Chapter 4 Problem solution Homework: 4.34.44.104.114.144.154.24 4.254.324.354.364.374.43
1 Chapter 4 Problem Solution Homework: 4.3 4.4 4.10 4.11 4.14 4.15 4.24 4.25 4.32 4.35 4.36 4.37 4.43
Chapter 4 Problem solution 4.4 Determine the inverse Fourier transform (a)x, a)=2rdo+rdo 4z+rda+4r) x;(a)= 1+cos4丌t Y, Go) 20≤≤2 )xjo)=1-2-2≤O<0 -20 2 0a>2 2 2()=4sim2 t
2 Chapter 4 Problem Solution 4.4 Determine the inverse Fourier transform: (a) ( ) 2 () ( 4 ) ( 4 ) X1 j = + − + + x (t) 1 cos4πt 1 = + ( ) ( ) − = 0 2 - 2 2 0 2 0 2 b 2 X j -2 0 2 ω 2 -2 X (j) 2 ( ) t j t x t 2 2 4sin =
Chapter 4 Problem solution 4.10.(a)Determine the Fourier transform of x(t) sint (b) Determine the numerical value of sint A dt Solution j/2r xGo 20 2 j/2T 2/1 sint + 2 Iodo 2丌 2丌
3 Chapter 4 Problem Solution 4.10. (a) Determine the Fourier transform of (b) Determine the numerical value of ( ) 2 sin = t t x t t dt t t A t 4 2 sin = + − Solution ( ) ( ) 4 2 2 3 sin 1 1 b 2 2 t A t dt X j d t + + − − = = = X(j) -2 0 2 ω j / 2 − j / 2
Chapter 4 Problem solution 4.11 Given the relationship y(o)=x()*h(t) and g(t)=x(3r)+h(3r) nd given x(< F>xo h(<F >hla Show that g(t)=Ay (Bt Determine the value ofa and B Solution 8(1 y(3t)A=1B=3 4
4 Chapter 4 Problem Solution g(t) y(3t) 3 1 = 3 3 1 A = B = 4.11 Given the relationship and and given Show that y(t) = x(t)h(t) g(t) = x(3t)h(3t) g(t) = Ay(Bt) Determine the value of A and B. x(t)⎯→X(j) F h(t)⎯→H(j) F Solution
Chapter 4 Problem solution 4.12 Consider the fourier transform pair o lt F 2 1+a (a) Find the Fourier transform of te 4t (b) Determine the Fourier transform of +t j40 te 4t F 2 nyse +t 5
5 Chapter 4 Problem Solution 4.12 Consider the Fourier transform pair (a) Find the Fourier transform of (b) Determine the Fourier transform of 2 1 2 + ⎯→ − t F e t te− ( ) 2 2 1 4 t t + ( ) 2 2 1 4 + − ⎯→ − j te t F ( ) 2 1 4 2 2 − ⎯→− + j e t t F