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\text { For the complex vector } z=\left[\begin{array}{l} z_{1} \\ z_{2} \end{array}\right] \text { in the previous question }

[2-а.What is the PDF, mean and covariance matrix of the vector q = U z where U is a deterministic(constant) unitary matrix. Unitary matrix is a matrix that has the property U" U = I

b. In part (a), did you find out that the two vectors q and z have the same PDF? If Yes, we call zisotropic vector. Read through the Summary Box (Summary A1) in Appendix A

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Now assume two independent complex random variables z, and z. Each one of them is Gaussian withzero mean and variance o² = 1

\text { a. What is the mean vector and covariance matrix of the vector } z=\left[\begin{array}{l} z_{1} \\ z_{2} \end{array}\right]

[22]b. What is the mean (expectation) of y = eJ® zwhere ele is a complex exponential with aconstant (deterministic) angle 0

d. In part (c), if the correct answer is (i) then the vector z is called circular symmetric vector. If the correct answer is (ii) then the vector z is not circular symmetric vector. Read through the Appendix A to learn more about this statement.

\text { i. } E(y)=e^{j \theta} E(z)

\text { ii. } E(V) \neq e^{i \theta} E(z)

In part (b), you calculated E(y). Which one of the following is correct:

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Now assume two independent real random variables x, and x2. Each one of them is Gaussian with zeromean and variance o2 = 1

\text { a. What is the mean vector and covariance matrix of the vector } x=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]

b. What is the mean vector and covariance matrix of the vector z = Ax + b where A is a 2x2deterministic (constant) matrix and b is a 2x1 deterministic (constant) vector.

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Assume x is a real Gaussian random variable with zero mean and o? variance

Write the probability density function (PDF) of the x

b. Write the PDF of the random variable y = a x + b where a and b are two deterministic real values.

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The message signal m(t) has the Fourier transform shown in Figure P-3.11(a). This signal is applied to the system shown in Figure P-3.11(b) to generate the signal y(t).

1. Plot Y (f), the Fourier transform of y(t).

2. Show that if y(t) is transmitted, the receiver can pass it through a replica of the system shown in Figure P-3.11 (b) to obtain m(t) back. This means that this system can be used as a simple scrambler to enhance communication privacy.

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The message signal m(t), whose spectrum is shown in Figure P-3.18, is passedthrough the system shown in that figure.

The bandpass filter has a bandwidth of 2W centered at fo, and the lowpass filter has a bandwidth of W. Plot the spectra of the signals x (t), yı (t), y2(t), y3 (t), and y4 (1).What are the bandwidths of these signals?

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A DSB-SC AM signal is modulated by the signal

m(t)=2 \cos 2000 \pi t+\cos 6000 \pi t .

The modulated signal is

u(t)=100 m(t) \cos 2 \pi f_{c} t

\text { where } f_{c}=1 \mathrm{MHz}

1. Determine and sketch the spectrum of the AM signal.

2. Determine the average power in the frequency components.

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The message signal m(t) = 2 cos 400t + 4 sin(500r +) modulates the carrier signal c(t) = A cos(8000 rt), using DSB amplitude modulation. Find the time-domain and frequency-domain representations of the modulated signal and plot the spectrum (Fourier transform) of the modulated signal. What is the power content of the modulated signal?

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\text { Prove that the power spectrum density of the output } y(t) \text { is } S_{y}(f)=S_{x}(f)|H(f)|^{2}

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\text { For the complex vector } z=\left[\begin{array}{l} z_{1} \\ z_{2} \end{array}\right] \text { in the previous question }

a. What is the PDF, mean and covariance matrix of the vector g = U z where U is a deterministic(constant) unitary matrix. Unitary matrix is a matrix that has the property U"U = I

b. In part (a), did you find out that the two vectors q and z have the same PDF? If Yes, we call zisotropic vector. Read through the Summary Box (Summary A1) in Appendix A

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