Digital Signal Processing - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 40

Dirac comb
The Fourier transform of the Dirac comb is a Dirac comb:

1

ˆ T

() ()

Ξ= Ξ f f

[2.64]

1/ T

T

This result is immediately obtained from formula [2.15] and the linearity of the
Fourier transform.

Kronecker sequence
From formula [2.21], we immediately obtain:
() ()

ˆ

δ v = 1 v

[2.65]

ℜ

Discrete time rectangular window
The Fourier transform of the gateway 1 0, N- 1 is expressed according to Dirichlet's
kernel (formula [2.17]):

( )

− j π N v − 1

1

() ()

vNe = diric v

[2.66]

0, N − 1 N

It is equal to N for all integer values of the abscissa.

Unit series
The Fourier transform of the constant series l z (k) is the Dirac comb Ξ 1 :
() ()

1

Ξ

[2.67]

v = v

Z 1

This result is obtained using formula [2.15].

Discrete time cisoid
We have the following transformation:
()

j 2 π vk

() ( )

ˆ

x ke =

0

xf = Ξ vv −

[2.68]

1 0

This means that the Fourier transform of the cisoid of frequency v 0 is a comb
centered on v 0 . Using the linearity of the Fourier transform, we obtain the Fourier
transform of real sine waves.

Unit comb
The Fourier transform of the unit comb is a Dirac comb:

1

ˆ Ξ

() ()

[2.69]

v = Ξ v

N

1

N

N

This result is obtained using formula [2.24].