2.4 Signal Processing Transformations

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37

other. In some cases of signal processing, the consideration in the frequency domain

is much clearer or simpler. This is particular the case at filters in relation to signals

that are already available in the form of their spectrum.

s(t)

h(t)

y(t) = s(t) ∗h(t)

S(jω)

H(jω)

Y (jω) = S(jω) H(jω)

Fig. 2.21: The relationship between the input and output of an LTI system is described by its impulse

response h(t) or its transfer function H(jω). The output quantity y(t) is calculated by the convolution

of the input quantity s(t) with the impulse response h(t), or in the frequency domain by the multi-

plication of the respective spectra.

The reverse transformation from the frequency to the time domain is done by the in-

verse Fourier transformation:

T⁻¹ {S(jω)} = s(t) = ¹

2π ^{∫S(jω)ejωtdω .}

(2.53)

In the previous considerations it was assumed that the Fourier transform exists. There

is no necessary condition for this. A sufficient condition regarding the convergence of

the Fourier transform is the absolute integrability of s(t), which is also known as the

Dirichlet condition¹⁴:

∫|s(t)|dt < ∞.

(2.54)

ne This condition is fulfilled by energy signals. The continuous Fourier transforms of

common signals are listed in Table 2.4. For signals that do not fulfil Equation 2.54, the

Tab. 2.4: Fourier transforms of important deterministic signals.

Signal

Time domain s(t)

Frequency domain S(jω)

Rectangular pulse

rect(t)

sin(ω/2)

ω/2

= si( ^ω

2 ⁾

Si-function

1

2π ^{si( t}

2⁾

rect(jω)

Dirac pulse

δ(t)

1

Constant

1

2πδ(jω)

Dirac pulse sequence

∑δ(t −kt0); k = . . . −1, 0, 1, . . .

ω0 ∑δ(ω −kω0); ω0 = ^2π

t0

Cosine-function

cos(ω0t)

π [δ(j(ω −ω0)) + δ(j(ω + ω0))]

Sine-function

sin(ω0t)

jπ [−δ(j(ω −ω0)) + δ(j(ω + ω0))]

Step-function

u(t)

πδ(ω) −j ¹

ω

󵄨󵄨󵄨󵄨ω ̸=0

Exponential pulse

1

t0 ^u(t)e−t

t0

1

1+jωt0

14 Dirichlet: German mathematician (1805–1859).