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Unformatted text preview: UCLA Spring 20092010 Systems and Signals Lecture 17: Frequency Response, Bode Plots, and Filters May 26, 2010 EE102: Systems and Signals; Spr 0910, Lee 1 Frequency Response The Laplace transform H ( s ) exists for any s = σ + jω such that the Laplace transform integral converges. One important special case is when σ = 0 , and s = jω . This corresponds to the Fourier transform. It is also known as the frequency response H ( jω ) of the system. ℜ ℑ s = j ω EE102: Systems and Signals; Spr 0910, Lee 2 The jω axis may not be in the region of convergence, and the frequency response may not exist. If it does, the frequency response characterizes the system after the transients have died out, and the system is in steady state . We will consider two applications of frequency response: • Filter Design where we want to design a system with a specified frequency response • Feedback Control where we want to modify the frequency response of an existing system EE102: Systems and Signals; Spr 0910, Lee 3 Types of Filters Basic idea: Pass some signals at some frequencies, suppress others ω  H ( j ω )  ω  H ( j ω )  ω  H ( j ω )  ω  H ( j ω )  Lowpass Highpass Bandpass Bandstop, or Notch EE102: Systems and Signals; Spr 0910, Lee 4 Filter Terms  H ( j ω )  ω G p G s ω s Passband Transition Band Stopband Minimum Passband Gain Maximum Stopband Gain ω c EE102: Systems and Signals; Spr 0910, Lee 5 Ideal Filter Ideal lowpass filter is distortionless over a frequency band:  H ( f )  f ∠ H ( f ) f Unity passband with linear phase. A signal within the passband is delayed, but undistorted in amplitude or phase. EE102: Systems and Signals; Spr 0910, Lee 6 Impulse response (inverse Fourier transform) h ( t ) t but this is not causal. Two possible solutions for causal filter are: • Truncate response symmetrically: linear phase and increased transition width t h ( t ) Common for discrete time filters (next quarter). EE102: Systems and Signals; Spr 0910, Lee 7 • Nonlinear phase (here, minimum phase) t h ( t ) Common for continuous time filters (this quarter). To fix up phase, can follow with an allpass filter .  H ( f )  f ∠ H ( f ) f Truncated, Linear Phase  H ( f )  f ∠ H ( f ) f Nonlinear Phase EE102: Systems and Signals; Spr 0910, Lee 8 Functional Forms for Filters • Conceptually, we can consider any impulse response to be a filter. • Practically, we are going to consider filters that can be implemented as a discrete component circuit. • This means that the frequency response is a rational function ....
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 Spring '09
 Levan
 Digital Signal Processing, Frequency, Lowpass filter

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