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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 31 Copyright © Orchard Publications Fourier Series Method for Approximating an FIR Amplitude Response Figure E.34. Normalized frequency plots for the hann window created with the MATLAB function fir1 Figure E.35. Magnitude response for the hunn window created with the MATLAB function fvtool c. Hamming We recall from (E.8) that the Hamming window function is defined as (E.30) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -800 -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 -150 -100 -50 0 50 Normalized Frequency ( rad/sample) Magnitude (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Normalized Frequency ( ×π rad/sample) Magnitude Response (dB) ft () Hamm 0.54 0.46 2 π t τ -------- cos + for |t| τ 2 -- < = 0 otherwise =

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Appendix E Window Functions E 32 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Letting and we obtain (E.31) 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)=0.54+0.46.*(cos(pi.*m./10)); fprintf('%2.0f\t %12.5f\n',wm') and MATLAB outputs m wm ============== 0 1.00000 1 0.97749 2 0.91215 3 0.81038 4 0.68215 5 0.54000 6 0.39785 7 0.26962 8 0.16785 9 0.10251 10 0.08000 From (E.19) (E.32) Then, Cm=[0.2500 0.2251 0.1592 0.0750 0.0000 0.0450 0.0531 0.0322 0.0000 0.0250 0.0318]; Wm=[1.00000 0.97749 0.91215 0.81038 0.68215 0.54000 0.39785 0.26962 0.16785 0.10251 0.08000]; Next, D=Cm.*Wm and MATLAB outputs tm T = τ 20T = wm () Hamm 0.54 0.46 2 π mT ------------------ cos + 0.54 0.46 π m 10 ---------- cos + for |m| 10 == 0 otherwise = c' m w m c m =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 33 Copyright © Orchard Publications Fourier Series Method for Approximating an FIR Amplitude Response D = 0.2500 0.2200 0.1452 0.0608 0 -0.0243 -0.0211 -0.0087 0

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