5.1 Discretisation of Continuous Signals

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151

gF(t) = A ⋅rect ( ^t

∆T ⁾

(5.2)

of the pulse shaper, which generates a rectangular pulse rect(t)¹ (see Figure 5.4):

f∆T(t) = fTa(t) ∗gF(t) =

∞

∫

τ=−∞

∞

∑

k=−.∞

f(kTa) ⋅δ(τ −kTa) ⋅A ⋅rect ( ^{t −τ}

∆T ^{) dτ}

= A

∞

∑

k=−∞

f(kTa)

∞

∫

τ=−∞

δ(τ −kTa) ⋅rect ( ^{t −τ}

∆T ^{) dτ}

= A

∞

∑

k=−∞

f(kTa) ⋅rect ( ^{t −kTa}

∆T

) .

(5.3)

Since a multiplication in the time domain corresponds to a convolution in the fre-

quency domain and, conversely, a convolution in the time domain corresponds to a

multiplication in the frequency domain, the associated spectrum can be determined

by a convolution of the input signal spectrum F(f) with the spectrum of a Dirac-pulse

train and then a multiplication of this spectrum with the spectrum of a square pulse:

fTa(t) =

∞

∑

k=−∞

f(kTa) ⋅δ(t −kTa) ,

FTa(f) = F(f) ∗F{

+∞

∑

ν=−∞

δ(t −νTa)}

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

∑^∞

ν=−∞

1

Ta ^δ(f−ν

Ta ⁾

=

+∞

∫

ψ=−∞

F(f −ψ) ⋅( ¹

Ta

+∞

∑

ν=−∞

δ (ψ −^ν

Ta

)) ⋅dψ

= ¹

Ta

+∞

∑

ν=−∞

F (f −^ν

Ta

)

and further using the relation for the rectangular function rect(t) and its Fourier trans-

form Frec(f):

Frec(f) = F {rect(t)} = si(πf)

(5.4)

and the similarity theorem

F {f(at)} = ¹

|a| ^{F ( j2πf}

a

)

with

a

̸= 0 , a = const.

(5.5)

1 rect(t) := 1 from a Dirac pulse for −0.5 ≤t ≤0.5; otherwise rect(t) := 0.