Signal transmission and filtering section 3.2

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Signal transmission and filtering section 3.2

  1. 1. SIGNAL DISTORTION IN TRANSMISSION 11/16/2011 11:08 AM• Distortionless Transmission• Linear Distortion• Equalization 1
  2. 2. Distortionless Transmission 11/16/2011 11:08 AM Distortionless transmission means that the output signal has the same “shape” as the input. The output is undistorted if it differs from the input only by a multiplying constant and a finite time delayAnalytically, we have distortionless transmission if where K and td are constants. 2
  3. 3. 11/16/2011 11:08 AM Now by definition of transfer function, Y(f) = H(f)X(f) , soA system giving distortionless transmission must have constantamplitude response and negative linear phase shift, so 3
  4. 4. Major types of distortion: 11/16/2011 11:08 AM1. Amplitude distortion, which occurs when2. Delay distortion, which occurs when3. Nonlinear distortion, which occurs when the systemincludes nonlinear elements 4
  5. 5. Linear Distortion 11/16/2011 11:08 AMLinear distortion includes any amplitude or delay distortion associatedwith a linear transmission system. 5
  6. 6. 6 11/16/2011 11:08 AM
  7. 7. 11/16/2011 11:08 AM Shifting each component by one-fourth cycle, θ = –90°.The peak excursions of the phase-shifted signal are substantially greater (byabout 50 percent) than those of the input test signalThis is not due to amplitude response, it is because the components of thedistorted signal all attain maximum or minimum values at the same time, which 7was not true of the input.
  8. 8. Equalization 11/16/2011 11:08 AMLinear distortion—both amplitude and delay—is theoretically curablethrough the use of equalization networks.Since the overall transfer function is H(f) = HC(f)Heq(f) the final output willbe distortionless if HC(f)Heq(f) = Ke-jωtd, where K and td are more or lessarbitrary constants. Therefore, we require that 8 wherever X(f) ≠ 0
  9. 9. FILTERS AND FILTERING 11/16/2011 11:08 AM• Ideal Filters• Bandlimiting and Timelimiting• Real Filters 9
  10. 10. Ideal Filters 11/16/2011 11:08 AMThe transfer function of an ideal bandpass filter (BPF) is The filter’s bandwidth is 10
  11. 11. an ideal lowpass filter (LPF) is defined by an ideal highpass filter (HPF) has 11/16/2011 11:08 AMIdeal band-rejection or notch filters provide distortionless transmission over allfrequencies except some stopband, say 11
  12. 12. an ideal LPF whose transfer function, shown in Fig. above,can be written as 11/16/2011 11:08 AM H.W. Explain why the LPF is noncausal 12
  13. 13. Bandlimiting and Timelimiting 11/16/2011 11:08 AMA strictly bandlimited signal cannot be timelimited.Conversely, by duality, a strictly timelimited signal cannot bebandlimited. Perfect bandlimiting and timelimiting are mutually incompatible. 13
  14. 14. 11/16/2011 11:08 AMA strictly timelimited signal is not strictly bandlimited, itsspectrum may be negligibly small above some upperfrequency limit W.A strictly bandlimited signal may be negligibly small outside acertain time interval t1 ≤ t ≥ t2. Therefore, we will oftenassume that signals are essentially both bandlimited andtimelimited for most practical purposes. 14
  15. 15. Real Filters 11/16/2011 11:08 AM15
  16. 16. nth-order Butterworth LPFThe transfer function with has the form 11/16/2011 11:08 AM where B equals the 3 dB bandwidth and Pn(jf/B) is a complex polynomial 16
  17. 17. 11/16/2011 11:08 AMnormalized variable p= jf/B 17
  18. 18. 11/16/2011 11:08 AM18
  19. 19. EXAMPLE Second-order LPF 11/16/2011 11:08 AM 19
  20. 20. From Table 3.4–1 with p = jf/B , we want 11/16/2011 11:08 AMThe required relationship between R, L, and C that satisfies the equationcan be found by setting 20
  21. 21. QUADRATURE FILTERS AND HILBERT 11/16/2011 11:08 AMTRANSFORMS 21
  22. 22. A quadrature filter is an allpass network that merely shiftsthe phase of positive frequency components by -90° and 11/16/2011 11:08 AMnegative frequency components by +90°.Since a ±90° phase shift is equivalent to multiplying by ,The transfer function can be written in terms of the signum function as 22
  23. 23. The corresponding impulse response is 11/16/2011 11:08 AM applying duality to which yields so 23
  24. 24. Now let an arbitrary signal x(t) be the input to a quadrature filter. 11/16/2011 11:08 AM defined as the Hilbert transform of x(t)denoted by the spectrum of 24
  25. 25. Assume that the signal x(t) is real.1. A signal x(t) and its Hilbert transform have the same 11/16/2011 11:08 AMamplitude spectrum. In addition, the energy or power in asignal and its Hilbert transform are also equal.2. If is the Hilbert transform of x(t), then –x(t) is theHilbert transform of 25
  26. 26. 3. A signal x(t) and its Hilbert transform areorthogonal. 11/16/2011 11:08 AM 26
  27. 27. EXAMPLE Hilbert Transform of a Cosine Signal If the input is 11/16/2011 11:08 AM 27

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