Signals and Systems
Fall 2003
Lecture #9
2 October 2003
1,The Convolution Property of the CTFT
2,Frequency Response and LTI Systems Revisited
3,Multiplication Property and
Parseval’s Relation
4,The DT Fourier Transform
The CT Fourier Transform Pair
Last lecture,some properties
Today,further exploration
(Synthesis Equation)
(Analysis Equation)
Convolution Property
A consequence of the eigenfunction property:
Synthesis equation
for y(t)
The Frequency Response Revisited
The frequency response of a CT LTI system is simply the Fourier
transform of its impulse response
Example #1:
impulse response
frequency response
Example #2,A differentiator
1) Amplifies high frequencies (enhances sharp edges)
Larger at high ω
o phase shift
Differentiation property:
Example #3,Impulse Response of an Ideal Lowpass Filter
2) What is h(0)?
No.
Questions:
1) Is this a causal system?
3) What is the steady-state value of
the step response,i.e,s(∞)?
Example #4,Cascading filtering operations
H(jω)
Example #5:
Gaussian × Gaussian = Gaussian?
Gaussian? Gaussian = Gaussian
Example #6:
Example #2 from last lecture
Example #7:
Example #8,LTI Systems Described by LCCDE’s
(Linear-constant-coefficient differential equations)
Using the Differentiation Property
1) Rational,can use
PFE to get h(t)
2) If X(jω) is rational
e.g,
then Y(jω) is also rational
Parseval’s Relation
FT is highly symmetric,
We already know that:
Then it isn’t a
surprise that:
— A consequence of Duality
Convolution in ω
Multiplication Property
Examples of the Multiplication Property
For any s(t),..
Example (continued)
The Discrete-Time Fourier Transform
Define
DTFT Derivation (Continued)
DTFS synthesis eq.
DTFS analysis eq.
DTFT Derivation (Home Stretch)