Signals and SystemsFall 2003Lecture #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:

DT:

"System Function"

CT

DT"System Function"

Fourier Series: Periodic Signals and LTI Systems

Soor powers of signals getmodified through filter/system Includes both amplitude & phase

Includes both amplitude & phase

The Frequency Response of an LTI System

CT Frequency response:

DT Frequency response:

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.

Note for DT:

Stopband Passband Stopband

Highpass Filters

Remember:

high frequenc

y

high frequenc

y

highest frequency in DT

Bandpass Filters

Demo: Filtering effects on audio signals

Idealized Filters

CT

DT

Note: |H| = 1 and ∠H = 0 for the ideal filters in the passbands, no need for the phase plot.

ωc — cutoff frequency

Stopband Passband Stopband

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

Original sign Filtered sign

D

olla

rs

Year

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

HPF

BPF #1

BPF #M

LPF

Demo: Apply different filters to two-dimensional image signals.

Face of a monkey.

Note: To really understand these examples, we need to understand frequency contents of aperiodic signals ⇒ the Fourier Transform

Image removed do to copyright considerations