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Homework Answers
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summary:
1. G(jw) is a continous function. but in the given x[n],w[n] are distrete time functions so the convolution of x[n],w[n] are also in discrete functions with complex variables.
sketch of G[jw] is complex.
2. y[n] = { 0 n-->even no
{-2/[(pi*n)^2] * [1/(root(2)]^n] n-->odd
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Q1. Consider the system displayed below 8[n] blul y[n] W[n] sin(in/2) where x[n] = sin(itn/4) πη...
2TT sinn (1) a) Let x1 [n] = πη Find the Discrete Time Fourier transform of...
2TT sinn (1) a) Let x1 [n] = πη Find the Discrete Time Fourier transform of this signal and plot it with all its critical values. (you can use only transform tables from the book). b) Now, define xzlv) = (**) GHS) Using transform properties, find the Discrete Time Fourier transform of x2[n] and plot it with all its critical values. In your calculations be sure to show your steps ! 2TT sinn sinn sinwon c) Let y[n] [( )...
Name: 10. [8 points] Consider a discrete-time LTI system with input x[n] and out- put y[n]....
Name: 10. [8 points] Consider a discrete-time LTI system with input x[n] and out- put y[n]. When the input signal x[n] = (6)" is applied to the system, the output signal is y[n] = 0 for all n When the input signal xn] (3)" u[n] is applied to the system, the output signal is y[n] = A 8[n] + 2 (5)" u[n] for all n, where A is a constant number a) Find A. b) Find the impulse response of...
1. Using appropriate properties and the table of Fourier transforms, obtain and sketch the sin(at) Fourier...
1. Using appropriate properties and the table of Fourier transforms, obtain and sketch the sin(at) Fourier transform of the signal x()cn(31-4 marks) 2fX(a), determine the Fourier transform of the signal y(t)dx( F.T. dx(2t) dt (3 marks) 3. Find the Fourier transform of x(t)-cos(2t/4). (3 marks) 4. Let x(t) be the input to a linear time-invariant system. The observed output is y(t) 4x(t 2). Find the transfer function H() of the system. Hence, obtain and sketch the unit-impulse response h(t) of...
5. (12 points) Consider a continuous-time LTI system whose frequency response is sin(w) H(ju) 4w If...
5. (12 points) Consider a continuous-time LTI system whose frequency response is sin(w) H(ju) 4w If the input to this system is a periodic signal 0, -4<t<-1 x(t)=1, -1st<1 0, 1st<4 with period T= 8 (a) (2 points) sketch r(t) for -4ts4 (b) (5 points) determine the Fourier series coefficients at of x(t), (c) (5 points) determine the Fourier series coefficients be of the corresponding system output y(t)
5. (12 points) Consider a continuous-time LTI system whose frequency response is...
(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...
a causal discrete time LTI system is implemented using the difference equation y(n)-0.5y(n-1)=x(n)+x(n-1) where x(n) is...
a causal discrete time LTI system is implemented using the difference equation y(n)-0.5y(n-1)=x(n)+x(n-1) where x(n) is the input signal and y(n) the output signal. Find and sketch the impulse response of the system
8. a. Evaluate and sketch the output of the system if h[n] G un] x(t) -2-1...
8. a. Evaluate and sketch the output of the system if h[n] G un] x(t) -2-1 0 Find the Fourier series coefficients of the periodic signal shown above. [15 marks b.
Consider the DT LTI system defined by the mpulse response h[n] = ?[n] The input to...
Consider the DT LTI system defined by the mpulse response h[n] = ?[n] The input to this system is the signal rn: ?[n-1) (a) Sketch h[n] and r[n] (b) Determine the output of the systern, ylnj, using convolution. Sketch y[n] (c) Determine the DTFTs H(e) and X(e. Make fully-labeled sketches of the magni- tudes of these DTFTs (d) Recall that the discrete Fourier transform (DFT) is simply defined as samples of the discrete-time Fourier transform (DTFT). Compute the 4-point (N-4)...
QUESTION 2 [25 Marks Determine the Fourier Transform, H(2), of the discrete impulse response h[n]. where...
QUESTION 2 [25 Marks Determine the Fourier Transform, H(2), of the discrete impulse response h[n]. where ?[n] represents a discrete unit impulse: a. [6 marks] h[n] ?[n+3] + ?[n+2] + ?[n+1 ] + ?[n] + ?[n-1 ] + ?[n-2] + ?[n-3] The sequence h[n] implement a digital filter. Determine the nature of the filter sketch H(Q)). What is then the cut-off frequency if the sampling frequency is 8 kHz? b. [6 marks] v c. Predict the spectral coefficients a of...
Objective Conduct DTFT, DTFS on a periodic discrete signal. Task: Consider the system with impulse response...
Objective Conduct DTFT, DTFS on a periodic discrete signal. Task: Consider the system with impulse response Tth sin 8 h(n) S(n) Tn (1) Find the Fourier-series representation for the output y(n) when the input x(n) is the periodic extension of the sequence 3/2, -1,0, -3/2, 1,0 Plot the x(n), h(n), y(n) and Fourier coefficient bk using Matlab or handwriting (Example 7.2.6 irse material) in cour (2) Find the output y(n) of the system with the input 1 Tn Tn x(п)...
2TT sinn (1) a) Let x1 [n] = πη Find the Discrete Time Fourier transform of this signal and plot it with all its critical values. (you can use only transform tables from the book). b) Now, define xzlv) = (**) GHS) Using transform properties, find the Discrete Time Fourier transform of x2[n] and plot it with all its critical values. In your calculations be sure to show your steps ! 2TT sinn sinn sinwon c) Let y[n] [( )...
Name: 10. [8 points] Consider a discrete-time LTI system with input x[n] and out- put y[n]. When the input signal x[n] = (6)" is applied to the system, the output signal is y[n] = 0 for all n When the input signal xn] (3)" u[n] is applied to the system, the output signal is y[n] = A 8[n] + 2 (5)" u[n] for all n, where A is a constant number a) Find A. b) Find the impulse response of...
1. Using appropriate properties and the table of Fourier transforms, obtain and sketch the sin(at) Fourier transform of the signal x()cn(31-4 marks) 2fX(a), determine the Fourier transform of the signal y(t)dx( F.T. dx(2t) dt (3 marks) 3. Find the Fourier transform of x(t)-cos(2t/4). (3 marks) 4. Let x(t) be the input to a linear time-invariant system. The observed output is y(t) 4x(t 2). Find the transfer function H() of the system. Hence, obtain and sketch the unit-impulse response h(t) of...
5. (12 points) Consider a continuous-time LTI system whose frequency response is sin(w) H(ju) 4w If the input to this system is a periodic signal 0, -4<t<-1 x(t)=1, -1st<1 0, 1st<4 with period T= 8 (a) (2 points) sketch r(t) for -4ts4 (b) (5 points) determine the Fourier series coefficients at of x(t), (c) (5 points) determine the Fourier series coefficients be of the corresponding system output y(t)
5. (12 points) Consider a continuous-time LTI system whose frequency response is...
(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...
8. a. Evaluate and sketch the output of the system if h[n] G un] x(t) -2-1 0 Find the Fourier series coefficients of the periodic signal shown above. [15 marks b.
Consider the DT LTI system defined by the mpulse response h[n] = ?[n] The input to this system is the signal rn: ?[n-1) (a) Sketch h[n] and r[n] (b) Determine the output of the systern, ylnj, using convolution. Sketch y[n] (c) Determine the DTFTs H(e) and X(e. Make fully-labeled sketches of the magni- tudes of these DTFTs (d) Recall that the discrete Fourier transform (DFT) is simply defined as samples of the discrete-time Fourier transform (DTFT). Compute the 4-point (N-4)...
QUESTION 2 [25 Marks Determine the Fourier Transform, H(2), of the discrete impulse response h[n]. where ?[n] represents a discrete unit impulse: a. [6 marks] h[n] ?[n+3] + ?[n+2] + ?[n+1 ] + ?[n] + ?[n-1 ] + ?[n-2] + ?[n-3] The sequence h[n] implement a digital filter. Determine the nature of the filter sketch H(Q)). What is then the cut-off frequency if the sampling frequency is 8 kHz? b. [6 marks] v c. Predict the spectral coefficients a of...
Objective Conduct DTFT, DTFS on a periodic discrete signal. Task: Consider the system with impulse response Tth sin 8 h(n) S(n) Tn (1) Find the Fourier-series representation for the output y(n) when the input x(n) is the periodic extension of the sequence 3/2, -1,0, -3/2, 1,0 Plot the x(n), h(n), y(n) and Fourier coefficient bk using Matlab or handwriting (Example 7.2.6 irse material) in cour (2) Find the output y(n) of the system with the input 1 Tn Tn x(п)...
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