Question 1. Consider the signal

x(0) = {-A 0 < t ≤  T/2

          { A/2 T/2 < t ≤ T

(a) Determine the impulse response h(t) of a filter matched to the signal x(t). Sketch x(t) and h(t) as functions of time.

(b) Suppose the signal x(t) is applied to the matched filter h(t). Graphically determine and plot the filter output y(t) as a function of time.

(c) If two signals, s₁(t) = x(t) and s₂(1)= -x(t), are selected as transmitted signals for a binary communication system operating over an additive white Gaussian noise (AWGN) channel, design and draw an opitman receiver to detect the incoming signals. The power spectral density (PSD) ftmction of AWGN is assumed to be N₀/2.

(d) Determine the average energy .645 of s, (t) and 32(). Derive the bit-error rate (BER) of the binary communication system in terms of &function. El, and No.

(d) Suppose the input signal is changed to

x(t)   A     0 < t ≤ T/2

        -A/2  T/2 < t ≤ T

Discuss the impact of the change on the BER of the binary communication system.

Question 2. Duobinary signalling can be extended to polybinary signalling with j levels, where j> 3. The signalling levels are labelled consecutively from 0 to (j - 1). The encoding process consists of two steps. First, the intermediate binary sequence {y_k} is obtained from the original
binary sequence {x_k} as

Y_k = x_k ⊕ Y_k-1 ⊕ Y_k-2 ⊕.... ⊕Y_(k-(j-2)

where ⊕ denotes modulo-2 addition. Hence, the value of y_k is either 0 or 1. Next, the polybinary pulse train {x_k } is computed by the following equation:

Z_k = Y_k + Y_k-1+ ........ + Y_k-(j-2)

Note that the terms on the RHS of the above equation are added algebraically. Without noise, it is easy to see that {x_k} can be recovered by observing the even or odd values of {z_k}.

(a) Use your class number from the attendance list to generate the data sequence as follows:

{x_k} = {a_k}⊕{c_k}

where {a_k} is the sequence {0001 0010 0011 0100 0101 0110} and {c_k} is the extended BCD of your class number. For example, if your class number is 6 or 59, then the BCD of it is 0110 or 0101 1001. The extended BCD sequence is thus 0110 0110 0110 0110 0110 0110 or 0101 1001 0101 1001 0101 1001, the same length as {a_k}.

Suppose that the binary data sequence is {x_k}, where the bits on the left are received earlier). How many reference or startup bits are needed for j= 6? Determine the binary sequence {y_k}. Although the selection of reference bits are arbitrary (i.e.. it can be 0 or 1 for each bit), let us choose each bit to be zero for convenience.

(c) Derive the polybinary sequence {z_k} and decode the binary sequence {x^_k}

Question 3. The probability density function (PDF) of a random variable Θ is described by

                1/2Π   -Π ≤ θ < Π

P_Θ(θ) =

                 0    Otherwise.

(a) Determine the mean m_Θ and the variance σ²; of the random variable Θ. Find the probability P_r(0 ≤ Θ ≤ Π/2).

(b) Suppose the random variable Θ is approximated as Gaussian distributed with mean m_e, and variance σ²_Θ. Express the approximate PDF and find the probability Pr(0 ≤ Θ ≤ Π/2). You may use the table of Q-finction to compute the probability. Do you think that the approximate PDF is a good choice for the true PDF? Justify your answer.

(c) A sinusoidal signal with random phase is defined as X(t)= Acos(2Πf_ct + Θ), where the amplitude A and the carrier frequency f_c are constants. The random phase Θ is uniformly distributed over the range [-Π, Π). Obtain the autocorrelation function and the power spectral density of X(t).