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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 23 Copyright © Orchard Publications Fourier Series Method for Approximating an FIR Amplitude Response Figure E.29. Normalized frequency plots for the rectwin window created with the MATLAB function fir1 Example E.2 Compare the frequency response in Example E.1, Figure E.27, when it is multiplied by the follow- ing window functions: a .t r i an gu l a r b . Hanning c . Hamming Solution: a. triangular We recall from (E.4) that the triangular window function is defined as (E.24) Letting and we obtain (E.25) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -600 -400 -200 0 Normalized Frequency ( ×π rad/sample) Phase (degrees) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 Normalized Frequency ( rad/sample) Magnitude (dB) ft () triang 1 2t τ ------- for |t| τ 2 -- < = 0 otherwise = tm T = τ 20T = Wm 1 2mT -------------- 1 m 10 for |m| 10 == otherwise =

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Appendix E Window Functions E 24 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications As before, the window coefficients are applied to the non causal form of the transfer function centered at the origin, and thus the 10th coefficient is considered as the origin. For the com- putations we use the MATLAB script below. disp('m wm') disp('=================') m=0:10; wm=zeros(11,2); wm(:,1)=m'; m=m+(m==0).*eps; wm(:,2)=1 m/10; fprintf('%2.0f\t %12.5f\n',wm') MATLAB outputs the values shown below. m wm =============== 0 1.00000 1 0.90000 2 0.80000 3 0.70000 4 0.60000 5 0.50000 6 0.40000 7 0.30000 8 0.20000 9 0.10000 10 0.00000 From (E.19) (E.26) For this example, Cm=[0.2500 0.2251 0.1592 0.0750 0.0000 0.0450 0.0531 0.0322 0.0000 0.0250 0.0318]; Wm=[1.0000 0.9000 0.8000 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000 0.0000]; B=Cm.*Wm and MATLAB outputs the modified coefficients below. 0.2500 0.2026 0.1274 0.0525 0 0.0225 0.0212 0.0097 0 0.0025 0 and we add these coefficients in the table below.
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