This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1
[p. 3] Analog signal definition[p. 4] Periodic signal[p. 5] One-sided signal[p. 6] Finite length signal[p. 7] Impulse function[p. 9] Sampling property[p.11] Impulse properties[p.17] Continuous time system building blocks
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 17
CT Building Blocks
DELAY by to
INTEGRATOR (CIRCUITS)
DIFFERENTIATOR
MULTIPLIER & ADDER
Others…
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 18
Ideal Delay:
Mathematical Definition:
To find the IMPULSE RESPONSE of a system, h(t), let x(t) be an impulse, so
( ) ?h t =
y ( t ) = x ( t − t d )
System Sx(t) y(t)
System Sx(t) y(t)
x(t)=δ(t) y(t)=h(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 19
Output of Ideal Delay of 1 sec
y(t) = x(t −1) = e−(t−1)u(t −1)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 20
Integrator:
Mathematical Definition:
To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so
y ( t ) = x (τ−∞
t∫ )d τ
h(t) = δ (τ−∞
t
∫ )dτ = u(t)
Running Integral
System Sx(t) y(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 21
Integrator:
Integrate the impulse
IF t<0, we get zeroIF t>0, we get one
Thus we have h(t) = u(t) for the integrator
y ( t ) = x (τ−∞
t∫ ) d τ
δ (τ−∞
t
∫ )d τ = u ( t )
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 22
Graphical Representation
δ ( t ) =du ( t )
dt
1 0( ) ( )
0 0
t tu t d
tδ τ τ
−∞
≥⎧= = ⎨ <⎩
∫
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 23
Output of Integrator
( ) ( )t
y t x dτ τ− ∞
= ∫
y(t)=
System Sx(t) y(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 24
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 25
Differentiator Output:y ( t ) =
dx ( t )dt
y(t)=
Differentiator:System S
x(t) y(t)
Example:
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 26
Linear and Time-Invariant (LTI) Systems
If a continuous-time system is both linear and time-invariant, then the output y(t) is related to the input x(t)by a convolution integralconvolution integral
where h(t) is the impulse responseimpulse response of the system.
System Sx(t)
y(t)
x(t)=δ(t) h(t)=y(t)
06/12/10 2003rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 27