Digital Signal Processing - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 50

W 0 , T (t) (0 ≤ t ≤ T)

Rectangular 1

t
T

Hamming

0.54 - 0.46 cos (2π )

t
T

Hanning

0.50 - 0.50 cos (2π )

t
T

t
T

Blackman

0.42 - 0.50 cos (2π ) + 0.08 cos (4π )

2 t

T

1 −− 1

Bartlett

Table 2.2. Continuous time truncation windows

The first characteristic of a truncation window is the width of the main lobe. This
width can be measured by the -3 dB bandwidth, that is B -3dB ( T ), that is to say the
frequency interval for which the Fourier transform module

()

ˆ

w f

is greater

0, N − 1

w divided by 2 (or, similarly, the interval for which the
spectral energy density is greater than half its maximum value). The modulus of the
Fourier transform of a window with real values is even. This interval is thus
()

than its maximum value

0

()

⎡

BT BT

⎤ − 3dB − 3dB

− ,

⎦ . The equation to be solved to obtain this bandwidth is thus:
()

⎢ ⎥

2 2

⎣

()

ˆ

⎛ BT ⎞ = w 0

0, T

− 3dB

ˆ

w

[2.79] ⎜ ⎟

0, T

2

2

⎝ ⎠

()

ˆ 0, wf

Rectangular

− j π fT

e T sinc (π f T )

Hamming

− j π fT

e T {0.54 sinc (π f T ) + 0.23 sinc (π f T + π)

+ 0.23 sinc (π f T - π)}

Hanning

− j π fT

e T {0.50 sinc (π f T ) + 0.25 sinc (π f T + π)

+ 0.25 sinc (π f T - π)}

Blackman

− j π fT

e T {0.42 sinc (π f T ) + 0.25 sinc (π f T + π)

+ 0.25 sinc (π f T - π)

+ 0.04 sinc (π f T + 2 π)
+ 0.04 sinc (π f T - 2 π)}

Bartlett

T

− j π fT

e

2

Table 2.3. Transform of continuous time truncation windows