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5. Consider the 'top-hat' function which is zero everywhere except between -1 and 1 where it take...
1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when...
1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when t = 1. Calculate its Fourier transform, and the convolution N (t) n (t) (the convolution with itself). n(t/T) 1 T 2 2
1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when t = 1....
. Let n (t) be the function being 1 between 1 and and 0 2 2...
. Let n (t) be the function being 1 between 1 and and 0 2 2 everywhere else, that is, the function in the graph when T= 1. Calculate its Fourier transform, and the convolution n (t) n (t) (the convolution with itself). (t/T) 1 T 2 T2
. Let n (t) be the function being 1 between 1 and and 0 2 2 everywhere else, that is, the function in the graph when T= 1. Calculate its Fourier transform,...
A certain wavefunction is zero everywhere except between x = 0 and x = L, where...
A certain wavefunction is zero everywhere except between x = 0 and x = L, where it has the constant value A. Normalize the wavefunction.
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by V(eJ"). Define y[nl-v[Nn] Express Y(e) as a function of V(e). Hint : If confused, start with N-2 (b) Consider the discrete-time signal r[n] with discrete-time Fourier transform X(e). Now, let z[n] be formed by inserting two zeroes between any two samples of x[n]. Give a formula...
(1) Consider the following continuous-time signal: (1) 2ua(-t+t)ua(t), where its energy is 20 milli Joules (2...
(1) Consider the following continuous-time signal: (1) 2ua(-t+t)ua(t), where its energy is 20 milli Joules (2 x 103Joules). The signal ra(t) is sampled at a rate of 500 samples/sec to yield its discrete-time counter part (n) (a) Find ti, and hence sketch ra(t). (b) From part (a), plot r(n) and finds its energy (c) Derive an expression for the Fourier transform of a(n), namely X(ew). (d) Plot the magnitude spectrum (1X(e)) and phase spectrum 2(X(e). (e) Consider the signal y(n)...
1. Consider the function f(x)-e- (a) Find its Fourier transform. (b) Use the result of part (a) t...
1. Consider the function f(x)-e- (a) Find its Fourier transform. (b) Use the result of part (a) to find the value of the integral o0 cos kx dk 0 1 +k2 (c) Show explicitly that Parseval's theorem is satisfied for eand its Fourier transform
1. Consider the function f(x)-e- (a) Find its Fourier transform. (b) Use the result of part (a) to find the value of the integral o0 cos kx dk 0 1 +k2 (c) Show explicitly that Parseval's...
3. Consider a function F(t) which is zero for negative t, and takes the value exp(-t/2...
3. Consider a function F(t) which is zero for negative t, and takes the value exp(-t/2 ) for > 0. Find its Fourier transforms, C(w) and S(w), defined in 200 F(t) = C(w) cos(wt) dw + Sw) sin(wt) do. J-00 J-00 [Hint: Use Euler's theorem.] 4. Demonstrate that Sr?)dt = 2* ["icºw) +8?(]dio, J-00 J-00 where the relation between F(t), C(w), and S(w) is defined above. This result is known as Parseval's theorem.
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and poss...
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.)
the function y=f(x)={ 0-4), 14x+16, x20 x
2. Consider the function sx if x EQ, f(x) = { 1-x if x ER\Q. a)...
2. Consider the function sx if x EQ, f(x) = { 1-x if x ER\Q. a) Prove that f(x) is discontinuous everywhere except at 1. b) Hence, or otherwise, find a bijection g : [0, 1] → [0, 1] which is discontinuous everywhere in (0,1).
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of...
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when t = 1. Calculate its Fourier transform, and the convolution N (t) n (t) (the convolution with itself). n(t/T) 1 T 2 2
1. Let M (t) be the function being 1 between and 1 1 and 0 2 2 everywhere else, that is, the function in the graph when t = 1....
. Let n (t) be the function being 1 between 1 and and 0 2 2 everywhere else, that is, the function in the graph when T= 1. Calculate its Fourier transform, and the convolution n (t) n (t) (the convolution with itself). (t/T) 1 T 2 T2
. Let n (t) be the function being 1 between 1 and and 0 2 2 everywhere else, that is, the function in the graph when T= 1. Calculate its Fourier transform,...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by V(eJ"). Define y[nl-v[Nn] Express Y(e) as a function of V(e). Hint : If confused, start with N-2 (b) Consider the discrete-time signal r[n] with discrete-time Fourier transform X(e). Now, let z[n] be formed by inserting two zeroes between any two samples of x[n]. Give a formula...
(1) Consider the following continuous-time signal: (1) 2ua(-t+t)ua(t), where its energy is 20 milli Joules (2 x 103Joules). The signal ra(t) is sampled at a rate of 500 samples/sec to yield its discrete-time counter part (n) (a) Find ti, and hence sketch ra(t). (b) From part (a), plot r(n) and finds its energy (c) Derive an expression for the Fourier transform of a(n), namely X(ew). (d) Plot the magnitude spectrum (1X(e)) and phase spectrum 2(X(e). (e) Consider the signal y(n)...
1. Consider the function f(x)-e- (a) Find its Fourier transform. (b) Use the result of part (a) to find the value of the integral o0 cos kx dk 0 1 +k2 (c) Show explicitly that Parseval's theorem is satisfied for eand its Fourier transform
1. Consider the function f(x)-e- (a) Find its Fourier transform. (b) Use the result of part (a) to find the value of the integral o0 cos kx dk 0 1 +k2 (c) Show explicitly that Parseval's...
3. Consider a function F(t) which is zero for negative t, and takes the value exp(-t/2 ) for > 0. Find its Fourier transforms, C(w) and S(w), defined in 200 F(t) = C(w) cos(wt) dw + Sw) sin(wt) do. J-00 J-00 [Hint: Use Euler's theorem.] 4. Demonstrate that Sr?)dt = 2* ["icºw) +8?(]dio, J-00 J-00 where the relation between F(t), C(w), and S(w) is defined above. This result is known as Parseval's theorem.
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.)
the function y=f(x)={ 0-4), 14x+16, x20 x
2. Consider the function sx if x EQ, f(x) = { 1-x if x ER\Q. a) Prove that f(x) is discontinuous everywhere except at 1. b) Hence, or otherwise, find a bijection g : [0, 1] → [0, 1] which is discontinuous everywhere in (0,1).
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
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