Given that ? ??0? ? ↔ 2??(? − ?0 ), if ?(?) = 3 cos(50?) −...
Given that ? ??0? ? ↔ 2??(? − ?0 ), if ?(?) = 3 cos(50?) − 2sin(150?),
1. Showing all working, derive the Fourier transform ?(?).
2. Sketch the amplitude spectrum |?(?)| and phase spectrum ∠?(?), clearly showing values for all key points in the sketch.
Homework Answers
Consider a sinusoidal signal?(?) = ? cos(?0? + ?). Here ?0 is the angular frequency of the signal and ? is the initial phase shift.
Consider a sinusoidal signal?(?) = ? cos(?0? + ?). Here ?0 is the angular frequency of the signal and ? is the initial phase shift.a) Draw the time domain signal. (10 Marks) b) Given that the Fourier transform pair is ? ??? ℱ ↔ 2??(? − ?), find the Fourier Transform ?(?) of ?(?). Here ?(?) is the direct delta function. (12 Marks) c) Draw the Magnitude and phase spectrums of ?(?). (8 Marks) d) Find the Fourier Transform of ?(?) = ?(?)?(?)...
Please finish these questions. Thank you Given find the Fourier transform of the following: (a) e dt 2T(2 1) 4 cos (2t) (Using properties of Fourier Transform to find) a) Suppose a signal m(t) is giv...
Please finish these questions. Thank you
Given find the Fourier transform of the following: (a) e dt 2T(2 1) 4 cos (2t) (Using properties of Fourier Transform to find) a) Suppose a signal m(t) is given by m()-1+sin(2 fm) where fm-10 Hz. Sketch the signal m(t) in time domain b) Find the Fourier transform M(jo) of m(t) and sketch the magnitude of M(jo) c) If m(t) is amplitude modulated with a carrier signal by x(t)-m(t)cos(27r f,1) (where fe-1000 Hz), sketch...
2. Determine and sketch the spectrum, the Fourier transform, of x() where -2l +cos(0)+ jsin for...
2. Determine and sketch the spectrum, the Fourier transform, of x() where -2l +cos(0)+ jsin for -<t<
1. (a) Evaluate the Fourier coefficients a, an, ba for the function defined as f)-2 cos() for-π/2...
1. (a) Evaluate the Fourier coefficients a, an, ba for the function defined as f)-2 cos() for-π/2 s sn2 and zero else over the period of 2T, do NOT use MATLAB or a calculator for integrations. All the steps should be shown. Write a few terms of the Fourier series expansion Plot 2 or 3 cycles of the Fourier series using MATLAB and verify whether the plot matches the given waveform Find Co and Cn and plot the amplitude spectrum...
4. A periodic signal x (t) is represented by a trigonometrie Fourier series X(t) = 8...
4. A periodic signal x (t) is represented by a trigonometrie Fourier series X(t) = 8 + 4 cos (2t + 60°) + 2sin (3t+30°) - cos (4t + 150°) = 0 * +30°) - cos (4t+150°) = 3 +4 Cos(216)+2 Cart ( 6) Col413 (a) Sketch the trigonometric Fourier series spectra (both magnitude and phase). O i 2 3 (b) Sketch the exponential Fourier series spectra (both magnitude and phase). + Dol -3 -2 -1 0 1 2 3...
Question 4 (35 points). Two message signals, ?1 (?) = 200??(100??) cos(103πt) volts and ?2 (?)...
Question 4 (35 points). Two message signals, ?1 (?) =
200??(100??) cos(103πt) volts and ?2 (?) = 900??2 (450??)
cos(103πt) volts, are required to be combined and then transmitted
over a channel around 2 MHz using amplitude double-side modulation.
We assume that the channel is ideal with no delay and no
attenuation. Design a system consisting of mixers, filters, etc. to
transmit and restore perfectly these two signals (i.e. ?1 (?) and
?2 (?)) at the receiving side after the...
Exercises: u used to the instructor b the end of next lab. 20 102 Plot the f(t)-sinc(20r) cos(300...
Exercises: u used to the instructor b the end of next lab. 20 102 Plot the f(t)-sinc(20r) cos(300t)sinc (10t) cos(100t) Use the fast Fourier transform to find the magnitude and phase spectrum of the signal and plot over an appropriate range. Use appropriate values for the time interval and the sampling interval. Note that in Matlab sinc(x)-, so we need to divide the argument by n to make it match the given function. Le, sinc(20t/pi) Hint: Use the parameters from...
2. (25p) Find the Fourier transform of the following signal. Sketch the amplitude phase spectrum of...
2. (25p) Find the Fourier transform of the following signal. Sketch the amplitude phase spectrum of the signal. x(1) 1 -2 2
(a) Based on the following discrete-time signal x[n], [n] →n -2 -1 0 1 2 3...
(a) Based on the following discrete-time signal x[n], [n] →n -2 -1 0 1 2 3 4 i. [5%] determine the Fourier transform (i.e., X(ein)) and sketch the magnitude spectrum. ii. [4%] Given the following signal Xp[n], which is the periodic version of x[n] with period 4. Derive the Fourier series coefficients of yn], i.e., {ax}. xp[n] -1 1 2 3 4 5 iii. [4%] Hence, derive the Fourier transform of ap[n], i.e., Xp(es"). iv. [5%] Based on the results...
(a) Determine the Fourier transform of x(t) 26(t-1)-6(t-3) (b) Compute the convolution sum of the following signals, (6%) [696] (c) The Fourier transform of a continuous-time signal a(t) is given bel...
(a) Determine the Fourier transform of x(t) 26(t-1)-6(t-3) (b) Compute the convolution sum of the following signals, (6%) [696] (c) The Fourier transform of a continuous-time signal a(t) is given below. Determine the [696] total energy of (t) 4 sin w (d) Determine the DC value and the average power of the following periodic signal. (6%) 0.5 0.5 (e) Determine the Nyquist rate for the following signal. (6%) x(t) = [1-0.78 cos(50nt + π/4)]2. (f) Sketch the frequency spectrum of...
Please finish these questions. Thank you
Given find the Fourier transform of the following: (a) e dt 2T(2 1) 4 cos (2t) (Using properties of Fourier Transform to find) a) Suppose a signal m(t) is given by m()-1+sin(2 fm) where fm-10 Hz. Sketch the signal m(t) in time domain b) Find the Fourier transform M(jo) of m(t) and sketch the magnitude of M(jo) c) If m(t) is amplitude modulated with a carrier signal by x(t)-m(t)cos(27r f,1) (where fe-1000 Hz), sketch...
2. Determine and sketch the spectrum, the Fourier transform, of x() where -2l +cos(0)+ jsin for -<t<
1. (a) Evaluate the Fourier coefficients a, an, ba for the function defined as f)-2 cos() for-π/2 s sn2 and zero else over the period of 2T, do NOT use MATLAB or a calculator for integrations. All the steps should be shown. Write a few terms of the Fourier series expansion Plot 2 or 3 cycles of the Fourier series using MATLAB and verify whether the plot matches the given waveform Find Co and Cn and plot the amplitude spectrum...
4. A periodic signal x (t) is represented by a trigonometrie Fourier series X(t) = 8 + 4 cos (2t + 60°) + 2sin (3t+30°) - cos (4t + 150°) = 0 * +30°) - cos (4t+150°) = 3 +4 Cos(216)+2 Cart ( 6) Col413 (a) Sketch the trigonometric Fourier series spectra (both magnitude and phase). O i 2 3 (b) Sketch the exponential Fourier series spectra (both magnitude and phase). + Dol -3 -2 -1 0 1 2 3...
Question 4 (35 points). Two message signals, ?1 (?) =
200??(100??) cos(103πt) volts and ?2 (?) = 900??2 (450??)
cos(103πt) volts, are required to be combined and then transmitted
over a channel around 2 MHz using amplitude double-side modulation.
We assume that the channel is ideal with no delay and no
attenuation. Design a system consisting of mixers, filters, etc. to
transmit and restore perfectly these two signals (i.e. ?1 (?) and
?2 (?)) at the receiving side after the...
Exercises: u used to the instructor b the end of next lab. 20 102 Plot the f(t)-sinc(20r) cos(300t)sinc (10t) cos(100t) Use the fast Fourier transform to find the magnitude and phase spectrum of the signal and plot over an appropriate range. Use appropriate values for the time interval and the sampling interval. Note that in Matlab sinc(x)-, so we need to divide the argument by n to make it match the given function. Le, sinc(20t/pi) Hint: Use the parameters from...
2. (25p) Find the Fourier transform of the following signal. Sketch the amplitude phase spectrum of the signal. x(1) 1 -2 2
(a) Based on the following discrete-time signal x[n], [n] →n -2 -1 0 1 2 3 4 i. [5%] determine the Fourier transform (i.e., X(ein)) and sketch the magnitude spectrum. ii. [4%] Given the following signal Xp[n], which is the periodic version of x[n] with period 4. Derive the Fourier series coefficients of yn], i.e., {ax}. xp[n] -1 1 2 3 4 5 iii. [4%] Hence, derive the Fourier transform of ap[n], i.e., Xp(es"). iv. [5%] Based on the results...
(a) Determine the Fourier transform of x(t) 26(t-1)-6(t-3) (b) Compute the convolution sum of the following signals, (6%) [696] (c) The Fourier transform of a continuous-time signal a(t) is given below. Determine the [696] total energy of (t) 4 sin w (d) Determine the DC value and the average power of the following periodic signal. (6%) 0.5 0.5 (e) Determine the Nyquist rate for the following signal. (6%) x(t) = [1-0.78 cos(50nt + π/4)]2. (f) Sketch the frequency spectrum of...
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