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FIR Filter Design Comparison Tool

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Description

This MicroSim designs two odd-length, linear-phase FIR filters, A (solid blue) and B (dashed maroon), for the same specification and overlays them so that you can compare design methods directly.

  • Filter type (low-pass, high-pass, band-pass) and the cutoff sliders \(\omega_{c1}\), \(\omega_{c2}\) (in units of \(\pi\) rad/sample) set the shared specification. The grey line is the ideal brick-wall response.
  • Each design has its own method, taps \(N\) (odd, 11-127) and one shape parameter:
    • Window: \(h[n] = h_d[n]\,w[n]\) with \(h_d[n] = \sin(\omega_c(n-M))/(\pi(n-M))\), \(M = (N-1)/2\), and a rectangular, Hann, Hamming, Blackman or Kaiser window (slider \(\beta\) for Kaiser).
    • Frequency sampling: \(N\) amplitude samples \(H[k]\) at \(\omega_k = 2\pi k/N\) (1 in the pass band, 0 in the stop band) followed by an inverse DFT. The transition sample \(T_1\) sets the first stop-band sample next to each band edge.
    • Parks-McClellan: the optimal equiripple design for pass and stop bands separated by a transition band of width \(\Delta\omega\) centered on the cutoff, with equal weights.
  • The right plot shows \(|H(e^{j\omega})|\) evaluated at 2049 frequencies in dB, on a linear scale, or zoomed on the passband. The left plot shows the impulse response \(h[n]\), the zeros of \(H(z)\), the phase or the group delay.
  • The table reports each design's transition width \(\Delta\omega\), passband ripple and stop-band attenuation. For a Kaiser window it also prints the prediction of Kaiser's formulas.

This MicroSim supports Chapter 10: Advanced Filter Design and Implementation.

How the read-outs are measured

  • \(\delta_p\) is the largest passband deviation \(\big||H|-1\big|\) and \(\delta_s\) the largest stop-band \(|H|\). Passband ripple \(= 20\log_{10}\frac{1+\delta_p}{1-\delta_p}\) dB and stop-band attenuation \(A_s = -20\log_{10}\delta_s\) dB.
  • The pass band ends at the last ripple peak before the cutoff and the stop band starts at the first null after it. \(\Delta\omega\) runs from the frequency where \(|H|\) falls below \(1-\delta_p\) to the frequency where it reaches \(\delta_s\) (the coloured bars at the top of the magnitude plot).
  • Kaiser's formulas: \(\beta = 0.1102(A-8.7)\) for \(A > 50\) dB, \(\beta = 0.5842(A-21)^{0.4} + 0.07886(A-21)\) for \(21 \le A \le 50\) dB, and \(\Delta\omega \approx (A-7.95)/(2.285\,(N-1))\).
  • Every design is symmetric, so \(H(e^{j\omega}) = A(\omega)e^{-j\omega M}\): the phase is linear and the group delay is \(M = (N-1)/2\) samples at every frequency.
  • Parks-McClellan limits: the sketch narrows the requested \(\Delta\omega\) if the band edges would remove more than half of a pass band or stop band, or if Kaiser's estimate \(2.285\,\Delta\omega\,(N-1) + 7.95\) exceeds 130 dB (a wider band would need a ripple smaller than double-precision arithmetic can resolve). The table and a footnote show the width used.
  • Hann and Blackman windows are zero at both ends, and some cutoffs make the end taps of \(h_d[n]\) zero (for example \(\omega_c = 0.40\pi\) with \(N = 51\), because \(\sin(10\pi) = 0\)). Such a filter has fewer than \(N-1\) zeros.

Lesson Plan

Learning Objectives

By the end of this lesson, students will be able to:

  1. Apply the window method to design a low-pass, high-pass or band-pass FIR filter and read its transition width, passband ripple and stop-band attenuation
  2. Compare how the rectangular, Hann, Hamming, Blackman and Kaiser windows trade transition width against stop-band attenuation at a fixed number of taps
  3. Predict the stop-band attenuation and transition width of a Kaiser-window design from \(\beta\) and \(N\) with Kaiser's formulas, and check the prediction against the measurement
  4. Explain how the transition sample \(T_1\) changes the stop-band attenuation of a frequency-sampling design, and find the value that maximizes it
  5. Evaluate which design method meets a given specification with the fewest taps, using Parks-McClellan as the optimal benchmark

Target Audience

Undergraduate engineering and computer science students in a digital signal processing course (typically third year). The activities take about 40 minutes.

Prerequisites

  • The z-transform and the frequency response \(H(e^{j\omega})\) of an FIR filter
  • The ideal low-pass impulse response (the sinc function) and the DFT
  • Decibels and the idea of passband and stop-band tolerances

Activities

All activities start from the default settings: low-pass, \(\omega_c = 0.40\pi\), \(N = 51\) for both designs, A = Hamming window, B = frequency sampling with \(T_1 = 0.40\). The table reports the values quoted below.

Activity 1: Window comparison (10 min). Set Design A's window to each option in turn and copy the table row.

Window (N = 51) Transition \(\Delta\omega\) Passband ripple Stop-band attenuation
Rectangular 0.037π 1.60 dB 21.1 dB
Hann 0.125π 0.110 dB 43.9 dB
Hamming 0.133π 0.030 dB 53.7 dB
Blackman 0.223π 0.0031 dB 75.3 dB

Every step down the table buys more attenuation and less ripple with a wider transition band. Switch the left plot to Zeros: with the Blackman window the stop-band zeros on the unit circle start at a higher angle than with the rectangular window, matching its later stop-band edge.

Activity 2: Taps versus window shape (10 min). Keep A on Hamming and set \(N\) = 21, 51 and 101. The transition width falls from 0.333π to 0.133π to 0.067π, so \(\Delta\omega\,(N-1) \approx 6.7\pi\) stays constant, while the attenuation stays near 54 dB (53.6, 53.7 and 53.9 dB). Now choose the Kaiser window with \(N = 51\). At \(\beta = 5.0\) the table shows 54.0 dB and 0.129π, and the footnote predicts 54.1 dB and 0.128π. At \(\beta = 8.0\) it shows 80.9 dB and 0.204π (predicted 81.3 dB and 0.204π). The shape parameter \(\beta\), not \(N\), sets the attenuation.

Activity 3: The transition sample (10 min). Watch Design B (frequency sampling, \(N = 51\)). Set \(T_1 = 0\): the attenuation drops to 16.4 dB and the passband ripple grows to 2.29 dB, even though every frequency sample is exactly 0 or 1. Sweep \(T_1\) upward: the attenuation peaks at 42.9 dB for \(T_1 = 0.38\) (39.7 dB at 0.40) and falls again to 22.0 dB at \(T_1 = 0.70\). Use Passband zoom to see the ripple between the samples.

Activity 4: Which method needs the fewest taps? (10 min). Keep A at Hamming, \(N = 51\) (53.7 dB, \(\Delta\omega =\) 0.133π). Set B to Parks-McClellan with \(\Delta\omega = 0.13\pi\). At \(N = 51\), B reaches 58.4 dB with 0.021 dB ripple. Lower B's taps: at \(N = 47\) it still gives 54.7 dB with 0.032 dB ripple, beating A with four fewer taps and a slightly narrower transition band. At \(N = 45\) it falls to 53.1 dB. Switch the left plot to Group delay: B's delay drops from 25 to 23 samples.

Assessment

  1. A Hamming design with \(N = 51\) has \(\Delta\omega = 0.133\pi\) and 53.7 dB attenuation. Predict both values for \(N = 101\). (Answer: the transition width scales as \(1/(N-1)\), so \(0.133\pi \times 50/100 \approx 0.067\pi\); the attenuation is set by the window and stays near 54 dB. The sim shows 0.067π and 53.9 dB.)
  2. You need at least 60 dB of stop-band attenuation with \(N = 51\) taps. Which Kaiser \(\beta\) do you choose, and what transition width do you expect? (Answer: \(\beta = 0.1102(60-8.7) = 5.65\), so set \(\beta = 5.7\); \(\Delta\omega \approx (60-7.95)/(2.285 \times 50) = 0.456\) rad \(= 0.145\pi\). The sim measures 60.1 dB and 0.146π.)
  3. Why does a frequency-sampling design with \(T_1 = 0\) reach only about 16 dB of attenuation, and how does \(T_1 = 0.38\) help? (Answer: between the samples the response is an interpolation of the abrupt 1-to-0 step, which oscillates like the response of a rectangular window. A transition sample makes the step gradual, so the interpolation ripple shrinks and the attenuation rises to 42.9 dB at the cost of a wider transition band.)
  4. A 51-tap linear-phase filter runs at \(f_s = 8\) kHz. How long is its delay, and does it depend on frequency? (Answer: \(M = 25\) samples, or \(25/8000 = 3.125\) ms, at every frequency, because the symmetric impulse response gives a phase of \(-25\omega\).)
  5. Why does a 47-tap Parks-McClellan design beat a 51-tap Hamming design? (Answer: Parks-McClellan spreads the error evenly (equiripple) across both bands, while a window design's ripples decay away from the band edge, so most of its stop band has more attenuation than the specification needs and the taps are used less efficiently.)

Common Misconceptions

"More taps give more stop-band attenuation." With the window method the window shape sets the attenuation and the number of taps sets the transition width. In Activity 2 the Hamming design stays between 53.6 and 53.9 dB from \(N = 21\) to \(N = 101\) while \(\Delta\omega\) shrinks by a factor of five.

"A frequency-sampling filter equals the ideal response because it matches it at every sample." It matches only at \(\omega_k = 2\pi k/N\). With \(T_1 = 0\) the samples are exactly 1 and 0, yet the response between them ripples by 2.29 dB in the pass band and rises to −16.4 dB in the stop band.

References

  1. Finite impulse response - Wikipedia - FIR filters, the window method and frequency sampling
  2. Window function - Wikipedia - definitions and side-lobe levels of the rectangular, Hann, Hamming and Blackman windows
  3. Kaiser window - Wikipedia - the Bessel-function window and the formulas that relate \(\beta\), attenuation and filter length
  4. Parks-McClellan filter design algorithm - Wikipedia - the optimal equiripple design used for method three
  5. Remez algorithm - Wikipedia - the exchange algorithm behind Parks-McClellan
  6. The Scientist and Engineer's Guide to Digital Signal Processing - Steven W. Smith - free online textbook with chapters on windowed-sinc and custom FIR filters