Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 28

Fig. 1.17 Region of
convergence (shaded) of the
Laplace transform for
x(t) = e -a tu(t)

ω

j

5

ROC {LT[e − α t u ( t )]}

σ

0

−5

−5 0 5

σ = − α

LTI Analog System

X ( f ),
Input
Spectrum

Y ( f ),
Output
Spectrum

H ( f )

Y ( f ) = H ( f ) • X ( f )

Fig. 1.18 Frequency-domain representation of an LTI system

1.2.3.4 Mathematical Frequency-Domain Representation

An analog time-domain signal x(t) can be represented equivalently in the fre-
quency domain by its Fourier transform. Similarly, an LTI system with impulse
response h(t) can be represented equivalently in the frequency domain by its
transfer function, H(f) (see Fig. 1.18 ).
Recall that the system output can be described in the time domain as the
convolution of the input signal and the impulse response of the system:

y ð t Þ¼ h ð t Þ x ð t Þ

Recall also that convolution in the time domain is transformed into multiplication
in the frequency domain, and vise versa (see Tables, Laplace Transform Pairs and
Theorems). In the frequency domain, therefore, the system output is given by the
multiplication of the transfer function by the Fourier transform of the input signal:

Y ð f Þ¼ H ð f Þ X ð f Þ

where Y ð f Þ¼Ff y ð t Þg and X ð f Þ¼Ff x ð t Þg . Similar results are obtained if one
uses the Laplace transform:

Y ð s Þ¼ H ð s Þ X ð s Þ