applied implicitly to all samples. The samples outside the FFT's scope are not
considered, and thus their e

fi

ective weight is zero, resulting in a “rectangular”-shaped win-
dow. This ultimately leads to discontinuities that cause leakage. A number of windows have
been devised that reduce the amplitude of samples at the edges of the window, while
increase the relevance of samples toward its center. By doing so, these windows reduce the
discontinuity to zero, thus lowering the amplitude of the sidelobes that surround a peak in
the PSD. In addition, use of a nonrectangular window increases the bandwidth of each bin,
which results in a decreased scalloping error.
Some typical window functions and their characteristics are presented in Table 5.6. In
essence, these functions produce N weights w 0 , w 1 , ... , w N- 1 which are weighted (multi-
plied) one to one with their corresponding data samples x 0 , x 1 , ... , x N- 1 before subjecting
them to an FFT:

ff

N 1

X ( f ) ∆ t w n x n e 2π jf n ∆ t

n 0