EECS 564________________________PROBLEM SET #6________________________Winter 1999
ASSIGNED: February 19, 1999 READ: Catch up on Sections 4.3-4.6 of Srniath (used this week).
DUE DATE: February 26, 1999 THIS WEEK: Detection of known signals in white noise processes.
- We observe r(t)=h(t)*x(t)+n(t),0 < t < T, where h(t)=Ae^{-At}, t > 0
and x(t) is a known causal signal. Under hypothesis H_{i}, A=A_{i}, i=0,1.
n(t) is white Gaussian noise (WGN) with power spectral density (PSD) N_{0}/2.
Compute the LRT. Then, compute P_{D} and P_{F} for x(t)=impulse and large T.
- Under hypothesis H_{i} we observe r(t)=s_{i}(t)+w(t) for 0 < t < T and i=0,1,
w(t) is white Gaussian noise (WGN) with power spectral density (PSD) N_{0}/2.
The signals s_{i}(t) for various modulation schemes are as follows:
signal | ASK | FSK | PSK | definition |
s_{0}(t) | 0 | A sin(w_{1}t) | A sin(w_{2}t) |
A=(2E/T)^{½} |
s_{1}(t) | A sin(w_{1}t) | A sin(w_{2}t) |
-A sin(w_{2}t) | w_{i}=2&pi i/T |
- Draw signal spaces for each scheme using two basis functions f_{i}(t), i=1,2.
This means depict s_{0}(t)=c_{1}f_{1}(t)+c_{2}f_{2}(t) for some
constants c_{i}; similary for s_{1}(t).
- Compute d^{2} and Pr[error] for each scheme, assuming Pr[H_{0}]=Pr[H_{1}]=½.
- Compare the schemes in terms of accuracy and energy required.
- Suboptimum receivers, or "You can't always get what you want..."
Often, in the real world, we can't afford to use the optimal matched filter (which is h(t)=s(T-t)).
Instead, we may use h(t)=e^{-at},t > 0 as a simpler, 1-pole, suboptimal matched filter.
Let the actual signal be s(t)=T^{-½},0 < t < T. We observe r(t)=s(t)+w(t) or r(t)=w(t).
w(t) is white Gaussian noise (WGN) with power spectral density (PSD) N_{0}/2.
- Choose a in the mismatched filter h(t) to maximize the Fisher discriminant d^{2}.
- Show that the performance loss can be compensated by a small increase in signal energy.
How many decibels of energy? "You get what you need..." (The Rolling Stones).
- A set of M signals s_{i}(t) is represented in signal space (see problem #2) by s_{i}.
Linear translation of the vectors s_{i} does not affect Pr[error] if a MAP receiver is used.
We can save energy by translating these vectors, which now represent different signals.
- What translation vector m minimizes average energy Pr[H_{1}]|s_{1}-m|^{2}
+...+Pr[H_{M}]|s_{M}-m|^{2}?
- Interpret your answer physically in terms of center of mass and moment of inertia.
- Now let s_{i} represent orthogonal equal-energy equally-likely signals.
Sketch the signal spaces for M=2,3,4. These are called "simplex signals."
- How much energy is saved by using simplex signals, as a function of M?