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Filtering: Numerical Experiment (Frequency Response of First-Order Filter)

Module by: Louis Scharf. E-mail the author

Note:

This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

Consider the exponential moving average filter

xn=k=0akun-k;a=0.98.xn=k=0akun-k;a=0.98.
(1)
  1. Write out a few terms of the sum to show how the filter works.
  2. Write xn as a recursion and discuss the computer memory required to implement the filter.
  3. Compute the complex frequency response H(ejθ)H(ejθ) for the filter.
  4. Write a MATLAB program to plot the magnitude and phase of the complex frequency response H(ejθ)H(ejθ) versus θ for θ=-πto+πθ=-πto+π in steps of 2π642π64 Do this for two values of a, namely, a=0.98a=0.98 and a=-0.98a=-0.98. Explain your findings.
  5. Write a MATLAB program to pass the following signals through the filter when a=0.98a=0.98:
    1. un=δnun=δn
    2. un=ξnun=ξn
    3. un=ξncos2π64nun=ξncos2π64n
    4. un=ξncos2π32nun=ξncos2π32n
    5. un=ξncos2π16nun=ξncos2π16n
    6. un=ξncos2π8nun=ξncos2π8n
    7. un=ξncos2π4nun=ξncos2π4n
    8. un=ξncos2π2nun=ξncos2π2n.

Plot the outputs for each case and interpret your findings in terms of the complex frequency response H(ejθ)H(ejθ). Repeat step 5 for a=-0.98a=-0.98. Interpret your findings.

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