A) Assume that a continuous-time aperiodic signal x(t) is real. Prove that the spectrum X(jω) satisfies...
A) Assume that a continuous-time aperiodic signal x(t) is real. Prove that the spectrum X(jω) satisfies the following property:
X(jω) =X(−jω)∗
where ∗ denotes conjugation.
B) Assume that a continuous-time aperiodic signal x(t) is real and even. Prove that the spectrum X(jω) is real and even.
Homework Answers
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A) Assume that a continuous-time aperiodic signal x(t) is real.
Prove that the spectrum X(jω) satisfies...
2) (35 points) Given the following continuous-time aperiodic signal. x(t) A t T 2 2 a....
2) (35 points) Given the following continuous-time aperiodic signal. x(t) A t T 2 2 a. (25 points) Compute the exponential Fourier transform. b. (10 points) Plot the Fourier spectrum.
Q2.) Consider the sampling of the continuous-time signal x(t) to obtain a discrete-time signal x[...
Q2.) Consider the sampling of the continuous-time signal x(t) to obtain a discrete-time signal x[n (1)-10cos(1000m + π/3) + 20cos(2000m + π/6). 110points! ], where x a) What is the maximum sampling interval (minimum sampling frequency) that could be used to ensure an aliasing free sampling of this signal? b) Plot the spectrum of the sampled signal if x() is sampled using a sampling frequency of (i) 2500 Hz (ii) 1800 Hz and state whether there will be an aliasing...
A continuous time system H has the frequency response H(jω) = 4π / (4π + jω)...
A continuous time system H has the frequency response H(jω) = 4π / (4π + jω) . a) Find and plot the magnitude as a function of radial frequency. b) Find and plot the phase as a function of radial frequency. c) Using H(jω), find the output y(t) for the input x(t) = 4cos(4πt) + 4cos(12πt)
Consider a sampler which samples the continuous-time input signal x(t) at a sampling frequency fs =...
Consider a sampler which samples the continuous-time input signal x(t) at a sampling frequency fs = 8000 Hz and produces at its output a sampled discrete-time signal x$(t) = x(nTs), where To = 1/fs is the sampling period. If the sampled signal is passed through a unity-gain lowpass filter with cutoff frequency of fs/2, sketch the magnitude spectrum of the resulting signal for the following input signals: (a) x(t) = cos(6000nt). (b) x(t) = cos(12000nt). (c) x(t) = cos(18000nt).
1 x(t) is a continuous-time real sinal. Let (t) and (t) the even and the odd...
1 x(t) is a continuous-time real sinal. Let (t) and (t) the even and the odd components of x(t) respectively, show the total energy E of x(t) is given by 2. In signal analysis, scaling and shifting of a function are quite common (a) Show that the functions r(t) and t to) have the same areas and energies in other words (b) Show that r(T)dr
A continuous time, si spectrum illustrated below: 2 s, has the magnitude spectrum illustrated berioic signal,x(),...
A continuous time, si spectrum illustrated below: 2 s, has the magnitude spectrum illustrated berioic signal,x(), with a fundamental period 7 - 2 -3 2 a) Plot the magnitude spectrum of x (3t). b) Plot the magnitude spectrum of eltx(t) c) Plot the magnitude spectrum of 2x(t) + d) Determine the average power of x(3t) co δ(t-3m)
1. (25 points) A continuous-time periodic signal x (t) is real valued and has a fundamental...
1. (25 points) A continuous-time periodic signal x (t) is real valued and has a fundamental period T-8. The nonzero Fourier series coefficients for x(t) are Express x(t) in the fornm
QUESTION 3 What statement best describes the following magnitude spectrum: X,(2) max max S max O A spectrum of a train of complex triangular pulses O A spectrum of a complex continuous-time non-bandl...
QUESTION 3 What statement best describes the following magnitude spectrum: X,(2) max max S max O A spectrum of a train of complex triangular pulses O A spectrum of a complex continuous-time non-bandlimited signal O A spectrum of a real discrete-time signal uniformly sampled with the rate above Nyquist O A spectrum of a complex continuous-time bandlimited sig O A spectrum of a train of real rectangular pulses O A spectrum of a complex discrete-time signal uniformly sampled with the...
10. Find the Fourier transform of a continuous-time signal x(t) = 10e Su(t). Plot the magnitude...
10. Find the Fourier transform of a continuous-time signal x(t) = 10e Su(t). Plot the magnitude spectrum and the phase spectrum. If the signal is going to be sampled, what should be the minimum sampling frequency so that the aliasing error is less than 0.1 % of the maximum original magnitude at half the sampling frequency. 11. A signal x(t) = 5cos(2nt + 1/6) is sampled at every 0.2 seconds. Find the sequence obtained over the interval 0 st 3...
Q1) For the continuous time signal below, x(t)=1+t (a) Determine the even and odd parts of...
Q1) For the continuous time signal below, x(t)=1+t (a) Determine the even and odd parts of the signal. (b) Sketch the signal from t = -3 to t = 3. (c) Explain why the signal does not possess BIBO stability.
2) (35 points) Given the following continuous-time aperiodic signal. x(t) A t T 2 2 a. (25 points) Compute the exponential Fourier transform. b. (10 points) Plot the Fourier spectrum.
Q2.) Consider the sampling of the continuous-time signal x(t) to obtain a discrete-time signal x[n (1)-10cos(1000m + π/3) + 20cos(2000m + π/6). 110points! ], where x a) What is the maximum sampling interval (minimum sampling frequency) that could be used to ensure an aliasing free sampling of this signal? b) Plot the spectrum of the sampled signal if x() is sampled using a sampling frequency of (i) 2500 Hz (ii) 1800 Hz and state whether there will be an aliasing...
Consider a sampler which samples the continuous-time input signal x(t) at a sampling frequency fs = 8000 Hz and produces at its output a sampled discrete-time signal x$(t) = x(nTs), where To = 1/fs is the sampling period. If the sampled signal is passed through a unity-gain lowpass filter with cutoff frequency of fs/2, sketch the magnitude spectrum of the resulting signal for the following input signals: (a) x(t) = cos(6000nt). (b) x(t) = cos(12000nt). (c) x(t) = cos(18000nt).
1 x(t) is a continuous-time real sinal. Let (t) and (t) the even and the odd components of x(t) respectively, show the total energy E of x(t) is given by 2. In signal analysis, scaling and shifting of a function are quite common (a) Show that the functions r(t) and t to) have the same areas and energies in other words (b) Show that r(T)dr
A continuous time, si spectrum illustrated below: 2 s, has the magnitude spectrum illustrated berioic signal,x(), with a fundamental period 7 - 2 -3 2 a) Plot the magnitude spectrum of x (3t). b) Plot the magnitude spectrum of eltx(t) c) Plot the magnitude spectrum of 2x(t) + d) Determine the average power of x(3t) co δ(t-3m)
1. (25 points) A continuous-time periodic signal x (t) is real valued and has a fundamental period T-8. The nonzero Fourier series coefficients for x(t) are Express x(t) in the fornm
QUESTION 3 What statement best describes the following magnitude spectrum: X,(2) max max S max O A spectrum of a train of complex triangular pulses O A spectrum of a complex continuous-time non-bandlimited signal O A spectrum of a real discrete-time signal uniformly sampled with the rate above Nyquist O A spectrum of a complex continuous-time bandlimited sig O A spectrum of a train of real rectangular pulses O A spectrum of a complex discrete-time signal uniformly sampled with the...
10. Find the Fourier transform of a continuous-time signal x(t) = 10e Su(t). Plot the magnitude spectrum and the phase spectrum. If the signal is going to be sampled, what should be the minimum sampling frequency so that the aliasing error is less than 0.1 % of the maximum original magnitude at half the sampling frequency. 11. A signal x(t) = 5cos(2nt + 1/6) is sampled at every 0.2 seconds. Find the sequence obtained over the interval 0 st 3...
Q1) For the continuous time signal below, x(t)=1+t (a) Determine the even and odd parts of the signal. (b) Sketch the signal from t = -3 to t = 3. (c) Explain why the signal does not possess BIBO stability.
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