84.4 The Concept of Filtering Now, the frequency response of the Fir filter is given by H(e)=小[0]+小[1e10+h21e-20 C(1+e j20 e e10+e-J0 2a le /u tBe (20c0s0+B)
§4.4 The Concept of Filtering • Now, the frequency response of the FIR filter is given by ω − ω − ω = + + 2 ( ) ]0[ ]1[ ]2[ j j j H e h h e h e − ω − ω = α + + β j j 1( e ) e 2 − ω − ω ω − ω + β + = α j j j j e e e e 2 2 − ω = α ω+ β j 2( cos )e
84.4 The Concept of Filtering The magnitude and phase functions are H(ejo)=2aCoS@+R 0(0)=-0 In order to block the low-frequency component, the magnitude function at Q=0.1 should be equal to zero Likewise, to pass the high-frequency component, the magnitude function at @=0.4 should be equal to one
§4.4 The Concept of Filtering • The magnitude and phase functions are |H(ejω)|=2αcosω+β θ(ω) = - ω • In order to block the low-frequency component, the magnitude function at ω = 0.1 should be equal to zero • Likewise, to pass the high-frequency component, the magnitude function at ω = 0.4 should be equal to one
84.4 The Concept of Filtering Thus the two conditions that must be satisfied are H(10)=20co(0.1)+=0 H(4)=20c0(04)+=1 Solving the a bove two equations we get c=-6,76195 β=1345635
§4.4 The Concept of Filtering • Thus, the two conditions that must be satisfied are |H(ej0.1)|=2αcos(0.1)+β = 0 |H(ej0.4)|=2αcos(0.4)+β = 1 • Solving the above two equations we get α = -6.76195 β = 13.456335
84.4 The Concept of Filtering Thus the output-input relation of the FIR flter is given by y=-676195(xn]+xn-2|)+13.456335n-2 where the input is x[n={co0.ln)+c0s(0.4 Program 4 2 can be used to verify the filtering action of the above system
§4.4 The Concept of Filtering • Thus the output-input relation of the FIR filter is given by y[n] = - 6.76195(x[n]+x[n-2])+13.456335x[n-2] where the input is x[n] = {cos(0.1n) + cos(0.4n)}µ[n] • Program 4_2 can be used to verify the filtering action of the above system
84.4 The Concept of Filtering igure below shows the plots generated by running this program x x. [] -1 80
§4.4 The Concept of Filtering • Figure below shows the plots generated by running this program