Chap6StudentSolutions

From the text for a moving average filter h f e jnf

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Unformatted text preview: 8/16/04 17. Find the minimum stop band attenuation of a moving-average filter with N = 3. Define 1 the stop band as the frequency region, Fc < F < , where Fc is the DT frequency of the 2 first null in the frequency response. From the text, for a moving-average filter H( F ) = e − jπNF sin(π ( N + 1) F ) N +1 sin(πF ) The first null in the frequency response occurs at π ( N + 1) F = π ⇒ F = 1 1 =. N +1 4 The phrase, “minimum stop band attenuation” refers to the point in the stop band at which the reduction in magnitude is the smallest. That is, the point in the stop band in which the transfer function is the largest. The biggest magnitude response after the null frequency is at the next maximum of H( F ) which occurs at π ( N + 1) F = 3π 3 3 ⇒F= =. 2 2( N + 1) 8 18. In the system below, x t ( t) = sinc( t) , f c = 10 and the cutoff frequency of the lowpass filter is 1 Hz. Plot the signals, x t ( t) , y t ( t) , y d ( t) and y f ( t) and the magnitudes and phases of their CTFT’s. x t(t) yt (t) = x r(t) cos(2πfct) x t ( t) = sinc( t) yd (t) LPF cos(2πfct) X t ( f ) = rect ( f ) Solutions 6-12 yf (t) M. J. Roberts - 8/16/04 Modulation |Xt( f )| xt(t) 1 1 -2 2 f Phase of Xt( f ) -4 4 π t -2 -0.5 2 f -π y t ( t) = sinc( t) cos(20πt) Fourier transforming, 1 [rect( f − 10) + rect( f + 10)] 2 Yt ( f ) = Modulated Carrier |Yt( f )| yt(t) 0.5 1 -10 -4 4 10 f Phase of Yt( f ) t π -10 -1 y d ( t) = sinc( t) cos2 (20πt) 10 f -π Yd ( f ) = 1 [rect( f − 20) + 2 rect( f ) + rect( f + 20)] 4 Solutions 6-13 M. J. Roberts - 8/16/04 Demodulated Carrier |Yd( f )| yd(t) 0.5 1 -20 20 f Phase of Yd( f ) -4 4 π t -20 20 -0.5 1 y f ( t) = sinc( t) 2 f -π 2 rect ( f ) 4 Demodulated and Filtered Carrier Yf ( f ) = |Xf( f )| xf(t) 0.5 0.5 -2 2 f Phase of Xf( f ) -4 4 π t -2 -0.25 2 f -π 19. In the system below, x t ( t) = sinc(10 t) ∗ comb( t) , m = 1, f c = 100 and the cutoff frequency of the lowpass filter is 10 Hz. Plot the signals, x t ( t) , y t ( t) , y d ( t) and y f ( t) and the magnitudes and phases of their...
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