7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0

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1 Fourier Transform Problems 1. Find the Fourier transform of the following signals: a) f 1 (t )=e 3 t sin(10 t)u (t) b) f 1 (t )=e 4 t cos(10 t)u (t) 2. Find the Fourier transform of the following signals: a) f (t)=cos(at π /3) b) g(t)=u (t+1)sin(π t) 3. Find the Fourier transforms of the following: a) f (t)=δ (t+3) δ (t 3) b) f (t)= 2δ (t 1)dt c) h(t )=(1+ A sin(at))cos(bt) d) i(t)=1 t, 0<t< 4 c) f (t)=δ (3 t) δ ' (2t ) 4. Determine the Fourier transforms of these functions: a) f (t)=8/t 2 b) g(t)=4/(4 +t 2 ) 5. Find the Fourier transforms of: a) x(t)=2 cos(2t) u(t) b) x(t)=0.5 sin(10t )u(t ) 6. Given that F(ω )=F [f (t )], prove the following, using the definition of the Fourier transform: a) F [f (t t 0 )]=e jω t 0 F(ω ) c) F [f ( t )]=F ( ω ) df (t) b) F[ dt ] = jω F (ω ) d) F [t f (t)]= j 7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= d d ω F (ω ) n= f n e j nω 0t has the Fourier transform F(ω )= f n δ (ω nω 0 ) where ω 0 =2 π /T. (b) Find the Fourier n= transform of the signal in Fig. PR-8. Figure PR-8 9. Show that (sin(aω)/(aω )) 2 dω=(π /a). Hint: Use the fact that F [u (t+ a) u (t a)]=2 a (sin( aω )/(aω )). 10. Prove F [f (t )sin(ω 0 t)]=( j/2)[ F(ω +ω 0 ) F (ω ω 0 )]. 11. If F(ω )=10/[(2+ jω )(5+ jω )] determine the following: a) f ( 3t) b) f (2t 1) c) f (t)cos(2t) d) d dt f (t) e) f (t) dt 12. Given that F [t f (t)]=( j/ω )(e jω 1), find the Fourier transforms of: a) x(t)=f (t)+3 b) y(t)=f (t 2) t c) h(t )=f ' (t) d) g(t)=4 f ( 2 3 t ) +10 f ( 5 3 t )

2 13. Obtain the inverse Fourier transform of the following signals: a) G(ω )= 5 10 c) X (ω )= jω 2 ( jω 1)( jω 2) b) H (ω )= 12 ω Determine the inverse Fourier transforms of the following: 1 b) a) H (ω )= G(ω )=2u(ω+1) 2u(ω 1) ( jω +4) Find the inverse Fourier transforms of the following: a) F(ω )= c) H (ω )= jω ( jω +10) ω 2 + j 40ω jω b) G(ω )= δ (ω ) d) Y (ω )= ( jω +2)( jω +3) ( jω +1)( jω +2) 16. Find the inverse Fourier transforms of: π δ (ω ) 20δ (ω 2) a) F(ω )= c) H (ω )= (5+ jω )(2+ jω ) (2+ jω )(3+ jω ) 10δ (ω +2) 5π δ (ω ) b) G(ω )= d) Y (ω )= jω ( jω +1) 5+ jω + 5 jω (5+ jω ) 17. Determine the inverse Fourier transforms of: a) F(ω )=4 δ (ω +3)+δ (ω )+4 δ (ω 3) c) H (ω )=6 cos(2ω ) b) G(ω )=4 u(ω +2) 4 u(ω 2) 18. For a linear system with input x(t) and output y(t), find the impulse response for the following cases: a) x(t)=e a t u(t), y(t)=u(t) u( t ) b) x(t)=e t u(t), y(t)=e 2 t u(t ) 19. For a linear system with output y(t) and impulse response h(t), find the corresponding input x(t) for the following cases: a) y(t)=t e a t u(t), h(t)=e a t u(t ) c) y(t)=e a t u(t), h(t )=sgn(t) b) y(t)=u (t+1) u(t 1), h(t)=δ (t ) 20. Determine the functions corresponding to the following Fourier transforms: a) F 1 (ω )= e j ω c) F jω +1 3 (ω )= δ (ω ) 1+ j2ω b) F1 (ω )= e j ω jω Find f(t) if: a) F(ω )=2 sin(π ω )[u(ω +1) u(ω 1)] b) F(ω )= 1 j ω (sin(2ω ) sin(ω ))+ ω (cos(2ω ) cos(ω )) 22. A signal f(t) has Fourier transform F(ω )=1/(2+ jω ). Determine the Fourier transform of the following signals: a) x(t)=f (3t 1) d) h(t )=f (t) f (t) b) y(t)=f (t )cos(5t) e) i(t)=t f (t) c) z(t)= d d t f (t) 23. The transfer function of a circuit is H (ω )=10 /(2+ jω ). If the input signal to the circuit is v s (t)=e 4 t u(t) V, find the output signal. Assume all initial conditions are zero.

3 24. Find the transfer function I O (ω )/I S (ω ) for the circuit in Fig. PR-24. Fig. PR Given the circuit in Fig. PR-25 with the given source voltage, find the Fourier transform of i(t). Figure PR Determine the current i(t) in the circuit of Fig. PR-26, given the source voltage as shown. Figure PR Determine the Fourier transform of v(t) in the circuit shown in Fig. PR-27. Figure PR Find the current i o (t) in the circuit of Fig. PR-28 if a) i(t)=sgn(t) A b) i(t)=4[u(t ) u(t 1)] A Figure PR-28

4 29. Find v o (t) in the circuit of Fig. PR-29 when i s (t)=5 e t u (t) A. Figure PR If the rectangular pulse v s (t) is applied to the circuit in Fig. PR-30 find v o (t) at t = 1 s. Figure PR Use the Fourier transform to find i(t) in the circuit of Fig. PR-31 if v s (t)=10e 2t u(t) V. Figure PR Determine the Fourier transform of i o (t) in the circuit of Fig. PR-32. Figure PR Find the voltage v o (t) in the circuit of Fig. PR-33. Let i s (t)=8 e t u (t) A. Figure PR-33

5 34. Find i o (t) in the op amp circuit of Fig. PR-34. Figure PR Use the Fourier transform method to obtain v o (t) in the circuit of Fig. PR-35. Figure PR Determine v o (t) in the transformer circuit of Fig. PR-36. Figure PR Find the energy dissipated by the resistor in the circuit of Fig. PR-37. Figure PR For F(ω )= 3 3+ jω find J= f 2 (t) dt. 39. If f (t)=e 2 t, find J= F (ω ) 2 dω. 40. Let f (t)=5 e 2(t 2 ) u(t). Find F(ω ) and use it to find the total energy in f(t). 41. The voltage across a 1 Ω resistor is v(t )=t e 2 t u(t ) V. (a) What is the total energy absorbed by the resistor? (b) What fraction of this energy is the frequency band -2 ω 2? 42. Let i(t)=2 e t u( t) A. Find the total energy carried by i(t) and the percentage of the 1 Ω energy in the frequency range -5 ω 5 rad/s.

6 43. When the input voltage to a certain linear system is v i (t)=2δ (t) V, the output is v o (t)=10e 2 t u(t) 6e 4 t u(t) V. Find the output when the input is v i (t)=4 e t u(t) V. 44. Find v(t) in the circuit shown in Fig. PR-44 when the input current is: i S (t)=10+8 cos(2π t+30 o )+5 cos(4 π t 150 o ) ma. Figure PR-44 Additional Problems 45. Find the Fourier transform of the function in Fig. PR-45. Figure PR Determine the Fourier transform of the function in Fig. PR-46. Figure PR-46

7 47. Find the Fourier transform of the waveform shown in Fig. PR-47. Figure PR Obtain the Fourier transform of the signal shown in Fig. PR-48. Figure PR Find the Fourier transforms of both functions in Fig. PR-49. Figure PR-49

8 50. Find the Fourier transforms of the signals in Fig. PR-50. Figure PR Find the Fourier transforms of the signals in Fig. PR-51. Figure PR Find the Fourier transforms of the signals in Fig. PR-52. Figure PR Find the Fourier transforms of the signals in Fig. PR-53. Figure PR-53

9 54. Find the Fourier transform of the sine-wave pulse shown in Fig. PR-54. Figure PR-54