endof-

More documents

Recommendations

Info

Other Window Functionssmaller side lobe levels.MATLAB provides the Kaiser window function asw=kaiser(L, beta)and this returns an L−point Kaiser window * in the column vector w.Figure E. 24 shows the Kaiser functions for L = 50 with β = 1 , β = 4 , and β = 9 . These functionswere plotted with the MATLAB script below.plot([kaiser(50, 1), kaiser(50,4), kaiser(50,9)]);10.8β=10.6β=40.40.2β=900 10 20 30 40 50Figure E.24. Kaiser window functions with L=50 and β = 1 , β = 4 , and β = 9Figure E.25 shows the corresponding frequency domain plots for the Kaiser window functions inFigure 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 FunctionsTable E.1 lists the window functions we’ve discussed in the previous sections, as well as othersalong with the MATLAB function names and are included in the MATLAB Signal ProcessingToolbox.* When the expression under the radical in (E.12) is negative, the function can be expressed in terms of thehyperbolic sine function.Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard PublicationsE−15
Appendix E Window FunctionsMH 1 ( z) =∑c m z m(E.16)m = – MRelation (E.16) represents an FIR transfer function but it is non−causal because it includes positivepowers of z, and this implies that the filter would produce an output advanced in time withrespect to the input, and this is, of course, impossible. We can be overcome this problem by introducinga delay of M samples. Accordingly, we define the transfer functionHz ( ) = z – MH 1 ( z) = z – M∑c m z m =∑c m z m – Mm = – M m = MIn (E.17) we let m – M = – i . Then, with the substitution c m– i = a iwe obtainM–M(E.17)2MHz ( ) =∑c m– i z – i =i = 0∑a i z – ii = 0(E.18)Let w mdenote the coefficients of a particular window function and let c' mdenote the coefficientsof the modified function, i.e., the given function multiplied by some window function. Themodified coefficients are then computed fromc' m=w m c m2M(E.19)Example E.1A low−pass FIR digital filter is to be designed using the Fourier series method. The desired amplituderesponse isAf () = 1 for 0 ≤f ≤125 Hz(E.20)0 elsewhere in the range 0 < f < f 0The sampling frequency is 1 KHz and the impulse response is to be limited to 20 delays. Derivethe transfer function using the Fourier series method.Solution:To make this low−pass filter an even function of frequency, we represent it as shown in FigureE.26 (a) in terms of the actual frequency. Let us express it in terms of the normalized frequencyscale. We do this by considering the sampling frequency which is given as f S = 1 KHz . Accordingly,the folding frequency * is f 0 = f S ⁄ 2 = 1000 ⁄ 2 = 500 Hz , and in terms of the normalized* We recall that the folding frequency is the highest frequency that can be processed by a given discrete−time systemwith sampling frequencyf SE−18Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications
Page 7:
4.2 Complex Impedance Z(s).........
Page 10 and 11: 8.6.4 The Transform of ft () = Acos
Page 12 and 13: 11.2.1 RC Low−Pass Filter .......
Page 14 and 15: MATLAB ComputingPages D−3, D−4,
Page 16 and 17: The Unit Step FunctionThus, the pul
Page 18 and 19: Chapter 1 Elementary Signals6. To d
Page 20 and 21: Chapter 2 The Laplace Transformatio
Page 22 and 23: Chapter 2 The Laplace Transformatio
Page 24 and 25: Chapter 3 The Inverse Laplace Trans
Page 26 and 27: Chapter 4Circuit Analysis with Lapl
Page 28 and 29: Solutions to End−of−Chapter Exe
Page 30 and 31: Chapter 5 State Variables and State
Page 32 and 33: A: [−1 −1; 3/4 0]B: [1 0]’C:
Page 38 and 39: Chapter 7 Fourier Seriesthat is, th
Page 40 and 41: Chapter 8 The Fourier Transform8.2
Page 42 and 43: Chapter 8 The Fourier TransformExam
Page 44 and 45: Chapter 9 Discrete−Time Systems a
Page 46 and 47: num=[0 1 0 0];% The coefficients of
Page 48 and 49: Chapter 10 The DFT and the FFT Algo
Page 50 and 51: The Fast Fourier Transform (FFT)The
Page 52 and 53: Basic Analog Filters11.2.3 RLC Band
Page 54 and 55: Chapter 11 Analog and Digital Filte
Page 56 and 57: Digital Filters1Four signals separa
Page 58 and 59: Chapter 11 Analog and Digital Filte
Page 60: Appendix E Window Functionsh[n]A(f)
Page 65 and 66: IndexSymbols Cauchy’s residue the
Page 67 and 68: Fourier transform - properties of f
Page 69 and 70: S for discrete-time systems 9-45 St
show all

endof-

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?