B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

TABLE 3.2
Properties of Fourier Transform Operations
3 .4 Signal Transmission Through a Linear System 91
Operation
g (t)
G(f)
Superposition
g1 (t) + g2 (t)
Scalar multiplication kg(t)
Duality
G(t)
Time scaling
g(at)
Time shifting
g(t - to)
Frequency shifting g (t)ei2Hfot
Time convolution g1 (t) * g2 (t)
Frequency convolution gJ (t)g2 (t)
Time differentiation
d n g(t)
dt 11
Time integration f oo g ( x ) dx
Gt (f) + G2 (f)
kG(f)
g (-f )
1 c(1)
TaT
a
G(f) e-j 2Hji0
G(f
to)
G1 (f)G2 (f)
G1 (f) * G2 (f)
(j2nf) n G(f)
(!if) + l G(0)8(f)
12nf 2
Figure 3.24
Signal
transmission
through a linear
time-invariant
system.
Input signal
Output signal
Time-domain x(t) LTI system y (t) = h(t) * x (t)
h(t)
Frequency-domain X(f) H(f)
Y (f) = H(f) - X (f)
A stable LTI system can be characterized in the time domain by its impulse response h(t), which
is the system response to a unit impulse input, that is,
y(t) = h(t) when x(t) = 8(t)
The system response to a bounded input signal x(t) follows the convolutional relationship
y(t) = h(t) * x(t) (3.53)
The frequency domain relationship between the input and the output is obtained by taking
Fourier transform of both sides of Eq. (3.53). We let
x(t) ¢=:> X (f)
y(t) ¢=:> y (f)
h(t) ¢=:> H (f)
Then according to the convolution theorem, Eq. (3.53) becomes
Y(f) =H(f) - X (f) (3.54)
Generally H (f), the Fourier transform of the impulse response h(t), is refen-ed to as the
transfer function or the frequency response of the LTI system. Again, in general, H (f) is