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Hossein Sameti Department of Computer Engineering Sharif University of Technology

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LTI Systems h(n) FIRIIR With rational transfer function No rational transfer function Determine coefficients of h(n) [or P(z) and Q(z)] 2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Design Stages 1.Specifications Application dependent 2.Design h(n) Determine coefficients of h(n) 3.Realization Direct form I,II, cascade and parallel 4.Implementation Programming in Matlab/C, DSP, ASIC,… Design of FIR filters ◦ Windowing 3 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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IDTFT of ideal low-pass filter: 4 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Multiply by a rectangular window It can be shown that if we have a linear-phase ideal filter and we multiply it by a symmetric window function, we end up with a linear- phase FIR filter. 5 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Windows are designed with linear phase in mind ◦ Symmetric around M/2 So their Fourier transform are of the form Will keep symmetry properties of the desired impulse response Assume symmetric desired response With symmetric window ◦ Periodic convolution of real functions 6 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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The steps in the design of FIR filters using windows are as follows: 1.Start with the desired frequency response results in the sinc function in time domain 2.Compute 3.Determine the appropriate window function w(n) 4.Calculate A finite-length window function 7 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Two properties should be considered: 1) The amplitude is unity in the pass band and it is zero in the stop band: 2) The phase is linear: 8 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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First, we have to decide on the type of the filter. Assume Type I filter (linear-phase) 9 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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IIR filter 10 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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It is a high-pass FIR filter with 7 taps that approximates the high-pass IIR filter. How can we quickly check that the resulting FIR filter has the desired properties that we were looking for? (i.e., it is a high-pass linear-phase filter)? 11 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Hossein Sameti, ECE, UBC, Summer 2012 Originally Prepared by: Mehrdad Fatourechi, 12

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What condition should we impose on W(ω) so that H (ω) looks like H d (ω) ? Impulse function in the frequency domain, means an infinitely-long constant in the time-domain Larger window means more computation 13 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Windowed frequency response The windowed version is smeared version of desired response If w[n]=1 for all n, then W(e j ) is pulse train with 2 period 14 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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(Oppenheim and Schaffer, 2009) Ideal filter Rectangular Window function 18 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Pass-band: Stop-band: Pass-band ripple: Stop-band ripple: Transition width: What is the ideal situation? (Oppenheim and Schaffer, 2009) 19 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Width of transition is not sharp! Ripples in the passband / stopband are proportional to the peaks of side lobes of the window. The width of transition depends on the width of the main lobe of the window. 21 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Q: How can we control the transition width (size of the main lobe)? A1: using the size of the window Uncertainty principle 22 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Q: How can we control the size of transition width (size of the main lobe)? A2: Shape of the window; in other words, windows with a fixed size that have different shapes can have different main lobe width. Rectangular window Smallest; and Blackman largest main lobe width 23 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Q: How can we control the peak of the side lobes so that we can get a good ripple behavior in the FIR filter? A: using the shape of the window 24 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Q: Can we control the peak of the side lobes by changing the size of the window? A: It can be shown that changes are not significant. 25 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Prefer windows that concentrate around DC in frequency ◦ Less smearing, closer approximation Prefer window that has minimal span in time ◦ Less coefficient in designed filter, computationally efficient So we want concentration in time and in frequency ◦ Contradictory requirements Example: Rectangular window 27 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Narrowest main lobe ◦ 4 /(M+1) ◦ Sharpest transitions at discontinuities in frequency Large side lobes ◦ -13 dB ◦ Large oscillation around discontinuities Simplest window possible 28 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Medium main lobe ◦ 8 /M Side lobes ◦ -25 dB Hamming window performs better Simple equation 29 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Medium main lobe ◦ 8 /M Side lobes ◦ -31 dB Hamming window performs better Same complexity as Hamming 30 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Medium main lobe ◦ 8 /M Good side lobes ◦ -41 dB Simpler than Blackman 31 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Large main lobe ◦ 12 /M Very good side lobes ◦ -57 dB Complex equation 32 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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rectangular Bartlett Hanning Hamming Blackman 33 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Shape of the window Main lobe Side lobe width of the window Main lobe Good design strategy: 1) Use shape to control the behavior of the side lobe. 2) Use width to control the behavior of the main lobe. 36 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Zero th order modified Bessel function of the first kind Number of taps Parameter to control the shape of the Kaiser window and thus the trade-off between the width of the main lobe and the peak of the side lobe. 37 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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M=20 38 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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1. Calculate the transition bandwidth 2. Calculate 3. Choose 4. Choose 41 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Specs: 42 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Specs: Type II filter Use Bessel equation to get w(n) 43 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Q: Does it satisfy the specs? 45 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Windowing method is a fast and efficient solution to design FIR filters. Using Kaiser windows, the window can be chosen automatically. 46 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Hossein Sameti Department of Computer Engineering Sharif University of Technology.

Hossein Sameti Department of Computer Engineering Sharif University of Technology.

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