4.2 Properties of Continuous-time Fourier Transform x(t) X(j F
ω) • Linearity
x(t) X(j F
ω) , y(t) F
Y(jω) ax(t) + by(t)
F
aX(jω) + bY(jω) • Time Shift x(t-t0) F
e
-jωto
X(jω)
linear phase shift (linear in frequency ) with amplitude unchanged • Conjugation x(t) *
X(-j F
*
ω)
if x(t) real , X(jω) conjugate symmetric X(-jω) = X( j*
ω) , x(t) real even/odd properties
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• Differentiation/Integration
F dx(t) jω X( jω)
dt
F 1 t
X(ω) + π X(0)δ(ω)
∫−∞ x(τ)dτ j ω
dc term from integration• Time/Frequency Scaling x(at) X( )
a | a|
F
x(-t) X(-jω)
See Fig. 4.11, p. 296 oftext
F 1 jω
- Inverse relationship between signal “width” in time/frequency domains - example: digital transmission (required bandwidth) α (bit rate) • Duality X(t) x(-ω)
F x(t) X( jω) expressed as x(t) X(ω)
F
F
time/frequency domains are kind of “symmetric” except for a sign change---“two domains”
See Fig. 4.17, p. 310 oftext
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