Questions on HW 4

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Question Problem #1:

You made the comment in lecture that the symbols are MSK encoded.  By this do you mean the using the constellation to encode for example pi/4 = 11, 3pi/4 = 01 5pi/4 = 00 and 7pi/4 = 10?  Then after they are encoding this way we up sample as done in com 1 and so on and so forth.  Am I seeing this correctly?

Response:

It was shown in Problem #3 in HW 3 that the MSK waveform has the form

s(t) = cos(2πf_ct)A_{2k−1} g(t − [2k − 1]T)(−1)^k − sin(2πf_ct)A_{2k} g(t − 2kT)(−1)^k

for nT ≤ t < (n + 1)T, where the encoded symbols A_n are given by

A_n = A_{n-1} I_n      (this is like differential encoding)

where I_n is the information symbol.  So, the information symbols I_n are encoded to A_n and then the even encoded symbols A_{2k} and odd encoded symbols A_{2k-1} are transmitted on the quadrature and in-phase carriers, respectively.  Also, note the (-1)^k.

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Question Problem #4:

I am not sure how Proakis (in section 3.4 eq 3.4-15) goes from

G_k(f) = E[S_l(f;I_k)S^*_l(f;I_0)] to
 = E[I_kI^*_0|G(f)|^2]

Maybe I am forgetting something, but I do not see how he pulls the I_k/I_0 out of the S_l functions.  Can you elaborate on this for me?

Response:

He uses equation 3.4-14 which indicates that

s_l(t,I_n) = I_n g(t).

Then the Fourier transform is

S_l(t,I_n) = I_n G(f).  I don't have a problem with that part.  However, 3.4-14 does not seem to follow from 3.4-13  unless the pulses g(t) are time limited to one symbol period.

Here is a better way to solve this problem.

1. Define the symbols to be I(n).
2. Define the pulse to be g(t).
3. The transmitted pulse train is s(t)=sum_n I(n) g(t-nT).  This is an infinite sum.
4. The autocorrelation function of s(t) is R(t;t-v) = E{s(t) s^*(t-v)}.  Show that this is a periodic function with period equal to the symbol period T.
5. Define the averaged autocorrelation function S(v) = (1/T) int_0^T R(t;t-v) dt, where int_A^B f(t) dt is the integral of f(t) on the interval (A,B).
6. Let P(f) be the continuous-time (CT) Fourier transform of S(v).  Interchange integrals and sums, do a few changes of variables and you will find that

P(f) = DTFT{autocorrelation function of the symbols evaluated at fT} |G(f)|^2 where f is continuous frequency in Hz.

Hope this helps.