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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.
Thursday, September 30, 2010
Wednesday, September 29, 2010
Monday, September 27, 2010
Class: 9/27/2010
- Discussed homework assignment 4.
- Discussed C-based implementation of Viterbi algorithm.
- No class on Friday, 1 October 2010.
Friday, September 24, 2010
Wednesday, September 22, 2010
Class: 9/22/2010
- Discussed several homework problems
- Homework due date extended to Friday
- Continued development the Viterbi algorithm
- (audio went out part way through class, sorry)
Monday, September 20, 2010
Class: 9/20/2010
- Set up for maximum likelihood sequence estimation via the Viterbi algorithm
- The channel acts like a state machine, state transition diagrams, trellis diagrams
- Input sequences of symbols are in one-to-one correspondence with paths through the trellis
- Decomposition and simplification of the joint likelihood function
Friday, September 17, 2010
Class: 9/17/2010
- Intersymbol interference (ISI)
- Nyquist's criterion review
- Frequency selective channel impairments and examples
- Introduction to equalization
Wednesday, September 15, 2010
Class: 9/15/2010
- CPFSK, modulation index, MSK, MSK/OQPSK equivalence
- Non-coherent detection of FSK
- QPSK, non-coherent detection of differentially encoded QPSK, and DQPSK
Monday, September 13, 2010
Class: 9/13/2010
- Generation of FSK and CPFSK
- Phase trajectories
- Pulse shapes
- Spectral properties of CPFSK
Friday, September 10, 2010
Class: 9/10/2010
Derived the probability of bit error (in terms of Eb/N0) for (orthogonal, coherent) M-FSK. Do the non-coherent case for homework.
Class: 9/8/2010
Devoted the whole hour to deriving and discussing the characteristics of complex low-pass noise given that the real bandpass noise characterization. For WSS bandpass noise, the cross-correlation of the real and imaginary parts is odd while the auto-correlations are the same. For white Gaussian bandpass noise, the real and imaginary parts of the low-pass noise are uncorrelated (and independent). There is twice as much total power in the low-pass noise as in the bandpass noise. If the bandpass noise is "white" and Gaussian with PSD N0/2 for |f-fc|<W, then the low pass noise is "white" and Gaussian with PSD 2N0 for |f|<W.
Friday, September 3, 2010
9/3/2010
The following topics were discussed.
- Frequency shift keying.
- Minimum frequency separation for orthogonality: 1/T (low pass), 1/(2T) (bandpass).
- Bandwidth efficiency of FSK compared to PSK, QAM.
- Receiver structure for optimal (coherent and noncoherent) detection of FSK transmitted over AWGN channel.
Wednesday, September 1, 2010
Class: 9/1/2010
The discussion in class today focused on the following topics.
- Low pass equivalent signals and systems.
- Two ways to convert band pass signals to low pass: sin/cosine modulation + low pass filtering, Hilbert transform + complex down conversion.
- Low pass to band pass conversion.
- Band pass and low pass signal spaces.
- QAM has two dimensional band pass signal space, but one dimensional low pass signal space.
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