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≤O2 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(xo h(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
Chapter 4 Problem solution 4. 14 Consider a signal x(t) be a signal with Fourier transformr(o Suppose we are given the following facts: (1). x(t) is real and nonnegative (2).Fu+ja)xljo)=Ae 2u(t) Where A is independent of t (3).X(o)ao=2 Determine a closed-form expression for x(o) x()=√1e-e2)l()
6 Chapter 4 Problem Solution 4.14 Consider a signal be a signal with Fourier transform . Suppose we are given the following facts: (1). is real and nonnegative. (2). Where A is independent of t . (3). Determine a closed-form expression for . x(t) X(j) x(t) x(t) F ( j )X(j ) Ae u(t) 1 2t 1 − − + = ( ) 2 2 = + − X j d x(t) (e e )u(t) t t 12 − −2 = −
Chapter 4 Problem solution 4. 15 Let x(t) be a signal with Fourier transform x(ja).Suppose we are given the following facts: (1). x(t)is real (2)-x(t)=0ort≤0 (32-Refx(io)felo do=ite" Determine a closed-form expression for x(t) x(t)=2te u(t) 7
7 Chapter 4 Problem Solution 4.15 Let be a signal with Fourier transform . Suppose we are given the following facts: (1). is real. (2). for . (3). Determine a closed-form expression for . x(t) X(j) x(t) x(t)= 0 t 0 ( ) 1 Re 2 j t t X j e d t e + − − = x(t) x(t) te u(t) −t = 2
Chapter 4 Problem solution 4.24 (a) Determine which,if any, of the real signals depicted in Figure P4.24 have Fourier transform that satisfy each of the following conditions: 1.Re{Xo)}=0(a),(d) 2.Im{X(o)}2=0(e),( 3. There exists a real a such that eid@ xja)is real (a),(b),(e),(f) 4.」X(o)o=0(a),(b,o),(,(0 5.∫aX(o)o=0(),(o,(), 6. X Gjo) is periodic(b)
8 Chapter 4 Problem Solution 4.24 (a) Determine which ,if any , of the real signals depicted in Figure P4.24 have Fourier transform that satisfy each of the following conditions: 1. ReX(j)= 0 2. ImX(j)= 0 ( ) = 0 + − 4. X j d ( ) = 0 + − 5. X j d 3. There exists a real such that is real. ( ) e X j j 6. is periodic X(j) (a) , (d) (e) , (f) (a) , (b) , (e) , (f) (a) , (b) , (c) , (d) , (f) (b) , (c) , (e) , (f) (b)
Chapter 4 Problem solution (b) Construct a signal that has properties (1),(4), and (5)and does not have the others 1.Re{X(io)}=0→x(-)=-x 4.」X(o)o=0→x()-=0 5.axiom=0s dxr(t) 0 dt t=0 3. There does not exist a real a such that ejdoxlja) is real x(t+a) is not even 6. XGo) is not periodic For example, x(t =e3
9 Chapter 4 Problem Solution (b) Construct a signal that has properties (1),(4),and (5) and does not have the others. 1. ReX(j)= 0 x(−t) = −x(t) ( ) = 0 + − 4. X j d ( ) 0 0 = t= x t ( ) = 0 + − 5. X j d ( ) 0 0 = t= dt dx t 3. There does not exist a real such that is real. ( ) e X j j x(t +) is not even. 6. is not periodic X(j) For example, x(t) = t 3 , t 5 , t 7
Chapter 4 Problem solution 4. 25 Let x vja) denote the Fourier transform of the signal x() (a)Find∠X(o 2↑x (b)Find X(jo) 1012 (c)FindX(joda; (d) Evaluate(jo) 、2SnOa20d0 (e) Evaluate xio)da (f) Sketch the inverse Fourier transform of Rex(jo))
10 Chapter 4 Problem Solution X(j) x(t) 4.25 Let denote the Fourier transform of the signal . (a) Find X(j) −1 0 1 2 3 1 2 x(t) t (b) Find X(j0) (c) Find X(j)d; + − (d) Evaluate ( ) X j e d 2sin j2 + − (e) Evaluate ( ) ; 2 X j d + − (f) Sketch the inverse Fourier transform of ReX(j)
