1. Determine the Fourier transform of the function...Make a sketch of ...{E(x)}. Discuss its relationship to Fig. 11.11. Get solution
2. Determine the Fourier transform of...Make a sketch of it. Get solution
3. Determine the Fourier transform of...Make a sketch of F(to), then sketch its limiting form as T -> ±∞. Get solution
4. Show that = ... Get solution
5. Determine the Fourier transform of the function f(x) = A cosk0x. Get solution
6. Get solution
7. Get solution
8. Get solution
9. Given that ...{f(x)} = F(k) and ...{h(x}} = H(k) if a and b are constants, determine ...{af(x) + bh(x) Get solution
10. Figure P.l 1.7 shows two periodic functions, f(x) and h(x), which are to be added to produce g(x). Sketch g(x); then draw diagrams of the real and imaginary frequency spectra, as well as the amplitude spectra for each of the three functions.Figure P.11.7... Get solution
11. Compute the Fourier transform of the triangular pulse shown in ' Fig. P. 11.8. Make a sketch of your answer, labeling all the pertinent values on the curve.Figure P.11.8... Get solution
12. Given that ...{f(x)} = F(k), introduce a constant scaling factor 1/a and determine the Fourier transform of/f(x/a). Show that the transform of f(-x) is F(-k). Get solution
13. Show that the Fourier transform of the transform, ...{f(x)}, equals 2пf(-x), and that this is not the inverse transform of the transform, which equals f(x). This problem was suggested by Mr. D. Chapman while a student at the University of Ottawa. Get solution
14. The rectangular function is often defined as...where it is set equal to ½ at the discontinuities (Fig. P.11.11). Determine the Fourier transform of...Notice that this is just a rectangular pulse, like that in Fig. 11.16, shifted a distance x0 from the origin.Figure P.11.11... Get solution
15. With the last two problems in mind, show that ...{(l/2п)sinc (½x)} - rect(k). starting with the knowledge that ...{rect(x)} - sinc(½k),in other words, Eq.(7.58) with L = a, where a = 1. Get solution
16. Utilizing Eq. (11.38), show that ...-l{...{f(x)}} = f(x). Get solution
17. Given ...{f(x)}, show that ...{f(x - x0)} differs from it only by a linear phase factor. Get solution
18. Prove that f...h = h .../directly. Now do it using the convolution theorem. Get solution
19. Get solution
20. Get solution
21. Suppose we have two functions, f(x, y) and h(x, y), where both have a value of 1 over a square region in the xy-plane and are zero everywhere else (Fig. P.11.16). If g(X, Y) is their convolution, make a plot of g(X, 0).Figure P.11.16... Get solution
22. Referring to the previous problem, justify the fact that the convolution is zero for ... when h is viewed as a spread function. Get solution
23. Use the method illustrated in Fig. 11.23 to convolve the two functions depicted in Fig. P.11.18.Figure P.11.18... Get solution
24. Given that ... show that after shifting one of the functions an amount x0, we get ... Get solution
25. Get solution
26. Get solution
27. Prove analytically that the convolution of any function f(x) with a delta function, δ(x), generates the original function f(X). You might make use of the fact that δ(x) is even. Get solution
28. Prove that δ(x - x0) ... f(x) =f(X - x0) and discuss the meaning of this result. Make a sketch of two appropriate functions and convolve them. Be sure to use an asymmetrical f(x). Get solution
29. Show that ...{f(x) cos k0x} = [F(k - k0) + F(k + k0)]/2 and that ...{f(x) sin k0x] = [F(k-k0) - F(k + k0)]/2i. Get solution
30. Figure P.11.23 shows two functions. Convolve them graphically and draw a plot of the result. Get solution
31. Get solution
32. Given the function...determine its Fourier transform. (See Problem 11.11.) Get solution
33. Given the function f(x) = δ(x + 3) + δ (x - 2) + δ (x-5), convolve it with the arbitrary function h(x).Figure P.11.23... Get solution
34. Make a sketch of" the function arising from the convolution of the two functions depicted in Fig. P.l 1.26.Figure P.l 1.26... Get solution
35. Figure P. 11.27 depicts a rect function (as defined above) and a periodic comb function. Convolve the two to get g(x). Now sketch the transform of each of these functions against spatial frequency k/2π = 1/λ. Check your results with the convolution theorem. Label all the relevant points on the horizontal axes in terms of d—like the zeros of the transform of f(x).Figure P.11.27... Get solution
36. Figure P.l 1.28 shows, in one dimension, the electric field across an illuminated aperture consisting of several opaque bars forming a grating. Considering it to be created by taking the product of a periodic rectangular wave h(x) and a unit rectangular function f(x), sketch the resulting electric field in the Fraunhofer region.... Get solution
37. Show (for normally incident plane waves) that if an aperture has a center of symmetry (i.e., if the aperture function is even), then the diffracted field in the Fraunhofer case also possesses a center of 1 symmetry. Get solution
38. Suppose a given aperture produces a Fraunhofer field pattern E(Y, Z). Show that if the aperture's dimensions are altered such that the aperture function goes from ... (y, z) to ... (αy, βz), the newly diffracted field will be given by... Get solution
39. Show that when f(t) - A sin (ωt + ε), Cff(T) - (A2/2) cos ωt, which confirms the loss of phase information in the autocorrelation. Get solution
40. Suppose we have a single slit along the y-direction of width b where the aperture function is constant across it at a value of ... What is the diffracted field if we now apodize the slit with a cosine function amplitude mask? In other words, we cause the aperture function to go from ... at the center to 0 at ±b/2 via a cosinusoidal dropoff. Get solution
41. Get solution
42. Get solution
43. Get solution
44. Get solution
45. Show, from the integral definitions, that ....... Get solution
46. Get solution
47. Get solution
48. Get solution
49. Figure P . l 1.34 shows a transparent ring on an otherwise opaque mask. Make a rough sketch of its autocorrelation function, taking l to be the center-to-center separation against which you plot that function.Figure P.11.34... Get solution
50. Consider the function in Fig. 11.35 as a cosine carrier multiplied by an exponential envelope. Use the frequency convolution theorem to evaluate its Fourier transform. Get solution
2. Determine the Fourier transform of...Make a sketch of it. Get solution
3. Determine the Fourier transform of...Make a sketch of F(to), then sketch its limiting form as T -> ±∞. Get solution
4. Show that = ... Get solution
5. Determine the Fourier transform of the function f(x) = A cosk0x. Get solution
6. Get solution
7. Get solution
8. Get solution
9. Given that ...{f(x)} = F(k) and ...{h(x}} = H(k) if a and b are constants, determine ...{af(x) + bh(x) Get solution
10. Figure P.l 1.7 shows two periodic functions, f(x) and h(x), which are to be added to produce g(x). Sketch g(x); then draw diagrams of the real and imaginary frequency spectra, as well as the amplitude spectra for each of the three functions.Figure P.11.7... Get solution
11. Compute the Fourier transform of the triangular pulse shown in ' Fig. P. 11.8. Make a sketch of your answer, labeling all the pertinent values on the curve.Figure P.11.8... Get solution
12. Given that ...{f(x)} = F(k), introduce a constant scaling factor 1/a and determine the Fourier transform of/f(x/a). Show that the transform of f(-x) is F(-k). Get solution
13. Show that the Fourier transform of the transform, ...{f(x)}, equals 2пf(-x), and that this is not the inverse transform of the transform, which equals f(x). This problem was suggested by Mr. D. Chapman while a student at the University of Ottawa. Get solution
14. The rectangular function is often defined as...where it is set equal to ½ at the discontinuities (Fig. P.11.11). Determine the Fourier transform of...Notice that this is just a rectangular pulse, like that in Fig. 11.16, shifted a distance x0 from the origin.Figure P.11.11... Get solution
15. With the last two problems in mind, show that ...{(l/2п)sinc (½x)} - rect(k). starting with the knowledge that ...{rect(x)} - sinc(½k),in other words, Eq.(7.58) with L = a, where a = 1. Get solution
16. Utilizing Eq. (11.38), show that ...-l{...{f(x)}} = f(x). Get solution
17. Given ...{f(x)}, show that ...{f(x - x0)} differs from it only by a linear phase factor. Get solution
18. Prove that f...h = h .../directly. Now do it using the convolution theorem. Get solution
19. Get solution
20. Get solution
21. Suppose we have two functions, f(x, y) and h(x, y), where both have a value of 1 over a square region in the xy-plane and are zero everywhere else (Fig. P.11.16). If g(X, Y) is their convolution, make a plot of g(X, 0).Figure P.11.16... Get solution
22. Referring to the previous problem, justify the fact that the convolution is zero for ... when h is viewed as a spread function. Get solution
23. Use the method illustrated in Fig. 11.23 to convolve the two functions depicted in Fig. P.11.18.Figure P.11.18... Get solution
24. Given that ... show that after shifting one of the functions an amount x0, we get ... Get solution
25. Get solution
26. Get solution
27. Prove analytically that the convolution of any function f(x) with a delta function, δ(x), generates the original function f(X). You might make use of the fact that δ(x) is even. Get solution
28. Prove that δ(x - x0) ... f(x) =f(X - x0) and discuss the meaning of this result. Make a sketch of two appropriate functions and convolve them. Be sure to use an asymmetrical f(x). Get solution
29. Show that ...{f(x) cos k0x} = [F(k - k0) + F(k + k0)]/2 and that ...{f(x) sin k0x] = [F(k-k0) - F(k + k0)]/2i. Get solution
30. Figure P.11.23 shows two functions. Convolve them graphically and draw a plot of the result. Get solution
31. Get solution
32. Given the function...determine its Fourier transform. (See Problem 11.11.) Get solution
33. Given the function f(x) = δ(x + 3) + δ (x - 2) + δ (x-5), convolve it with the arbitrary function h(x).Figure P.11.23... Get solution
34. Make a sketch of" the function arising from the convolution of the two functions depicted in Fig. P.l 1.26.Figure P.l 1.26... Get solution
35. Figure P. 11.27 depicts a rect function (as defined above) and a periodic comb function. Convolve the two to get g(x). Now sketch the transform of each of these functions against spatial frequency k/2π = 1/λ. Check your results with the convolution theorem. Label all the relevant points on the horizontal axes in terms of d—like the zeros of the transform of f(x).Figure P.11.27... Get solution
36. Figure P.l 1.28 shows, in one dimension, the electric field across an illuminated aperture consisting of several opaque bars forming a grating. Considering it to be created by taking the product of a periodic rectangular wave h(x) and a unit rectangular function f(x), sketch the resulting electric field in the Fraunhofer region.... Get solution
37. Show (for normally incident plane waves) that if an aperture has a center of symmetry (i.e., if the aperture function is even), then the diffracted field in the Fraunhofer case also possesses a center of 1 symmetry. Get solution
38. Suppose a given aperture produces a Fraunhofer field pattern E(Y, Z). Show that if the aperture's dimensions are altered such that the aperture function goes from ... (y, z) to ... (αy, βz), the newly diffracted field will be given by... Get solution
39. Show that when f(t) - A sin (ωt + ε), Cff(T) - (A2/2) cos ωt, which confirms the loss of phase information in the autocorrelation. Get solution
40. Suppose we have a single slit along the y-direction of width b where the aperture function is constant across it at a value of ... What is the diffracted field if we now apodize the slit with a cosine function amplitude mask? In other words, we cause the aperture function to go from ... at the center to 0 at ±b/2 via a cosinusoidal dropoff. Get solution
41. Get solution
42. Get solution
43. Get solution
44. Get solution
45. Show, from the integral definitions, that ....... Get solution
46. Get solution
47. Get solution
48. Get solution
49. Figure P . l 1.34 shows a transparent ring on an otherwise opaque mask. Make a rough sketch of its autocorrelation function, taking l to be the center-to-center separation against which you plot that function.Figure P.11.34... Get solution
50. Consider the function in Fig. 11.35 as a cosine carrier multiplied by an exponential envelope. Use the frequency convolution theorem to evaluate its Fourier transform. Get solution
