Solved: Question 4 (Counters): Design a sequential circuit with positive-edged

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question 4 counters design a sequential circuit with positive edged t

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Communications

Question 4 (Counters): Design a sequential circuit with positive-edged T flip-flops that counts up with clock pulses over Fibonacci numbers (1, 2, 3, 5, 8, 13) that can be represented using 4-bit binary values (A3A2A1A)2. The circuit (re)starts counting from 1 after it reaches 13. This circuit outputs Z = 1 when the counter value is 1, 5 or 8, for other cases it outputs Z = 0.

(4.a) Draw the state diagram of this circuit.

(4.b) Prepare the state table for this counter. Indicate every possible present state and map these states to a corresponding next state and output value.

.d) Draw the logic diagram.

(4.c) Derive the simplified flip-flop input equations and output equations.

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Question 44968

Communications

\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

Communications

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:

Communications

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.

Communications

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.

Communications

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.

Communications

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?

Communications

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.

Communications

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?

Communications

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

Communications

\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