Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 57

It can be seen that Equation 2.152 is achieved for a rectangular window w ( n )
defined as

w ( n )= 1 f0

≤ n ≤ N − 1

(2.154)

0 otherwise

The frequency response can also be obtained using the convolution theorem,
which gives

π

H (e jω )= 1
π

H I (e jθ )e j ( ω−θ ) d θ (2.155)

−π

Now, as the window size w ( n ) shrinks, the actual filter response H (e jω ) will
approach that of the ideal response H I (e jω ). For a rectangular window, the
number of samples N also affects the transfer characteristics of the filter, such
as the peak amplitude and oscillatory behavior. There are six popular types
of window (Akay, 1994), which are listed here.

1. Rectangular . For 0 ≤ n ≤ N − 1

w ( n )=1

2. Bartlett

⎧
⎨

2 n

N −

1

0 ≤ n ≤

N − 1 2

w ( n )=

⎩

2 n N − 1

2 − ≤ n ≤ N − 1

N − 1 2

3. Hanning . For 0 ≤ n ≤ N − 1, the window is defined as

1

cos 2 πn
N

w ( n )= 1
2

−

− 1

This is also referred to as the raised cosine window because it is
symmetric.

4. Hamming . For 0 ≤ n ≤ N − 1, the window is defined as

0 . 46 cos 2 πn
N

w ( n )=0 . 54 −

− 1

This window is also known as the raised cosine platform, which has N
nonzero terms.

5. Blackman . For 0 ≤ n ≤ N − 1, the window function is written as

0 . 5 cos 2 πn
N

+0 . 08 4 πn
N

w ( n )=0 . 42 −

− 1 − 1