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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