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Problem 2 Consider a continuous-time signal x(t), of which the Fourier transform is ( 21f #...
Q. 2 A continuous time signal x(t) has the Continuous Time Fourier Transform shown in Fig 2. Xc() -80007 0 80001 2 (rad/s) Fig 2 According to the sampling theorem, find the maximum allowable sampling period T for this signal. Also plot the Fourier Transforms of the sampled signal X:(j) and X(elo). Label the resulting signals appropriately (both in frequency and amplitude axis). Assuming that the sampling period is increased 1.2 times, what is the new sampling frequency 2? What...
Consider the continuous time signal: 2. , π (sin (2t) (Sin (8t) A discrete time signal x[n] -xs(t) -x(nTs) is created by sampling x() with sampling interval, 2it 60 a) Plot the Fourier Transform of the sampled signal, i.e. Xs (jo). b) Plot the DTFT of the sampled signal, ie, X(eja) o) Repeat (a) with 7, 2π d) Repeat (b) with , 18 Consider the continuous time signal: 2. , π (sin (2t) (Sin (8t) A discrete time signal x[n]...
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 4 Let x(t) be a continuous time signal whose Fourier transform has the property that Xe(ja)0 for lal 2 2,000. A discrete time signal aIn]x(n(0.5x 10-3)) is obtained. For each of the following constra ints on Xa(e/n), the Fourier transform of xaln], determine the coresponding constraint on Xe(ja) a) X(en) is real b) The maximum value of X4 (ea) over all is 1 c) Xa(ea)= Xa(e/ a-) Problem 4 Let x(t) be a continuous time signal whose Fourier transform...
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?
Problem 5. (Properties of Fourier transform) Consider a continuous time signal x(1) with the following Fourier transform: X(jw) = J 1 - if we l-207, 207] if|wl > 207 (3) Let y(t) = x(26) cos? (507). Sketch Y (w), i.e., the Fourier transform of y(t). (Note that 2 1 + cos(20) cos? (0) = 2
# 1 : Imagine that you have a continuous-time signal x(t) whose continuous-time Fourier transform is as given below -25 -20 f, Hz -10 10 20 25 (a) (10 pts) Imagine that this signal is sampled at the sampling rate of F, 65 Hz. Sketch the FT of the resulting signal that would be at the output of an ideal DAC (like we discussed in class) when given these samples. (b) (10 pts) Repeat part (a) for the case that...
(1) Consider the following continuous-time signal: (1) 2ua(-t+t)ua(t), where its energy is 20 milli Joules (2 x 103Joules). The signal ra(t) is sampled at a rate of 500 samples/sec to yield its discrete-time counter part (n) (a) Find ti, and hence sketch ra(t). (b) From part (a), plot r(n) and finds its energy (c) Derive an expression for the Fourier transform of a(n), namely X(ew). (d) Plot the magnitude spectrum (1X(e)) and phase spectrum 2(X(e). (e) Consider the signal y(n)...
Problem 1: (3 +2+3+2 10, sampling) Consider the continuous-time signal x(t) = 3 + cos(10?1+ 5) + sin(15?), t E R (a) Find the Fourier transform X-Fr. Hint: (F ejuot) (w) 2??(w-wo) (b) What is the Nyquist Frequency wn in radians/s of x? (c) Write an expression for the Fourier transform of the ideal sampling of x with sam- pling period T, = 2n/Cav.), i.e., ?00_ox(AZ)6(t-kZ) Hint: (F eiru>tz(t) (w) - X(w - rus) and recall Poisson's identity, CO eyru'st,...
-l 2. Consider the continuous-time signal: 0 x(t)- 1sts1 0, otherwise Find the Fourier transform X(a) of x(t). Simplify ths expression as much as po e simplest expression does not involve any complex numbers.) Draw a rough plot of o) as a function of w. Identify the peak value of X(w). Identify the location of the X( first null on either side of the vertical axis.