《信号与系统 Signals and Systems》课程教学资料（英文版）lecture 7 The Eigenfunction Property of Complex

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