4. (Ch. 7 and MATLAB) Consider the CT signal Ic(t) = cos (27 fot), with frequency fo=100 Hz and duration Ta=0.8 seconds. The signal is used as the transmit signal in an active radar to determine the position of an object. The received signal is corrupted by white Gaussian noise w(t) and is given by ye(t)=xe(t) + w(t). (a) Use MATLAB (1) to generate the signal xc(t), using sampling frequency fs= 1 kHz, and its Fourier transform (FT); and (ii) to plot the signal and the magnitude of its FT, correctly label the axis in time and frequency. Provide the code and plots. Hint: See Notes Module 3a for sample MATLAB code to generate a cosine signal and its FT. = (b) Plot the FT R(S) of the cosine signal r(t) =cos (27 fot). Note that, using frequency f in Hz, r(t) = cos (27 fot) "", R(S) = 0.5(8( fo) + 8( f + fo)) Is this plot the same as the one obtained in part (a) for Xc(f) in MATLAB? If not, explain your answer. Hint: Express xe(t) in terms of r(t), noting that the transmit signal xc(t) has finite duration (c) Repeat part (a) to generate and plot the received signal ye(t) using the same sampling frequency fs= 1 kHz. The following MATLAB code can be used to generate N samples of white Gaussian noise with 31 dB (decibels) signal-to-noise ratio (SNR), where N is the number of samples of Ze(t) in part (a). The code assumes that xc is the 1 x N row vector of the transmit signal. E=xc*xc.'; $ Energy of transmit signal SNR=31; : Signal-to-noise ratio in dB var=E/ (10 (SNR/10)); $ Noise variance from SNR w=sqrt (var) *randn (1,N); Gaussian noise with zero-mean and variance var yc=xc+w; % Received signal * (d) Repeat part (c) using an SNR value such that the transmit signal is barely detectable in the frequency domain. What SNR value did you use? (e) Repeat part (a) using the same sampling frequency fs= 1 kHz but the new transmit signal ze(t) = cos (24fit). You need to select and provide the value of frequency fi such that ze(t) is aliased and appears at 100 Hz in the frequency domain. Demonstrate the aliasing theoretically by deriving the FT of the continuous-but-sampled signal zes(t). 4. (Ch. 7 and MATLAB) Consider the CT signal Ic(t) = cos (27 fot), with frequency fo=100 Hz and duration Ta=0.8 seconds. The signal is used as the transmit signal in an active radar to determine the position of an object. The received signal is corrupted by white Gaussian noise w(t) and is given by ye(t)=xe(t) + w(t). (a) Use MATLAB (1) to generate the signal xc(t), using sampling frequency fs= 1 kHz, and its Fourier transform (FT); and (ii) to plot the signal and the magnitude of its FT, correctly label the axis in time and frequency. Provide the code and plots. Hint: See Notes Module 3a for sample MATLAB code to generate a cosine signal and its FT. = (b) Plot the FT R(S) of the cosine signal r(t) =cos (27 fot). Note that, using frequency f in Hz, r(t) = cos (27 fot) "", R(S) = 0.5(8( fo) + 8( f + fo)) Is this plot the same as the one obtained in part (a) for Xc(f) in MATLAB? If not, explain your answer. Hint: Express xe(t) in terms of r(t), noting that the transmit signal xc(t) has finite duration (c) Repeat part (a) to generate and plot the received signal ye(t) using the same sampling frequency fs= 1 kHz. The following MATLAB code can be used to generate N samples of white Gaussian noise with 31 dB (decibels) signal-to-noise ratio (SNR), where N is the number of samples of Ze(t) in part (a). The code assumes that xc is the 1 x N row vector of the transmit signal. E=xc*xc.'; $ Energy of transmit signal SNR=31; : Signal-to-noise ratio in dB var=E/ (10 (SNR/10)); $ Noise variance from SNR w=sqrt (var) *randn (1,N); Gaussian noise with zero-mean and variance var yc=xc+w; % Received signal * (d) Repeat part (c) using an SNR value such that the transmit signal is barely detectable in the frequency domain. What SNR value did you use? (e) Repeat part (a) using the same sampling frequency fs= 1 kHz but the new transmit signal ze(t) = cos (24fit). You need to select and provide the value of frequency fi such that ze(t) is aliased and appears at 100 Hz in the frequency domain. Demonstrate the aliasing theoretically by deriving the FT of the continuous-but-sampled signal zes(t)
