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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 15 Copyright © Orchard Publications Other Window Functions smaller side lobe levels. MATLAB provides the Kaiser window function as w=kaiser(L, beta) and this returns an L point Kaiser window * in the column vector w. Figure E. 24 shows the Kaiser functions for with , , and . These func- tions were plotted with the MATLAB script below. plot([kaiser(50, 1), kaiser(50,4), kaiser(50,9)]); Figure E.24. Kaiser window functions with L=50 and , , and Figure E.25 shows the corresponding frequency domain plots for the Kaiser window functions in Figure E.24. These plots were created with the MATLAB script below. n=50; w1 =kaiser(n, 1); w2=kaiser(n,4); w3=kaiser(n,9); [W1,f]=freqz(w1/sum(w1),1,512,2); [W2,f]=freqz(w2/sum(w2),1,512,2); [W3,f]=freqz(w3/sum(w3),1,512,2); plot(f,20*log10(abs([W1 W2 W3])));grid; E.3 Other Window Functions Table E.1 lists the window functions we’ve discussed in the previous sections, as well as others along with the MATLAB function names and are included in the MATLAB Signal Processing Toolbox. * When the expression under the radical in (E. 12) is negative, the function can be expressed in terms of the hyperbolic sine function. L5 0 = β 1 = β 4 = β 9 = 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 β =1 β =4 β =9 β 1 = β 4 = β 9 =

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Appendix E Window Functions E 16 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Figure E.25. Frequency domain plots for the Kaiser functions with L=50 and , , and TABLE E.1 MATLAB Window Functions Window Function MATLAB Function Bartlett bartlett Bartlett Hann (modified) barthannwin Blackman blackman Blackman Harris blackmanharris Bohman bohmanwin Chebyshev chebwin Flat Top flattopwin Gaussian gausswin Hamming hamming Hann or Hanning hann Kaiser kaiser Nuttall’s Blackman Harris nuttallwin Parzen (de la Valle Poussin) parzenwin Rectangular rectwin Tukey tukeywin Triangular triang 0 0.2 0.4 0.6 0.8 1 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 β =1 β =4 β =9 β 1 = β 4 = β 9 =
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition E 17 Copyright © Orchard Publications Fourier Series Method for Approximating an FIR Amplitude Response E.4 Fourier Series Method for Approximating an FIR Amplitude Response The Fourier series method for approximating an FIR amplitude response is relatively straightfor- ward, and is best used in conjunction with window functions. This is because the amplitude response corresponding to a Discrete Time Linear Time Invariant (DTLTI) impulse response is a periodic function of frequency and thus it can be expanded in a Fourier series in the frequency dornain.

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