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Communication Theory Homework 4

1. I have a baseband analog signal with a 20 kHz bandwidth. How fast must I sample to ensure
a 4 kHz guard band?
2. Suppose I have bandlimited noise with PSD 8 for frequency less than 200 kHz in absolute
value. Sampling this noise at Nyquist and applying a 16-level quantizer, what are the rate
and distortion (mean squared error)? What is the SQNR?
The next few problems will consider the following setup: Suppose I am doing a digital logic
design project, and I represent 1 as a 2.5 V pulse of duration A and 0 as a -2.5 V pulse of
duration A. My chips perform a least squares decision when they receive a signal. The noise
over a wire is largely thermal noise caused by statistical variations in charge carriers, modeled
as AWGN. This noise has variance 4kBT R where kB ≈ 1.38 × 10−23 J/K is the Boltzmann
constant, T is the absolute temperature (in Kelvin) and R is the total resistance of the wire.

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Communication Theory Homework 4

1. I have a baseband analog signal with a 20 kHz bandwidth. How fast must I sample to ensure
a 4 kHz guard band?
2. Suppose I have bandlimited noise with PSD 8 for frequency less than 200 kHz in absolute
value. Sampling this noise at Nyquist and applying a 16-level quantizer, what are the rate
and distortion (mean squared error)? What is the SQNR?
The next few problems will consider the following setup: Suppose I am doing a digital logic
design project, and I represent 1 as a 2.5 V pulse of duration A and 0 as a -2.5 V pulse of
duration A. My chips perform a least squares decision when they receive a signal. The noise
over a wire is largely thermal noise caused by statistical variations in charge carriers, modeled
as AWGN. This noise has variance 4kBT R where kB ≈ 1.38 × 10−23 J/K is the Boltzmann
constant, T is the absolute temperature (in Kelvin) and R is the total resistance of the wire.
A commonly used wire type (available in the lab at school) is 22 AWG solid copper wire, for
which 1 m of wire has 52.7 mΩ resistance. Recall that resistance is proportional to length.
3. Describe the operation of a least squares decision system for this problem.
4. Is the least squares decision rule the same as a matched filter decision rule in this case? Is it
the same as maximum likelihood? Is it the same as maximum a posteriori? If there is not
enough information to answer, give the conditions that would guarantee the equality. In each
case, don’t just say “yes” or “no”, provide a one-sentence explanation.
5. What is the name of this modulation scheme? What is the basis? Draw the geometrical
representation.
6. For this scheme, suppose I transmit a 1 75% of the time – what are the MAP decision regions?
7. Assume now that the symbols are equiprobable. In terms of A, T and wire length L, write a
formula for the SNR and probability of error.
8. Assume we are operating the system at room temperature – how long does the wire have
to be to yield an error probability of 1/10? Write this number in lightyears. Compare this
to the size of the Milky Way. This should give you an appreciation for how robust wired
communication is to thermal noise.
“Professor, my project doesn’t work, and I think it might be because of thermal noise!”
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