1) Consider the sinusoidal signal x(t) = sin(not) = If x(t) is sampled with frequency 2,...

 

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1) Consider the sinusoidal signal x(t) = sin(not) = If x(t) is sampled with frequency 2, = 2π/T rad/sec then the discrete-time signal x[n] x(nT) is equal to x[n] = sin(nonT) Assume the sampling frequency is fixed at 2, = 27(8192) rad/sec. a) Assume 2, = 27 (1000) rad/sec and define T = 1/8192. Create the vector n = [0:8191], so that t = n* T contains the contains the 8192 time samples of the interval 0 < t < 1. Create a vector x which contains the samples of x(t) at the time samples in t. b) Display the first fifty samples of x[n] versus n using stem. Display the first fifty samples of x(t) versus the sampling times using plot. (Use subplot to simultaneously display these two plots.) 2) In the following problems, you will use both bandlimited and linear interpolation to reconstruct the following signals T₁ (t): cos (³), I₂(t)= )={₁- 1-t\/2, t ≤2, 0, otherwise, from samples obtained at sample times t = nT with T = 1/2. a) Create a vector ts which contains the sampling times t = nT on the interval |t| ≤ 4. Store in the vectors xs1 and xs2 the samples of x₁ (t) and x₂ (t) at the corresponding times in ts. Use stem to plot xs1 and xs2 versus ts. To reconstruct x₁ (t) and x₂ (t) from these samples, note that the reconstructed signals can only be computed at a finite number of samples in MATLAB. Therefore, you will calculate the interpolated signals only at t = n/8 on the interval [t] ≤ 2. In other words, on the interval |t| ≤ 2 you will calculate three samples in between every sample contained in xs1 and xs2. The sampling interval of the interpolated signal is thus T, = 1/8. Call yıbl(t) and y2b (t) the signals given by interpolating the samples of x₁ (t) and x₂ (t) with the interpolating filter holf (t). Similarly, call yılin (t) and yzlin (t) the signals given by interpolating the samples of x₁ (t) and x₂ (t) with the linear interpolator hun (t). b) Set T₁ = 1/8, and create a vector of the interpolation times t₁ = [-2: T₁: 2]. Store in the vectors hbl and hlin the values of hof (t) and hun (t) at the interpolation times. Use plot to display these two impulse responses versus ti. What are the values of impulse responses at the sample times ts? The peak value of each impulse response should be at t = 0. = 1) Consider the sinusoidal signal x(t) = sin(not) = If x(t) is sampled with frequency 2, = 2π/T rad/sec then the discrete-time signal x[n] x(nT) is equal to x[n] = sin(nonT) Assume the sampling frequency is fixed at 2, = 27(8192) rad/sec. a) Assume 2, = 27 (1000) rad/sec and define T = 1/8192. Create the vector n = [0:8191], so that t = n* T contains the contains the 8192 time samples of the interval 0 < t < 1. Create a vector x which contains the samples of x(t) at the time samples in t. b) Display the first fifty samples of x[n] versus n using stem. Display the first fifty samples of x(t) versus the sampling times using plot. (Use subplot to simultaneously display these two plots.) 2) In the following problems, you will use both bandlimited and linear interpolation to reconstruct the following signals T₁ (t): cos (³), I₂(t)= )={₁- 1-t\/2, t ≤2, 0, otherwise, from samples obtained at sample times t = nT with T = 1/2. a) Create a vector ts which contains the sampling times t = nT on the interval |t| ≤ 4. Store in the vectors xs1 and xs2 the samples of x₁ (t) and x₂ (t) at the corresponding times in ts. Use stem to plot xs1 and xs2 versus ts. To reconstruct x₁ (t) and x₂ (t) from these samples, note that the reconstructed signals can only be computed at a finite number of samples in MATLAB. Therefore, you will calculate the interpolated signals only at t = n/8 on the interval [t] ≤ 2. In other words, on the interval |t| ≤ 2 you will calculate three samples in between every sample contained in xs1 and xs2. The sampling interval of the interpolated signal is thus T, = 1/8. Call yıbl(t) and y2b (t) the signals given by interpolating the samples of x₁ (t) and x₂ (t) with the interpolating filter holf (t). Similarly, call yılin (t) and yzlin (t) the signals given by interpolating the samples of x₁ (t) and x₂ (t) with the linear interpolator hun (t). b) Set T₁ = 1/8, and create a vector of the interpolation times t₁ = [-2: T₁: 2]. Store in the vectors hbl and hlin the values of hof (t) and hun (t) at the interpolation times. Use plot to display these two impulse responses versus ti. What are the values of impulse responses at the sample times ts? The peak value of each impulse response should be at t = 0. =

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