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)