Reference no: EM133026311

ECE 5377 Embedded Wireless Design - Southern Methodist University

(1) Consider an order L-tap frequency-selective channel. After matched-filtering and assuming perfect synchro- nization, the received signal is

          L
y[n] = ∑h[l]s[n - l] + v[n], (1)
         l=0

where s[n] is the transmitted symbols, h[l] is the channel coefficient, and v[n] is additive white Gaussian noise. Suppose that s[n] = t[n] for n = 0, 1, . . . , N_t is the known training data.

In class and the book we considered the use of a finite impulse response equalizer. Suppose instead that we instead apply an all-pole filter to equalize the channel.

The output of the equalizer is given by

r[n] = f[1]r[n - 1] + f [2]r[n - 2] + .... + f [K]r[n - K] + y[n] (2)

where the initial conditions are zero, e.g. r[-1] = r[-2] = ... = r[-K] = 0.

We would like the output of the equalizer to approximate r[n] ≈ s[n - n_d]. Formulate and solve a least squares problem to determine the unknown coefficients {f [1], f [2], . . . , f [K]}.

a) Write the squared error function (using sequences not matrices). This may be tricky but remember to use your knowledge of the training data.

b) Write the least-squares problem in matrix form.

c) Determine an expression for the equalizer coefficients and the resulting squared error.

(2) Consider a zero padded QAM system designed for frequency domain equalization. Instead of a cyclic prefix, the zero padded QAM system sends Lc zero values in place of the cyclic prefix. Suppose we want to transmit

N symbols {s[n]}^N-1_n=0. Let

ω[n] = 0 for n = 0, 1, 2, . . . , L_c - 1

and

ω[n] = s[n - L_c] for n = Lc, L_c + 1, . . . , N + L_c - 1.

After convolution with an L + 1 tap channel

          L
y[n] = ∑h[l]ω[n - l]
         l=0

For this problem we neglect received noise. Suppose that L ≤ L_c. Assume that the channel has been estimated perfectly.

Now, form a new signal

y˜[n] = y[n + L_c] + y[n + N + L_c] for n = 0, 1, . . . , L_c - 1

and

y˜[n] = y[n + L_c] for n = L_c, L_c + 1, . . . , N - 1.

a) First we work with the signal y[n]. Show how the zeros can be exploited to successively decode s[n] from y[n]. Hint: Start with n = 0 and show that s[0] can be derived from y[L_c]. Let sˆ[0] denote the detected symbol. Then show how, by subtracting off sˆ[0], you can detect sˆ[1]. Then assume it is true for a given n and show it works for n + 1.

b) Prove that

           L
y˜[n] = ∑h[l]s[((n - l))_N ]
           l=0

c) Mathematically describe a simple way to equalize y˜[n] in the frequency domain.

d) Comment briefly on the differences between the solutions in (a) and (b). Which would work better? Which is more sensitive to errors? Which has higher complexity?

e) Cyclic prefix and zero-padding are two types of the guard interval. In zero-padded QAM systems, during the guard interval, no signal is transmitted. Note that in cyclic prefix QAM systems, however, data samples are still transmitted. We consider a zero padded QAM system over that of a cyclic prefix QAM system that have the same values of N and Lc. Assuming that the transmission power per sample is fixed, determine the ratio of the total transmission powers the two systems as a function of N and Lc. Which system uses a lower total transmission power?

f) Draw a block diagram for a receiver that uses zero padded QAM and frequency domain equalization.

Attachment:- Embedded Wireless Design.rar