Signals and systems Fall 2003 Lecture #7 25 September 2003 1. Fourier Series and lti Systems Frequency Response and Filtering 3. Examples and demos
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 st h(se CT System Function H()= h(tes dt DT h DT System Function" H(a)=2hinlz-n
The Eigenfunction Property of Complex Exponentials DT: CT: CT "System Function" DT "System Function
Fourier Series: Periodic Signals and lti systems r() ∑ t)=∑ Howo hoJkwot ak→→H(k0)ak gain So|ak一→|H(jko) H(kwo)=H(kwo)le ∠H(kuo) or powers of signals get modified through filter/system includes both amplitude phase ∑akck y=∑H(ck kwon hInI k= k= k kwak jk H ∠H ncludes both amplitude phase
Fourier Series: Periodic Signals and LTI Systems
The Frequency response of an LTI System Hu) H( l weSt CT Frequency response: H(jw)=/ h(t)e utat H JejuN DT Frequency response: H(ebu ∑ hne y
The Frequency Response of an LTI System CT notation
Frequency shaping and filtering By choice of HGo(or H(e/o) as a function of @ we can shape the frequency composition of the output Preferential amplification Selective filtering of some frequencies Example #1: Audio System Adjustable Filter equalizer er Bass. Mid-range, Treble controls For audio signals, the amplitude is much more important than the phase
Frequency Shaping and Filtering • By choice of H(j ω) (or H(ej ω)) 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 Filter out signals outside of the frequency range of interest CT owpass filters IH(jo)I Only show amplitude here Stopband Passband Stopband Note for dt DT IH(ej o) H(eju )=H(ej(w+ low frequency frequency 2π 兀 兀
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 CT 0 Remember DT J丌7 T= highest frequency in dt H(ej h frequency frequenc 兀 2π 3π
Highpass Filters Remember: high frequency high frequency
Bandpass filters He/°儿 Demo: Filtering effects on audio signals
Demo: Filtering effects on audio signals Bandpass Filters
Idealized Filters CT frequency Stopband Passband Stopband DT 2 2π0 Note: H=l and zh=0 for the ideal filters in the passbands no need for the phase plot
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 JO) DT H(el 2π-2兀+0-兀一0 0l兀2兀-O270
Highpass CT DT
