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Frequency transformation


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Low Pass to High Pass and High Pass to Low Pass Frequency Transform

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Frequency transformation

  1. 1. Frequency Transformationi.e. Low Pass to High Pass Filter etc.
  2. 2. Frequency Transformations• We need to apply a suitable frequency transformation, if we wish to designbandpass, bandstop and high-pass filters, using the low-pass approximatingfunction analysis
  3. 3. Filter Transformations• We can use the concept of filter transformations to determine the new filter designs from a lowpass design. As a result, we can construct a 3rd-order Butterworth high-pass filter or a 5th-order Chebychev bandpass filter!
  4. 4. Filter Transformations
  5. 5. Normalized Lowpass Filter• When designing a filter, it is common practice to first design a normalized low- pass filter, and then use a spectral transform to transform that low-pass filter into a different type of filter (high-pass, band-pass, band-stop).• The reason for this is because the necessary values for designing lowpass filters are extensively described and tabulated. From this, filter design can be reduced to the task of looking up the appropriate values in a table, and then transforming the filter to meet the specific needs
  6. 6. Lowpass to Lowpass Transformation• Having a normalized transfer function, with cutoff frequency of 1 Hz, one can modify it in order to move the cutoff frequency to a specified value
  7. 7. Transfer Function
  8. 8. Analog Element Values
  9. 9. Lowpass to Highpass
  10. 10. Lowpass to HighpassNormalised HPF response
  11. 11. Lowpass to Highpass we define values for the transformed frequency Ω as Equivalent LPF response
  12. 12. Lowpass to Highpassbut with frequency transformation, we substitute for ω/ωp with In the Chebychev case, we apply the substitution
  13. 13. Conversion of Low-pass and High-passFilter transfer functions fromcontinuous time to discrete timedifference equations.• The following converts two filter transfer function that are represented in the Laplace Space• (Continuous time) into their discrete time equivalents in the Z-space using the Bilinear Transform• (AKA Tustin’s Method), then converts them to difference equations expressing the current output as a combination of previous inputs and outputs.
  14. 14. Definition of Values
  15. 15. Definition of Values
  16. 16. Low Pass Filter Conversion
  17. 17. Low Pass Filter Conversion
  18. 18. High Pass Filter Conversion
  19. 19. High Pass Filter Conversion
  20. 20. High Pass Filter Conversion