Signals and Systems
Fall 2003
Lecture #7
25 September 2003
1,Fourier Series and LTI Systems
2,Frequency Response and Filtering
3,Examples and Demos
The Eigenfunction Property of Complex Exponentials
DT:
CT:
CT
"System Function"
DT
"System Function"
Fourier Series,Periodic Signals and LTI Systems
The Frequency Response of an LTI System
CT notation
Frequency Shaping and Filtering
By choice of H(jω) (or H(e
jω
)) as a function of ω,we can shape
the frequency composition of the output
- Preferential amplification
- Selective filtering of some frequencies
Example #1,Audio System
Adjustable
Filter
Equalizer Speaker
Bass,Mid-range,Treble controls
For audio signals,the amplitude is much more important than the phase.
Example #2,Frequency Selective Filters
Lowpass Filters:
Only show
amplitude here.
— Filter out signals outside of the frequency range of interest
low
frequency
low
frequency
Highpass Filters
Remember:
high
frequency
high
frequency
Demo,Filtering effects on audio signals
Bandpass Filters
Idealized Filters
CT
ω
c
— cutoff
frequency
DT
Note,|H| = 1 and ∠H = 0 for the ideal filters in the passbands,
no need for the phase plot.
Highpass
CT
DT
Bandpass
CT
DT
lower cut-off upper cut-off
Example #3,DT Averager/Smoother
LPF
FIR (Finite Impulse
Response) filters
Example #4,Nonrecursive DT (FIR) filters
Rolls off at lower
ω as M+N+1
increases
Example #5,Simple DT,Edge” Detector
— DT 2-point,differentiator”
Passes high-frequency components
Demo,DT filters,LP,HP,and BP applied to DJ Industrial average
Example #6,Edge enhancement using DT differentiator
Courtesy of Jason Oppenheim.
Used with permission.
Courtesy of Jason Oppenheim.
Used with permission.
Example #7,A Filter Bank
Demo,Apply different filters to two-dimensional image signals.
Note,To really understand these examples,we need to understand
frequency contents of aperiodic signals? the Fourier Transform
HP
LP
BP
BP
LP
HP
Image removed do to
copyright considerations
Face of a monkey.
Fall 2003
Lecture #7
25 September 2003
1,Fourier Series and LTI Systems
2,Frequency Response and Filtering
3,Examples and Demos
The Eigenfunction Property of Complex Exponentials
DT:
CT:
CT
"System Function"
DT
"System Function"
Fourier Series,Periodic Signals and LTI Systems
The Frequency Response of an LTI System
CT notation
Frequency Shaping and Filtering
By choice of H(jω) (or H(e
jω
)) as a function of ω,we can shape
the frequency composition of the output
- Preferential amplification
- Selective filtering of some frequencies
Example #1,Audio System
Adjustable
Filter
Equalizer Speaker
Bass,Mid-range,Treble controls
For audio signals,the amplitude is much more important than the phase.
Example #2,Frequency Selective Filters
Lowpass Filters:
Only show
amplitude here.
— Filter out signals outside of the frequency range of interest
low
frequency
low
frequency
Highpass Filters
Remember:
high
frequency
high
frequency
Demo,Filtering effects on audio signals
Bandpass Filters
Idealized Filters
CT
ω
c
— cutoff
frequency
DT
Note,|H| = 1 and ∠H = 0 for the ideal filters in the passbands,
no need for the phase plot.
Highpass
CT
DT
Bandpass
CT
DT
lower cut-off upper cut-off
Example #3,DT Averager/Smoother
LPF
FIR (Finite Impulse
Response) filters
Example #4,Nonrecursive DT (FIR) filters
Rolls off at lower
ω as M+N+1
increases
Example #5,Simple DT,Edge” Detector
— DT 2-point,differentiator”
Passes high-frequency components
Demo,DT filters,LP,HP,and BP applied to DJ Industrial average
Example #6,Edge enhancement using DT differentiator
Courtesy of Jason Oppenheim.
Used with permission.
Courtesy of Jason Oppenheim.
Used with permission.
Example #7,A Filter Bank
Demo,Apply different filters to two-dimensional image signals.
Note,To really understand these examples,we need to understand
frequency contents of aperiodic signals? the Fourier Transform
HP
LP
BP
BP
LP
HP
Image removed do to
copyright considerations
Face of a monkey.
