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# lecture17n - Stanford University Winter 2009-2010 Signal...

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Stanford University Winter 2009-2010 Signal Processing and Linear Systems I Lecture 17: Frequency Response, Bode Plots, and Filters March 4, 2010 EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 1 Frequency Response The Laplace transform H ( s ) exists for any s = σ + j ω such that the Laplace transform integral converges. One important special case is when σ =0 ,and s = j ω . This corresponds to the Fourier transform. It is also known as the frequency response H ( j ω ) of the system. ! ! s = j ! EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 2

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The j ω 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 speciFed frequency response Feedback Control where we want to modify the frequency response of an existing system EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 3 Types of Filters Basic idea: Pass some signals at some frequencies, suppress others ! | H ( j ! ) ! ! ! Lowpass Highpass Bandpass Bandstop, or Notch EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 4
Filter Terms | H ( j ! ) ! G p G s ! s Passband Transition Band Stopband Minimum Passband Gain Maximum Stopband Gain ! c EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 5 Ideal Filter Ideal lowpass flter 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. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 6

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Impulse response (inverse Fourier transform) h ( t ) t but this is not causal. Two possible solutions for causal ±lter are: Truncate response symmetrically: linear phase and increased transition width t h ( t ) Common for discrete time ±lters (next quarter). EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 7 Non-linear phase (here, minimum phase) t h ( t ) Common for continuous time ±lters (this quarter). To ±x up phase, can follow with an allpass flter . | H ( f ) | f H ( f ) f Truncated, Linear Phase | H ( f ) | f H ( f ) f Non-linear Phase EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 8
Functional Forms for Filters Conceptually, we can consider any impulse response to be a flter. Practically, we are going to consider flters that can be implemented as a discrete component circuit. This means that the Frequency response is a rational function .

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lecture17n - Stanford University Winter 2009-2010 Signal...

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