EECS 564________________________PROBLEM SET #6________________________Winter 1999

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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.

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

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2. 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) N₀/2.
   The signals s_i(t) for various modulation schemes are as follows:
   

   signal	ASK	FSK	PSK	definition
   s0(t)	0	A 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 f_i(t), i=1,2.
      This means depict s₀(t)=c₁f₁(t)+c₂f₂(t) for some constants c_i; 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.

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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^-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) 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).

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4. 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.
   1. What translation vector m minimizes average energy Pr[H₁]|s₁-m|² +...+Pr[H_M]|s_M-m|²?
   2. Interpret your answer physically in terms of center of mass and moment of inertia.
   3. 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."
   4. How much energy is saved by using simplex signals, as a function of M?

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