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Unformatted text preview: 02 Solutions 63 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 halfpower DTfrequency 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 onehalf of its maximum value (the “halfpower” bandwidth).
H( jΩ) = The lowfrequency 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 DTfrequency 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 64 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 voltagedivision 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.
 Spring '09
 ATOUSA

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