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Lecture 29

Lecture 29. Review: Frequency response Frequency response examples Frequency response plots & signal spectra Filters Related educational modules: Section 2.8.1, 2.8.2. Frequency Response. Systems are characterized in the frequency domain, by their frequency response, H(j )

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Lecture 29

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  1. Lecture 29 Review: Frequency response Frequency response examples Frequency response plots & signal spectra Filters Related educational modules: Section 2.8.1, 2.8.2

  2. Frequency Response • Systems are characterized in the frequency domain, by their frequency response, H(j) • Magnitude response: the ratio of the output amplitude to the input amplitude as a function of frequency • Phase response: the difference between the output phase and the input phase, as a function of frequency

  3. Magnitude and phase responses • Output:

  4. Review: RC circuit frequency response • Determine the magnitude and phase responses of the circuit below. vin(t) is the input and vout(t) is the output

  5. Annotate previous slide to denote |H|=mag resp, <H = phase resp.

  6. Example: RL circuit frequency response • Determine the magnitude and phase responses of the circuit below. vS(t) is the input and v(t) is the output

  7. Frequency response plots • Frequency responses are often presented graphically in the form of two plots: • Magnitude response vs. frequency • Phase response vs. frequency

  8. RC circuit frequency response plots

  9. RL circuit frequency response plots

  10. Signal spectra • The frequency domain content of a signal is called the spectrum of the signal • Example: v(t) = 3cos(t+20) + 7cos(2t-60) • Spectrum:

  11. Plots of signal spectra • Signal spectra plotted like frequency responses • Amplitude and phase vs. frequency • For our previous example:

  12. Graphical interpretation of system response • Plots of the input spectrum and frequency response can combine to provide an output spectrum plot • Point-by-point multiplication of magnitude plots • Point-by-point addition of phase plots • Can provide conceptual insight into system behavior

  13. Example – RL circuit response to example input

  14. Frequency selective circuits and filters • Circuits are often categorized by the general “shape” of their magnitude response • The response in some frequency ranges will be high relative to the input; these frequencies are passed • H(j) is “large” in these frequency ranges • The response in some frequency ranges will be low relative to the input; these frequencies are stopped • H(j) is “small” in these frequency ranges

  15. Filters • Circuits which select certain frequency ranges to pass and other frequency ranges to stop are often called frequency selective circuits or filters • Example: audio system graphic equalizer • The range (or band) of frequencies that are passed by the filter is called the passband • The range (or band) of frequencies that are stopped by the filter is called the stopband

  16. Specific case I – Lowpass filters • Lowpass filters pass low frequencies and stop high frequencies • The boundary between the two bands is the cutoff frequency, c • “Low” frequencies are less than c, “high” frequencies are greater than c

  17. On previous slide, note that IDEAL filters absolutely remove all components outside the passband. • Also point out that these cannot be implemented in the real world (turns out that they would need to respond to the input before the input is applied – they need to see into the future)

  18. Specific case II – Highpass filters • Highpass filters pass high frequencies and stop low frequencies • The boundary between the two bands is (still) called the cutoff frequency, c

  19. Additional filter categories • Filters are often categorized by the order of the differential equation governing the circuit • e.g. First order filter, second order filter • Filters can also be bandpass or bandstop • A band of frequencies between two cutoff frequencies is either passed or stopped • Lowpass & highpass filters can be first or higher order • Bandpass & bandstop filters must be at least second order • We will only work with first order filters in this course

  20. Filter example 1 – Lowpass filter • RC circuit:

  21. Filter example 2 – Highpass filter • RL circuit:

  22. Non-ideal first order filters • Realizable filters do not have sharp transitions between the passband and stopband • So where is the cutoff frequency (c)? • Define the cutoff frequency where the magnitude • response is times the maximum magnitude • Why? • The power is (generally) the square of the signal  the cutoff frequency is where we have half of the maximum power (it is sometimes called the half power point)

  23. RC circuit cutoff frequency • Magnitude response:

  24. Annotate previous slide to calculate maximum value and frequency where we have 0.707 times maximum value

  25. RL circuit cutoff frequency • Magnitude response:

  26. Annotate to show calculation of cutoff frequency

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