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

Module by: Louis Scharf. E-mail the author

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Consider the exponential moving average filter

xn=∑k=0∞akun-k;a=0.98.xn=∑k=0∞akun-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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