FIR filter design - Continuous and Discrete Time Signals and Systems - page 669

Fig. 15.2. Commonly used
windows of length N .

w [ k ]

1

rectangular

Bartlett

Hanning

Blackman
Hamming

k

0 N − 1 N − 1

2

Generalized Hamming window For 0 <α< 1,

2 π k
N

α − (1 − α ) cos 0 ≤ k ≤ N − 1

w gene [ k ] = − 1 (15.8)

0 otherwise.

Hamming window

2 π k
N − 1

0 . 54 − 0 . 46 cos 0 ≤ k ≤ N − 1

w hamm [ k ] = (15.9)

0 otherwise.

Hanning window

2 π k
N − 1

0 . 5 − 0 . 5 cos 0 ≤ k ≤ ( N − 1)

w hann [ k ] = (15.10)

0 otherwise.

Blackman window

2 π k
N − 1

4 π k
N − 1

0 . 42 − 0 . 5 cos + 0 . 08 cos 0 ≤ k ≤ N − 1

w blac [ k ] =

0 otherwise.
(15.11)
The shapes of the windows are shown in Fig. 15.2, where, for convenience of
illustration, continuous plots are used. In reality, the windows are a function of
the DT variable k . It may be noted that the Hamming and Hanning windows are
special cases of the generalized Hamming window. For the Hamming window,
variable α in Eq. (15.8) of the generalized Hamming window equals 0.54.
Similarly, for the Hanning window, variable α in Eq. (15.8) equals 0.5.
The DTFTs of the aforementioned windows are shown in Fig. 15.3, where
the vertical axis represents the magnitude of the DTFTs based on the decibel
(dB) scale. The two important parameters used in the FIR filter design are (i) the
width of the main lobes of the DTFT of the windows; (ii) the relative strength
of the highest value side lobe with respect to the main lobe. The width of the
main lobe is defined as the distance between the nearest zero crossings of the
main lobe, while the relative side lobe strength is defined as the difference in
dB between the magnitudes of the highest value side lobe and the main lobe.