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Moving to another question will save this respose. Question 5 of 23 Question 5 10 points Save Answer A student warnts to visualize the spectrum of the following CT signal using MATLAB, xt) cos(2 p2*:t in seconds He computes a 1024-point FFT of 100 samples using a sampling frequency of 10 Hz. Which of the following recommendations would you give him? 12 to 13 1. The spectrum obtained will not match the theoretical spectrum due to the fact that a 1024-point FFT is not adequate. He should uso at enst 2412 points for the FFT computation. Even if a blackman window is used, sidelobes would still be visible on the linear spectra plot. R19 O 2. The spectrum obtained will not match the theoretical spectrum due to the fact that a rectangular window was used. Consequently, it will have sidelobes. The sidelobes can be reduced by using another window. For instance, with the y-axis in inear scale the sidelobes would not be visible ส a blackman widow is used, and the resu ting spectrum wold mpresent more acarately the spectrum of xit, However, it is still does not match the ideal due to the limited frequency resolution. A 1024 point FFT has sufficient frequency resolution in this example, since it already provides a frequency resolution of 0.0098 Hz. 29 O 3. The spectrum obtained will not match the theoretical spectrum due to the fact that a rectangular window was used. Consequently, it will have f a blackman window is used, and the resulting spectrum would ropresent more accurately the spectrum of xt). However, it is stial sidelobes. The sidelobes can be reduced by using another widow. For nstance, with the aus in inear scale the sideotes wa be odeed but does not match the ideal due to the limited frequency resolution. A 1024 point FFT has suflicient computational resolution in this example, since t already provides a computational resolution of 0.0098 He O 4. The spectrum obtained will not match the theoretical spectrum due to the fact that a rectangular window was used. Consequently, it will have sidelobes. f a blackman window is used, and the resulting spectrum would represent more accurately the spectrum of xit). However, it io still does not match the ideal due to the limited frequency resolution. A 1024 point FFT has sufficient provides a computational resolution of 0.0195 Hz The sidelobes can be reduced by using another window. For instance, with the y-axis in Inear scale the sidelobes would not be visible OS AM computational resolution in this example, since it already o s. The spectrum obtained vwll nof match the theorical spectrum due to the fact that a rectangular window was used. Consequently, it wil have shot 6.42 PM sidelobes. The sidelobes can be reduced by using another window. For instance, with the y-axis in inear scale the sidelobes would not be visible if a blackman window is used, and the matoh the ideal due to the limited frequency resolution. A 1024 point FFT has sufficient provides a computational resolution of 0.0098 Hz resulting spectrum would represent more accurately the spectrum of x0). However, it in stil doos not computational resolution in this example, since it already Shot 6.47 PM 9 3-DSP-Intervie... paf Show All

Answer 1

We have x(t) = cos(2*pi*2*t). The ideal spectrum should be an impulse at f = 2Hz. But to get the ideal spectrum, the signal should be considered for infinite interval. But it is sampled at Fs = 10Hz and only 100 samples are considered, i.e till time t=10s.

Taking a 1024 point FFT of the sampled signal leads to a frequency resolution $\Delta f = 10Hz/1024 = 0.0098Hz$

Taking the FFT of 100 samples is the same as using a rectangular window of length 100 and hence sidelobes will be observed in the spectrum as shown below(generated using Matlab) :-

1.6 1.4 1.2 0.8 0.6 0.4 0.2 0 -5 4-321 0 23 4 5 Frequency (Hz)

Since Blackman window offers an attenuation of around 50dB from the main lobe to the first sidelobe, on linear scale sidelobes are not visible in the spectrum as shown(generated using Matlab):-

  

0.7 0.6 0.5 0.4 2 0.3 0.2 0.1 -5 4-321 0 23 4 5 Frequency (Hz)

Therefore the answer is 5^th   option.