Description
Paper Assignment
- This problem examines conversions between various filter specifications.
Given the absolute specifications δs = 0.0001 and ω𝑝 = 0.3π, ω𝑠 = 0.5π, determine the relative specifications 𝐴𝑠 and ω𝑐, ∆𝜔.
- The Hann window function can be written as
w[n] = [0.5 − 0.5 cos(2πn/M)]𝑤𝑅[n].
where 𝑤𝑅[n] is the rectangular window of length M + 1.
- Express the DTFT of w[n] in terms of the DTFT of 𝑤𝑅[n].
- Explain why the Hann window has the wider mainlobe but lower sidelobes than the rectangular window of the same length.
- Consider an FIR filter with impulse response h[n] = u[n] − u[n − 4].
- Determine and sketch the magnitude response |𝐻(𝑒𝑗ω)|
- Determine and sketch the amplitude response 𝐴(𝑒𝑗ω). Compare this sketch with that in (a) and comment on the difference.
- Determine and sketch the phase response ∠𝐻(𝑒𝑗ω).
- Determine and sketch the angle response Ψ(𝑒𝑗ω). Compare this sketch with that in (c) and comment on the difference.
- Consider the type-IV linear-phase FIR filter characterized by antisymmetric impulse response and odd-M.
- Show that the amplitude response 𝐴(𝑒𝑗ω) is given by (10.38) with coefficients d[k] given
in (10.39).
- Show that the amplitude response 𝐴(𝑒𝑗ω) can be further expressed as (10.40) with coefficients d̂ [k] given in (10.41)
II Program Assignment
- A lowpass FIR filter is given by the specifications: ω𝑝 = 0.3π, ω𝑠 = 0.5π, and 𝐴𝑠 = 50 dB.
Use the fir2 function to obtain a minimum length linearphase filter. Use the appropriate window function in the fir2 function. Provide a plot similar to Figure 10.12.
- Design a highpass FIR filter to satisfy the specifications: ω𝑠 = 0.3π, ω𝑝 = 0.5π, and 𝐴𝑠
= 50 dB.
- Use Kaiser window to obtain a minimum length linear-phase filter. Provide a plot similar to Figure 10.12.
- Repeat (a) using the fir1
- In this problem we reproduce Figures 10.4 and 10.5. For each of the following linearphase FIR filters described by h[n], obtain impulse response, amplitude response, magnitude response, and pole-zero plots in one figure window. For frequency response plots use the interval −2π ≤ ω ≤ 2π.
- Type-I filter: h[n] = {1, 2, 3, −2, 5, −2, 3, 2, 1}.
- Type-II filter: h[n] = {1, 2, 3, −2, −2, 3, 2, 1}.
- Type-III filter: h[n] = {1, 2, 3, −2, 0, 2, −3, −2, −1}.
- Type-IV filter: h[n] = {1, 2, 3, −2, 2, −3, −2, −1 }.
- (18%) Consider a Blackman window of length L = 21.
- Compute and plot the log-magnitude response in dB over −π ≤ ω ≤ π. In the plot measure and show the value of the peak of the first sidelobes.
- Compute and plot the accumulated amplitude response in dB using the cumsum In the plot measure and show the value of the peak of the first sidelobe.
- Repeat (a) and (b) for L = 41.
- An ideal lowpass filter has a cutoff frequency of ω𝑐 = 0.4π. We want to obtain a length L = 40 linear-phase FIR filter using the frequency-sampling method.
- Let the sample at ω𝑐 be equal to 0.5. Obtain the resulting impulse response h[n]. Plot the log-magnitude response in dB and determine the minimum stopband attenuation.
- Now vary the value of the sample at ω𝑐 and find the largest minimum stopband attenuation. Obtain the resulting impulse response h[n] and plot the log-magnitude response in dB in the plot window of (a).
- Compare your results with those obtained using the fir2 function (choose hamming window).
III Reference
𝑀 𝑗𝑀
𝐻(𝑒𝑗 𝑗 𝑒
. (10.38)
𝑑[𝑘] = 2ℎ [𝑀 − 𝑘] . 𝑘 = 1,2,… , 𝑀 (10.39)
𝐴. (10.40)
- d. (10.41)
𝐻 . (10.42)
𝛹 . (10.43)
