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
CT:
CT
"System Function"
DT:
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(ejω)) 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
Equalizer Speaker
Filter
Bass, Mid-range, Treble controls
For audio signals, the amplitude is much more important than the phase.
Example #2: Frequency Selective Filters
— Filter out signals outside of the frequency range of interest
Lowpass Filters:
Only show
amplitude here.
low low
frequency frequency
Highpass Filters
Remember:
high high
frequency frequency
Bandpass Filters
Demo: Filtering effects on audio signals
Idealized Filters
CT
ωc — 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
lower cut-off upper cut-off
DT
Example #3: DT Averager/Smoother
FIR (Finite Impulse
Response) filters
LPF
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.
Face of a monkey.
HP
Image removed do to
copyright considerations LP
LP BP
HP BP
Note: To really understand these examples, we need to understand
frequency contents of aperiodic signals ⇒ the Fourier Transform
