3a. The signals b(t) = rect(t/2) and c(t) = rect(t/4) are convolved to give d(t) as:...
3a. The signals b(t) = rect(t/2) and c(t) = rect(t/4) are convolved to give d(t) as: d(t) = b(t)∗c(t) = b(x) c(t − x) dx −∞ +∞ ∫ On the graphs below, make neat and labeled sketches of the signals b(t), c(t), d(t)
b.Check your answer by confirming the width _____ and area _____ of d(t). Explain.
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3a. The signals b(t) = rect(t/2) and c(t) = rect(t/4) are
convolved to give d(t) as:...
1.On the graph below, make a neat and labeled sketch of the signal s(t) = 2...
1.On the graph below, make a neat and labeled sketch of the signal s(t) = 2 + cos( t/4 + 1 ),Describe s(t) as odd, even, or neither. What is the energy in s(t)? What is the power in s(t)? 2.On the graph below, make a neat and labeled sketch of the signal s(t) = 3 rect( t - 2 ) cos( π t ),Describe s(t) as odd, even, or neither. What is the energy in s(t)? What is the...
For the remainder of this problem, the signals (t) and y(t) denote the input and output, respecti...
For the remainder of this problem, the signals (t) and y(t) denote the input and output, respectively, of a stable LTI system whose (double-sided) frequency response is known to be w-4m 27T 4m H(w) = rect ( 2π with rect(t) denoting the unit-pulse function i.e., rect(t) 1 for lt| < 1/2 and is zero otherwise. Hint: Use sketches as a guide for answering each question most efficiently. (c) (15 points) Determine y(t) for all t given the applied input is...
Problem 2 In each step to follow the signals h(t) r (t) and y(t) denote respectively the impulse ...
Problem 2 In each step to follow the signals h(t) r (t) and y(t) denote respectively the impulse response. input, and output of a continuous-time LTI system. Accordingly, H(), X (w) and Y (w) denote their Fourier transforms. Hint. Carefully consider for each step whether to work in the time-domain or frequency domain c) Provide a clearly labeled sketch of y(t) for a given x(t)-: cos(mt) δ(t-n) and H(w)-sine(w/2)e-jw Answer: y(t) Σ (-1)"rect(t-1-n)
Problem 2 In each step to follow...
DIFFERENTIATION: For the signals x(t) in Problems (1-2), (a) Compute the fomula for and (b) sketch...
DIFFERENTIATION: For the signals x(t) in Problems (1-2), (a) Compute the fomula for and (b) sketch the signal's derivative x'(t) = x(t). If necessary, use the Differentiation Product Rule: (f.g)' = fg + fig', or "RUD", e. g u (t) = 8(t). In your plots, label both axes, and indicate key values of time and amplitude. (1) X(t) = 4 rect ). (Hint: express rect(t/10) in terms of the difference of Two unit step functions.) ( 10 points) (2) X(t)...
Let T:P1→P2T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2.T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T)range(T) and give a basis...
Let T:P1→P2T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2.T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T)range(T) and give a basis for range(T)range(T). (b) Find ker(T)ker(T) and give a basis for ker(T)ker(T). (c) By justifying your answer determine whether TT is onto. (d) By justifying your answer determine whether TT is one-to-one. (e) Find [T(7+x)]B[T(7+x)]B, where B={−1,−2x,4x2}B={−1,−2x,4x2}.
Let T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T) and give a basis...
Let T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T) and give a basis for range(T). (b) Find ker(T) and give a basis for ker(T) (c) By justifying your answer determine whether T is onto. (d) By justifying your answer determine whether T is one-to-one. (e) Find [T(7+x)]B, where B={−1,−2x,4x2} Please solve it in very detail, and make sure it is correct.
4 a. y(t)-x(t)cos(t/2 b. y(t)-x(t)cos(') x()cos(21) c. X (ju) 5. The signals y(t) in 4a-4e are...
4 a. y(t)-x(t)cos(t/2 b. y(t)-x(t)cos(') x()cos(21) c. X (ju) 5. The signals y(t) in 4a-4e are passed through a filter with unit impulse response h(t) so that the output is z(t)-h(t)*y(t) . Ifthe frequency response of the filter is sketch by hand the Fourier transforms Z(j for 4a-4e Fromjust observing your sketches of Z (jo), which z(1) if any in a-e equal to the original
Find the Nyquist rates for these signals: (a) X(t) = sinc (20) (b)x(t) = 4 sinca...
Find the Nyquist rates for these signals: (a) X(t) = sinc (20) (b)x(t) = 4 sinca (100t) (C) x(t) = 8 sin(50TTT) (d) x(t) = 4 sin(30TTt) + 3 cos(70nt) (e) X(t) = rect(300t) (f) X(t) = -10 sin(40nt) cos(300Tt) (g) X(t) = sinc(t/2)*710(t) (h) x(t) = sinc(t/2) 70.1() (i) X(t) = 8tri((t - 4)/12) (1) X(t) = 13e-201 cos(70TTt)u(t) (k) x(t) = u(t)-u(t-5)
CHAPTER 2-FOURIER TRANSFORMS (30) The table shows a sequence of signals or operations (rows A to...
CHAPTER 2-FOURIER TRANSFORMS (30) The table shows a sequence of signals or operations (rows A to E) in the time 1. domain. Note the symbols for multiplication X and convolution a. Draw the signals and the results in the time and frequency domains b. Draw to scale. Label and tick-mark all the graphs. c. Justify. Use the last column to back up your answer SIGNALS AND OPS FREQUENCY EXPLANATION (AMPLITUDE) TIME MATH Sine wave A Period T Pulse width 4×T...
For the remainder of this problem, the signals (t) and y(t) denote the input and output, respectively, of a stable LTI system whose (double-sided) frequency response is known to be w-4m 27T 4m H(w) = rect ( 2π with rect(t) denoting the unit-pulse function i.e., rect(t) 1 for lt| < 1/2 and is zero otherwise. Hint: Use sketches as a guide for answering each question most efficiently. (c) (15 points) Determine y(t) for all t given the applied input is...
Problem 2 In each step to follow the signals h(t) r (t) and y(t) denote respectively the impulse response. input, and output of a continuous-time LTI system. Accordingly, H(), X (w) and Y (w) denote their Fourier transforms. Hint. Carefully consider for each step whether to work in the time-domain or frequency domain c) Provide a clearly labeled sketch of y(t) for a given x(t)-: cos(mt) δ(t-n) and H(w)-sine(w/2)e-jw Answer: y(t) Σ (-1)"rect(t-1-n)
Problem 2 In each step to follow...
DIFFERENTIATION: For the signals x(t) in Problems (1-2), (a) Compute the fomula for and (b) sketch the signal's derivative x'(t) = x(t). If necessary, use the Differentiation Product Rule: (f.g)' = fg + fig', or "RUD", e. g u (t) = 8(t). In your plots, label both axes, and indicate key values of time and amplitude. (1) X(t) = 4 rect ). (Hint: express rect(t/10) in terms of the difference of Two unit step functions.) ( 10 points) (2) X(t)...
4 a. y(t)-x(t)cos(t/2 b. y(t)-x(t)cos(') x()cos(21) c. X (ju) 5. The signals y(t) in 4a-4e are passed through a filter with unit impulse response h(t) so that the output is z(t)-h(t)*y(t) . Ifthe frequency response of the filter is sketch by hand the Fourier transforms Z(j for 4a-4e Fromjust observing your sketches of Z (jo), which z(1) if any in a-e equal to the original
Find the Nyquist rates for these signals: (a) X(t) = sinc (20) (b)x(t) = 4 sinca (100t) (C) x(t) = 8 sin(50TTT) (d) x(t) = 4 sin(30TTt) + 3 cos(70nt) (e) X(t) = rect(300t) (f) X(t) = -10 sin(40nt) cos(300Tt) (g) X(t) = sinc(t/2)*710(t) (h) x(t) = sinc(t/2) 70.1() (i) X(t) = 8tri((t - 4)/12) (1) X(t) = 13e-201 cos(70TTt)u(t) (k) x(t) = u(t)-u(t-5)
CHAPTER 2-FOURIER TRANSFORMS (30) The table shows a sequence of signals or operations (rows A to E) in the time 1. domain. Note the symbols for multiplication X and convolution a. Draw the signals and the results in the time and frequency domains b. Draw to scale. Label and tick-mark all the graphs. c. Justify. Use the last column to back up your answer SIGNALS AND OPS FREQUENCY EXPLANATION (AMPLITUDE) TIME MATH Sine wave A Period T Pulse width 4×T...
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