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.

1. 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 Hi, A=Ai, i=0,1.
n(t) is white Gaussian noise (WGN) with power spectral density (PSD) N0/2.
Compute the LRT. Then, compute PD and PF for x(t)=impulse and large T.

2. Under hypothesis Hi we observe r(t)=si(t)+w(t) for 0 < t < T and i=0,1,
w(t) is white Gaussian noise (WGN) with power spectral density (PSD) N0/2.
The signals si(t) for various modulation schemes are as follows:
s0(t)0A sin(w1t)A sin(w2t) A=(2E/T)½
s1(t)A sin(w1t)A sin(w2t) -A sin(w2t)wi=2&pi i/T
1. Draw signal spaces for each scheme using two basis functions fi(t), i=1,2.
This means depict s0(t)=c1f1(t)+c2f2(t) for some constants ci; similary for s1(t).
2. Compute d2 and Pr[error] for each scheme, assuming Pr[H0]=Pr[H1]=½.
3. Compare the schemes in terms of accuracy and energy required.

3. 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,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) N0/2.
1. Choose a in the mismatched filter h(t) to maximize the Fisher discriminant d2.
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).

4. A set of M signals si(t) is represented in signal space (see problem #2) by si.
Linear translation of the vectors si does not affect Pr[error] if a MAP receiver is used.
We can save energy by translating these vectors, which now represent different signals.
1. What translation vector m minimizes average energy Pr[H1]|s1-m|2 +...+Pr[HM]|sM-m|2?
2. Interpret your answer physically in terms of center of mass and moment of inertia.
3. Now let si represent orthogonal equal-energy equally-likely signals.
Sketch the signal spaces for M=2,3,4. These are called "simplex signals."
4. How much energy is saved by using simplex signals, as a function of M?