5.2 Discrete Transformations of Signal Processing

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5.2.2 The Discrete Fourier Transform (DFT)

If the signal to be sampled f(t) (and thus also its samples f(k)) is limited in time up to

a maximum time duration tg, the discrete-time Fourier-transformation can be simpli-

fied according to Equation 5.9. In this case, the signal to be sampled f(t) is continued

periodically, whereby the period duration Tp is greater than the maximum time dur-

ation tg of the original signal f(t) (tg < Tp). However, the periodic signal fp(t) has

a discrete spectrum which has a value only for multiples of the fundamental period

fp = 1/Tp. On the other hand, the signal f(t) is sampled, i.e. it also has a spectrum

periodic with the sampling frequency fa = 1/Ta. Thus, for a period fa in the frequency

domain only N = fa/fp = Tp/Ta values of the spectrum are needed to describe the peri-

odically continued sampling signal fp(t). In the time domain, the original continuous

signal f(t) can then be restored by temporally cutting off the periodically continued

signal fp(t) after one period and subsequent low-pass interpolation according to Fig-

ure 5.7 if the sampling theorem (Equation 5.7) is fulfilled.

The fact that a periodically continued signal fp(t) can be described by the values

of the continuous spectrum F(f) of the original non-periodic signal can be explained

by applying the convolution theorem in the time domain. For this purpose, analogous

to Figure 5.3, the continuous spectrum F(f) of the original signal f(t) is described by

multiplication with a periodic sequence of Dirac pulses δ(f)

P(f) := ¹

Tp

∞

∑

ν=−∞

δ (f −^ν

Tp

)

(5.12)

as per

FP(f) = F(f) ⋅P(f)

(5.13)

"sampled" in the frequency domain. A multiplication in the frequency domain corres-

ponds to a convolution in the time domain. Because of the relations

p(t) =

+∞

∑

ν=−∞

δ(t −νTp) ∘−−∙P(f) = ¹

Tp

∞

∑

ν=−∞

δ (f −^ν

Tp

)

fp(t) =

∞

∫

−∞

f(t −τ) ⋅p(τ) dτ ∘−−∙FP(f) = F(f) ⋅P(f)

between time and frequency domain is obtained after substituting in Equation 5.13

fp(t) =

∞

∫

−∞

f(t −τ) ⋅(

+∞

∑

ν=−∞

δ(τ −νTp)) dτ

=

+∞

∑

ν=−∞

f(t −νTp)

(5.14)

a periodic time function fp(t). We therefore obtain the result that frequency scanning

of a spectrum in the time domain results in a periodic signal. Conversely, however, it

also follows from this: