![The sequence r[n] = cos (Fn). oo < n < oo was obtained by sampling the continuous-time signal ra(t) = cos (Rot), -oo < t < oo at a sampling rate of 1000 samples/sec, what are two possible values of Ω0 that could have resulted in the sequence zn]?](http://img.homeworklib.com/questions/c8ed1e90-b9d9-11ea-afdc-0bb48a97ef7d.png?x-oss-process=image/resize,w_560)
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The sequence r[n] = cos (Fn). oo < n < oo was obtained by sampling the...
THTCos (200Tt) -oo<t< oo. We want to convert = {0,1,2,...} Problem 1. Suppose we have a continuous-time signal r(...
THTCos (200Tt) -oo<t< oo. We want to convert = {0,1,2,...} Problem 1. Suppose we have a continuous-time signal r(t) (t) into a discrete-time signal yn] where we sample the signal every t= nT, seconds for n a. Determine the frequency fo of the cosine in Hertz and its corresponding period (T = 1/fo). b. Plot the continuous signal rt) for t = [0,4T] seconds so that four periods of the signal are plot command) plotted (use the (0, 1, 2,...
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...
Plot the signal s(t) = cos(2πt), and then illustrate the resulting samples with the following sampling...
Plot the signal s(t) = cos(2πt), and then illustrate the resulting samples with the following sampling intervals: [30 points] • Ts= 0.5 sec. • Ts= 0.75 sec • Ts =1 sec. (a) For each case, also sketch the reconstructed continuous time signal from the samples using linear interpolation (i.e. connecting samples by straight lines). (b) In which case the sampled signal has aliasing distortion? What is the minimal sampling frequency and the corresponding sampling interval needed to avoid aliasing? 3....
Problem 3: Sampling a Cosine (again) The continuous-time signal ra(t) = cos (150) is sampled with...
Problem 3: Sampling a Cosine (again) The continuous-time signal ra(t) = cos (150) is sampled with sampling period T, to obtain a discrete-time signal x[n] = XanT). 1. Compute and sketch the magnitude of the continuous-time Fourier transform of ra(t) and the discrete-time Fourier Transform of x[n] for T, = 1 ms and T, = 2 ms. 2. What is the maximum sampling period Ts max such that no aliasing occurs in the sampling process?
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a...
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
3: (Practice Problem)Consider the representation of the process of sampling followed by reconstruction shown below oce=nt)...
3: (Practice Problem)Consider the representation of the process of sampling followed by reconstruction shown below oce=nt) C) Assume that the input signal is Ia(t) = 2 cos(100nt – /4) + cos(300nt + 7/3) -0<t< The frequency response of the reconstruction filter is H.(12) = {T 121</T 10 1921 > A/T (a) Determine the continuous-time Fourier transform X (12) and plot it as a function of N. (b) Assume the fs = 1/T = 500 samples/sec and plot the Fourier transform...
MATLAB Fourier transform. Suppose that a signal x(t) is sampled with sampling frequency fs =100Hz. The sequence x[n] obtained after the sampling is given below: Take the DFT of the sampled sequence a...
MATLAB Fourier transform. Suppose that a signal x(t) is sampled
with sampling frequency fs =100Hz.
The sequence x[n] obtained after the sampling is given below:
Take the DFT of the sampled sequence and plot
its magnitude and phase.
What is the frequency resolution (Δf) of your plot?
N= 20, 100 Hz
N= 20, 100 Hz
2: (a) Consider a discrete-time sequence x[n] = cos(n+3). Find the fundamental period(N). (b) Consider the...
2: (a) Consider a discrete-time sequence x[n] = cos(n+3). Find the fundamental period(N). (b) Consider the sinusodal signal x(t) = 10 sin(21 Fot) with analog frequency F. Write an equa- tion for the discrete time signal n. (c) In part(b) if Fe = 400Hz and the sampling frequency F. = 4kHz, determine the fundamen- tal period of x[n].
3 Sampling and aliasing The aim of this part is to demonstrate the effects of aliasing...
3 Sampling and aliasing The aim of this part is to demonstrate the effects of aliasing arising from improper sampling. A given analog signal z(t) is sampled at a rate fs = 1/T, the resulting samples (nT) are then reconstructed by an ideal reconstructor into the analog signal rat). Improper choice of f, will result in different signals ra(t) + (t), even though they agree at their sample values, that is, tanT) = x(nT). The procedure is illustrated by the...
THTCos (200Tt) -oo<t< oo. We want to convert = {0,1,2,...} Problem 1. Suppose we have a continuous-time signal r(t) (t) into a discrete-time signal yn] where we sample the signal every t= nT, seconds for n a. Determine the frequency fo of the cosine in Hertz and its corresponding period (T = 1/fo). b. Plot the continuous signal rt) for t = [0,4T] seconds so that four periods of the signal are plot command) plotted (use the (0, 1, 2,...
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...
Problem 3: Sampling a Cosine (again) The continuous-time signal ra(t) = cos (150) is sampled with sampling period T, to obtain a discrete-time signal x[n] = XanT). 1. Compute and sketch the magnitude of the continuous-time Fourier transform of ra(t) and the discrete-time Fourier Transform of x[n] for T, = 1 ms and T, = 2 ms. 2. What is the maximum sampling period Ts max such that no aliasing occurs in the sampling process?
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
3: (Practice Problem)Consider the representation of the process of sampling followed by reconstruction shown below oce=nt) C) Assume that the input signal is Ia(t) = 2 cos(100nt – /4) + cos(300nt + 7/3) -0<t< The frequency response of the reconstruction filter is H.(12) = {T 121</T 10 1921 > A/T (a) Determine the continuous-time Fourier transform X (12) and plot it as a function of N. (b) Assume the fs = 1/T = 500 samples/sec and plot the Fourier transform...
MATLAB Fourier transform. Suppose that a signal x(t) is sampled
with sampling frequency fs =100Hz.
The sequence x[n] obtained after the sampling is given below:
Take the DFT of the sampled sequence and plot
its magnitude and phase.
What is the frequency resolution (Δf) of your plot?
N= 20, 100 Hz
N= 20, 100 Hz
2: (a) Consider a discrete-time sequence x[n] = cos(n+3). Find the fundamental period(N). (b) Consider the sinusodal signal x(t) = 10 sin(21 Fot) with analog frequency F. Write an equa- tion for the discrete time signal n. (c) In part(b) if Fe = 400Hz and the sampling frequency F. = 4kHz, determine the fundamen- tal period of x[n].
3 Sampling and aliasing The aim of this part is to demonstrate the effects of aliasing arising from improper sampling. A given analog signal z(t) is sampled at a rate fs = 1/T, the resulting samples (nT) are then reconstructed by an ideal reconstructor into the analog signal rat). Improper choice of f, will result in different signals ra(t) + (t), even though they agree at their sample values, that is, tanT) = x(nT). The procedure is illustrated by the...
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