Chap6StudentSolutions

A system has an impulse response t 001 h t 10 rect

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Unformatted text preview: 02 Solutions 6-3 M. J. Roberts - 8/16/04 What is its null bandwidth? 6. A system has an impulse response, n 7 h[ n ] = u[ n ] . 8 What is its half-power DT-frequency bandwidth? Using F α n u [ n ] ← → 1 1 − α e− jΩ the transfer function is 8 1 . = 7 − jΩ 8 − 7e − jΩ 1− e 8 This is a DT lowpass filter. Its maximum transfer function magnitude occurs at Ω = 0 . The -3 dB point must be the first frequency at which the square of the magnitude of the transfer function is one-half of its maximum value (the “half-power” bandwidth). H( jΩ) = The low-frequency gain is H(0) = 8 The -3 dB point occurs where H( jΩ−3 dB ) Solving, 2 82 = = 32 . 2 Ωhp = 0.1337 ± 2 nπ So the -3 dB DT-frequency bandwidth in radians is 0.1337. In cycles it is 0.0213. (Notice that the bandwidths are not in radians/s or in Hz. This is because they are DT bandwidths, not CT bandwidths.) 7. Determine whether or not the CT systems with these transfer functions are causal. (a) H( f ) = sinc( f ) (c) H( jω ) = rect (ω ) (d) H( jω ) = rect (ω )e − jω (e) H( f ) = A (f) H( f ) = Ae j 2πf (b) h( t) = H( f ) = sinc( f )e − jπf 1 t − 1 sinc Not Causal 2π 2π h( t) = Aδ ( t + 1) Solutions 6-4 Not Causal M. J. Roberts - 8/16/04 8. Determine whether or not the DT systems with these transfer functions are causal. (a) H( F ) = sin( 7πF ) sin(πF ) (c) H( F ) = sin( 3πF ) − j 2πF e sin(πF ) (d) H( F ) = rect (10 F ) ∗ comb( F ) H( F ) = (b) sin( 7πF ) − j 2πF e sin(πF ) h[ n ] = rect1[ n − 1] Causal 9. Find and sketch the frequency response of each of these circuits given the indicated excitation and response. (a) Excitation, v i ( t) - Response, v L ( t) R = 10 Ω C = 1 µF + + vi (t) vL(t) L = 1 mH - - Using voltage-division principles, VL ( jω ) Z L ( jω ) jω L −ω 2 LC H ( jω ) = = = = Vi ( jω ) Z L ( jω ) + ZC ( jω ) + Z R ( jω ) jω L + 1 + R 1 − ω 2 LC + jω RC jω C |H( jω )| 3 ω -150000 150000 Phase of H( jω ) π ω -150000 150000 -π (b) Excitation, v i ( t) - Resp...
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This note was uploaded on 06/19/2013 for the course ENSC 380 taught by Professor Atousa during the Spring '09 term at Simon Fraser.

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