Homework answers / question archive / 8) Consider the following basis set: Æ1(t) = {cos(pt/T)cos(2pfct) |t| £ T/2                0                                 |t| > T/2 Æ2(t) = {sin(pt/T)cos(2pfct)   |t| £ T/2                 0                                 |t| > T/2 Æ3(t) = {cos(pt/T)sin(2pfct)   |t| £ T/2               0                                   |t| > T/2 Æ4(t) = {sin(pt/T)sin(2pfct)     |t| £ T/2               0                                    |t| > T/2 Where: fc = k/2T for some integer k

8) Consider the following basis set: Æ1(t) = {cos(pt/T)cos(2pfct) |t| £ T/2                0                                 |t| > T/2 Æ2(t) = {sin(pt/T)cos(2pfct)   |t| £ T/2                 0                                 |t| > T/2 Æ3(t) = {cos(pt/T)sin(2pfct)   |t| £ T/2               0                                   |t| > T/2 Æ4(t) = {sin(pt/T)sin(2pfct)     |t| £ T/2               0                                    |t| > T/2 Where: fc = k/2T for some integer k

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8) Consider the following basis set:

Æ₁(t) = {cos(pt/T)cos(2pf_ct) |t| £ T/2

               0                                 |t| > T/2

Æ₂(t) = {sin(pt/T)cos(2pf_ct)   |t| £ T/2

                0                                 |t| > T/2

Æ₃(t) = {cos(pt/T)sin(2pf_ct)   |t| £ T/2

              0                                   |t| > T/2

Æ₄(t) = {sin(pt/T)sin(2pf_ct)     |t| £ T/2

              0                                    |t| > T/2

Where: f_c = k/2T for some integer k.

a. Show that the basis functions are all orthogonal.

b. Normalize the set so each basis function has unit energy over the T second interval.

c. Sketch a block diagram of the transmitter for a communication scheme which uses basis functions multipled by +1 or -1. Assume rectangular pulse shaping (i.e. don’t worry about pulse shapping).

d. For the communication scheme in (c), draw the single space if you can, or describe it in words: how many dimensions are there, where are the constellation points located, etc.

e. For the communication scheme in (c) what is the minimum distance between two singles in the constellation? How many nearest neighbors does a single have?

f. For the communication scheme in (c), how many bits can we use each transmitted symbol to represent?