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Frequency Domain Analysis of Systems - Analyzing CT and DT LTI Systems

Learn about the frequency domain analysis of CT and DT LTI systems, including responses to various input signals like sinusoidal, complex exponential, and periodic inputs. Understand how to analyze output signals and frequency responses using Fourier transforms.

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Frequency Domain Analysis of Systems - Analyzing CT and DT LTI Systems

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  1. Chapter 5Frequency Domain Analysis of Systems

  2. CT, LTI Systems • Consider the following CT LTI system: • Assumption: the impulse response h(t) is absolutely integrable, i.e., (this has to do with system stability (ECE 352))

  3. Response of a CT, LTI System to a Sinusoidal Input • What’s the response y(t) of this system to the input signal • We start by looking for the response yc(t) of the same system to

  4. Response of a CT, LTI System to a Complex Exponential Input • The output is obtained through convolution as

  5. The Frequency Response of a CT, LTI System is thefrequency response of the CT, LTI system = Fourier transform of h(t) • By defining it is • Therefore, the response of the LTI system to a complex exponential is another complex exponential with the same frequency

  6. Analyzing the Output Signal yc(t) • Since is in general a complex quantity, we can write output signal’s phase output signal’s magnitude

  7. Response of a CT, LTI System to a Sinusoidal Input • With Euler’s formulas we can express as and, by exploiting linearity, it is

  8. Response of a CT, LTI System to a Sinusoidal Input – Cont’d • Thus, the response to is which is also a sinusoid with the same frequency but with the amplitudescaled by the factor and with the phase shifted by amount

  9. DT, LTI Systems • Consider the following DT, LTI system: • The I/O relation is given by

  10. Response of a DT, LTI System to a Complex Exponential Input • If the input signal is • Then the output signal is given by where is thefrequency response of the DT, LTI system = DT Fourier transform (DTFT) of h[n]

  11. Response of a DT, LTI System to a Sinusoidal Input • If the input signal is • Then the output signal is given by

  12. Example: Response of a CT, LTI System to Sinusoidal Inputs • Suppose that the frequency response of a CT, LTI system is defined by the following specs:

  13. Example: Response of a CT, LTI System to Sinusoidal Inputs – Cont’d • If the input to the system is • Then the output is

  14. Example: Frequency Analysis of an RC Circuit • Consider the RC circuit shown in figure

  15. Example: Frequency Analysis of an RC Circuit – Cont’d • From ENGR 203, we know that: • The complex impedance of the capacitor is equal to where • If the input voltage is , then the output signal is given by

  16. Example: Frequency Analysis of an RC Circuit – Cont’d • Setting , it is whence we can write where and

  17. Example: Frequency Analysis of an RC Circuit – Cont’d

  18. Example: Frequency Analysis of an RC Circuit – Cont’d • The knowledge of the frequency response allows us to compute the response y(t) of the system to any sinusoidal input signal since

  19. Example: Frequency Analysis of an RC Circuit – Cont’d • Suppose that and that • Then, the output signal is

  20. Example: Frequency Analysis of an RC Circuit – Cont’d

  21. Example: Frequency Analysis of an RC Circuit – Cont’d • Suppose now that • Then, the output signal is

  22. Example: Frequency Analysis of an RC Circuit – Cont’d The RC circuit behaves as alowpass filter, by letting low-frequency sinusoidal signals pass with little attenuation and by significantly attenuating high-frequency sinusoidal signals

  23. Response of a CT, LTI System to Periodic Inputs • Suppose that the input to the CT, LTI system is a periodic signalx(t) having period T • This signal can be represented through its Fourier series as where

  24. Response of a CT, LTI System to Periodic Inputs – Cont’d • By exploiting the previous results and the linearity of the system, the output of the system is

  25. Example: Response of an RC Circuit to a Rectangular Pulse Train • Consider the RC circuit with input

  26. Example: Response of an RC Circuit to a Rectangular Pulse Train – Cont’d • We have found its Fourier series to be with

  27. Example: Response of an RC Circuit to a Rectangular Pulse Train – Cont’d • Magnitude spectrum of input signal x(t)

  28. Example: Response of an RC Circuit to a Rectangular Pulse Train – Cont’d • The frequency response of the RC circuit was found to be • Thus, the Fourier series of the output signal is given by

  29. Example: Response of an RC Circuit to a Rectangular Pulse Train – Cont’d filter more selective

  30. Example: Response of an RC Circuit to a Rectangular Pulse Train – Cont’d filter more selective

  31. Example: Response of an RC Circuit to a Rectangular Pulse Train – Cont’d filter more selective

  32. Response of a CT, LTI System to Aperiodic Inputs • Consider the following CT, LTI system • Its I/O relation is given by which, in the frequency domain, becomes

  33. Response of a CT, LTI System to Aperiodic Inputs – Cont’d • From , the magnitude spectrum of the output signal y(t) is given by and its phase spectrum is given by

  34. Example: Response of an RC Circuit to a Rectangular Pulse • Consider the RC circuit with input

  35. Example: Response of an RC Circuit to a Rectangular Pulse – Cont’d • The Fourier transform of x(t) is

  36. Example: Response of an RC Circuit to a Rectangular Pulse – Cont’d

  37. Example: Response of an RC Circuit to a Rectangular Pulse – Cont’d

  38. Example: Response of an RC Circuit to a Rectangular Pulse – Cont’d

  39. Example: Response of an RC Circuit to a Rectangular Pulse – Cont’d • The response of the system in the time domain can be found by computing the convolution where

  40. Example: Response of an RC Circuit to a Rectangular Pulse – Cont’d filter more selective

  41. Example: Attenuation of High-Frequency Components

  42. Example: Attenuation of High-Frequency Components

  43. Filtering Signals • The response of a CT, LTI system with frequency response to a sinusoidal signal • Filtering: if or then or is

  44. Four Basic Types of Filters lowpass highpass passband stopband stopband cutoff frequency bandpass bandstop (many more details about filter design in ECE 464/564 and ECE 567)

  45. Phase Function • Filters are usually designed based on specifications on the magnitude response • The phase response has to be taken into account too in order to prevent signal distortion as the signal goes through the system • If the filter has linear phase in its passband(s), then there is no distortion

  46. Linear-Phase Filters • A filter is said to have linear phase if • If is in passband of a linear phase filter, its response to is

  47. Ideal Linear-Phase Lowpass • The frequency response of an ideal lowpass filter is defined by

  48. Ideal Linear-Phase Lowpass – Cont’d • can be written as whose inverse Fourier transform is

  49. Ideal Linear-Phase Lowpass – Cont’d Notice: the filter is noncausal since is not zero for

  50. . . Ideal Sampling • Consider the ideal sampler: • It is convenient to express the sampled signal as where

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