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Fourier Transform Analysis of Signals and Systems
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  1. Fourier Transform Analysis of Signals and Systems Chapter 6

  2. Ideal Filters • Filters separate what is desired from what is not desired • In the signals and systems context a filter separates signals in one frequency range from signals in another frequency range • An idealfilter passes all signal power in its passband without distortion and completely blocks signal power outside its passband M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  3. Distortion • Distortion is construed in signal analysis to mean “changing the shape” of a signal • Multiplication of a signal by a constant or shifting it in time do not change its shape No Distortion Distortion M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  4. Distortion Since a system can multiply by a constant or shift in time without distortion, a distortionless system would have an impulse response of the form, or The corresponding transfer functions are or M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  5. Filter Classifications There are four commonly-used classification of filters, lowpass, highpass, bandpass and bandstop. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  6. Filter Classifications M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  7. Bandwidth • Bandwidth generally means “a range of frequencies” • This range could be the range of frequencies a filter passes or the range of frequencies present in a signal • Bandwidth is traditionally construed to be range of frequencies in positive frequency space M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  8. Bandwidth Common Bandwidth Definitions M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  9. Impulse Responses of Ideal Filters M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  10. Impulse Responses of Ideal Filters M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  11. Impulse Response and Causality • All the impulse responses of ideal filters are sinc functions, or related functions, which are infinite in extent • Therefore all ideal filter impulse responses begin before time, t = 0 • This makes ideal filters non-causal • Ideal filters cannot be physically realized, but they can be closely approximated M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  12. Impulse Responses and Frequency Responses of Real Causal Filters M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  13. Impulse Responses and Frequency Responses of Real Causal Filters M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  14. Causal Filter Effects on Signals M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  15. Causal Filter Effects on Signals M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  16. Causal Filter Effects on Signals M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  17. Causal Filter Effects on Signals M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  18. Two-Dimensional Filtering of Images Causal Lowpass Filtering of Rows in Image Causal Lowpass Filtering of Columns in Image M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  19. Two-Dimensional Filtering of Images “Non-Causal” Lowpass Filtering of Rows in Image “Non-Causal” Lowpass Filtering of Columns in Image M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  20. Two-Dimensional Filtering of Images Causal Lowpass Filtering of Rows and Columns in Image “Non-Causal” Lowpass Filtering of Rows and Columns in Image M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  21. The Power Spectrum M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  22. Noise Removal A very common use of filters is to remove noise from a signal. If the noise band is much wider than the signal band a large improvement in signal fidelity is possible. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  23. RC Lowpass Filter M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  24. RLC Bandpass Filter M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  25. Log-Magnitude Frequency-Response Plots Consider the two (different) transfer functions, When plotted on this scale, these magnitude frequency response plots are indistinguishable. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  26. Log-Magnitude Frequency-Response Plots When the magnitude frequency responses are plotted on a logarithmic scale the difference is visible. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  27. Bode Diagrams A Bode diagram is a plot of a frequency response in decibels versus frequency on a logarithmic scale. The Bel (B) is the common (base 10) logarithm of a power ratio and a decibel (dB) is one-tenth of a Bel. The Bel is named in honor of Alexander Graham Bell. A signal ratio, expressed in decibels, is 20 times the common logarithm of the signal ratio because signal power is proportional to the square of the signal. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  28. Bode Diagrams M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  29. Continuous-time LTI systems are described by equations of the general form, Fourier transforming, the transfer function is of the general form, Bode Diagrams M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  30. Bode Diagrams A transfer function can be written in the form, where the “z’s” are the values of jw at which the transfer function goes to zero and the “p’s” are the values of jw at which the transfer function goes to infinity. These z’s and p’s are commonly referred to as the “zeros” and “poles” of the system. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  31. Bode Diagrams From the factored form of the transfer function a system can be conceived as the cascade of simple systems, each of which has only one numerator factor or one denominator factor. Since the Bode diagram is logarithmic, multiplied transfer functions add when expressed in dB. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  32. Bode Diagrams System Bode diagrams are formed by adding the Bode diagrams of the simple systems which are in cascade. Each simple-system diagram is called a component diagram. One Real Pole M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  33. Bode Diagrams One real zero M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  34. Bode Diagrams Integrator (Pole at zero) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  35. Bode Diagrams Differentiator (Zero at zero) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  36. Bode Diagrams Frequency-Independent Gain (This phase plot is for A > 0. If A < 0, the phase would be a constant p or - p radians.) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  37. Bode Diagrams Complex Pole Pair M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  38. Bode Diagrams Complex Zero Pair M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  39. Practical Active Filters Operational Amplifiers The ideal operational amplifier has infinite input impedance, zero output impedance, infinite gain and infinite bandwidth. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  40. Practical Active Filters Active Integrator M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  41. Practical Active Filters Active RC Lowpass Filter M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  42. Practical Active Filters Lowpass Filter An integrator with feedback is a lowpass filter. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  43. Practical Active Filters Highpass Filter M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  44. Discrete-Time Filters DT Lowpass Filter M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  45. Discrete-Time Filters Comparison of DT lowpass filter impulse response with RC passive lowpass filter impulse response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  46. Discrete-Time Filters DT Lowpass Filter Frequency Response RC Lowpass Filter Frequency Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  47. Discrete-Time Filters Moving-Average Filter M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  48. Discrete-Time Filters Ideal DT Lowpass Filter Impulse Response Almost-Ideal DT Lowpass Filter Impulse Response Almost-Ideal DT Lowpass Filter Magnitude Frequency Response M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  49. Discrete-Time Filters Almost-Ideal DT Lowpass Filter Magnitude Frequency Response in dB M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

  50. Advantages of Discrete-Time Filters • They are almost insensitive to environmental effects • CT filters at low frequencies may require very large components, DT filters do not • DT filters are often programmable making them easy to modify • DT signals can be stored indefinitely on magnetic media, stored CT signals degrade over time • DT filters can handle multiple signals by multiplexing them M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl