½ÅÈ£¹× ½Ã½ºÅÛ ¼Ö·ç¼Ç Fundamentals of Signal and System Michael J. Roberts

?M. J. Roberts - 7/12/03

Chapter 2 - Mathematical Description of Signals
Solutions
1. If g( t) = 7e ?2 t ? 3 write out and simplify (a) (b) (c) (d) (e) g( 3) = 7e ?9 g(2 ? t) = 7e ?2( 2 ? t ) ? 3 = 7e ?7 + 2 t
t ? ?11 ? t ? 5 g? + 4ÿ = 7e ? 10 ?

g( jt) = 7e ? j 2 t ? 3 g( jt) + g(? jt) e ? j 2t + e j 2t = 7e ?3 = 7e ?3 cos(2 t) 2 2 ? ? jt ? 3? ? jt ? 3? g? ÿ ÿ + g? ? 2 ? ? 2 ? e ? jt + e jt =7 = 7 cos( t) 2 2

(f)

2. If g( x ) = x 2 ? 4 x + 4 write out and simplify (a) (b) (c) (d) g( z) = z 2 ? 4 z + 4 g( u + v ) = ( u + v ) ? 4 ( u + v ) + 4 = u 2 + v 2 + 2 uv ? 4 u ? 4 v + 4
2

g(e jt ) = (e jt ) ? 4 e jt + 4 = e j 2 t ? 4 e jt + 4 = (e jt ? 2)
2 2

2

g(g( t)) = g( t 2 ? 4 t + 4 ) = ( t 2 ? 4 t + 4 ) ? 4 ( t 2 ? 4 t + 4 ) + 4 g(g( t)) = t 4 ? 8 t 3 + 20 t 2 ? 16 t + 4

(e)

g(2) = 4 ? 8 + 4 = 0

3. What would be the numerical value of ¡°g¡± in each of the following MATLAB instructions? (a) (b) (c)
t = 3 ; g = sin(t) ; 0.1411 [-1,1,-1,1,-1]

x = 1:5 ; g¡¦(»ý·«)