Digital Signal Processing - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 42

By keeping in mind the definition of the Dirac comb Ξ, we obtain:

() () ()

c

x t = Txt Ξ T t

[2.70]

1 c

Thus, by the Fourier transform, using the formulae [2.31], [2.70], [2.46] and
[2.64] successively, we have for all values of f :

1

( ) ()

xf T ˆ = xf ˆ

e c I

T

c

n ()
()

= x Ξ f

T

c

ˆ

= x ˆ Ξ f

⊗

T

c

⎛ ⎞

1

()

= ⎜ x ˆ Ξ f

⊗

⎟

⎜

1
c

⎟

T

⎝ ⎠

c

T

Then, using the definition of the convolution sum [2.39], the definition of the
Dirac comb [2.14] and the formula [2.10], we obtain:

1

+∞

( )

∫

() ( )

ˆ ˆ Ξ

xf T = xg f − g g

e c

1
c

T −∞

T

c

+∞

⎛ ⎞

1

+∞

k

∑

∫

()

ˆ

= x g δ f − − g dg

⎜ ⎟

T T

−∞

⎝ ⎠

c c

k =−∞

+∞

1

+∞

⎛ k ⎞

∑ ∫

()

ˆ

= x gf δ − − g g

⎜ ⎟

T T

−∞

⎝ ⎠

c c

k =−∞

+∞

⎛ ⎞

1 k

∑

ˆ

= xf −

⎜ ⎟

T T

⎝ ⎠

c c

k =−∞

This means that the Fourier transform of the sampled signal approximated to the
factor T c , is the sum of the Fourier transform of the continuous time signal and its

1
T

k
T

translated versions by a multiple of We obtain a periodic function of period ;

c c

we call this phenomenon the periodization of the spectrum.

For example, let us observe what happens when the Fourier transform of a
continuous signal () is of finite support {

}

x

ff < f and has the shape of

f

max