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# Detection, Estimation modulation theory part 1 - Vantress H.

Vantress H. Detection, Estimation modulation theory part 1 - Wiley & sons , 2001. - 710 p.
ISBN 0-471-09517-6 Previous << 1 .. 137 138 139 140 141 142 < 143 > 144 145 146 147 148 149 .. 254 >> Next 1. Find an expression for —
2. Assume that
s(t) = V2kPe~kt, t > 0,
0, / < 0
and
ń t \ - 2ka°2
5-ń(") - ř2 + k2
Evaluate A as a function of Ë Ä Aac2lkNo.
Nonwhite Additive Gaussian Noise 393
Problem 4,3.17. Sensitivity to Noise Spectrum. Assume the same nominal model as in Problem 4,3.16.
1. Now let and where Find
Na = No 2 2
î / ÷ 2 kaOa2
S'C(W) = ^Tk?
ka = ę (1 + ó)
Oa2 = "ń2 (1 + Z)
„ ÄÄ, ,„d M5
y=o a
dd/dy
Ë Ä
.= 0 — Z'
2. Evaluate Äó and Ä2 for the signal shape in Problem 4.3.16.
Problem 4.3.18. Sensitivity to Delay and Gain. The received waveforms under the two hypotheses are
r(t) = VEs(t) + hsit — r) + w(t\ r{t) — bis{t — r) + w(t),
— oo < t < oo :HU
— oo < t < oo: #o,
where bi is N(0, oj) and w(t) is white with spectral height N0/2. The signal is
s(t) =(^)'4’ 0 < t < T,
= 0, elsewhere.
1. Design the optimum receiver, assuming that r is known.
2. Evaluate d2 as a function of r and oj.
3. Now assume
ňŕ = r (1 + äă).
Find an expression for d2 of the nominal receiver as a function of x. Discuss the implications of your results.
4. Now we want to study the effect of changing oIt Let
*,a2 = a/2 (1 + Ó) and find an expression for d2 as a function of y.
Linear Channels
Problem 4.3.19. Optimum Signals. Consider the system shown in Fig. 4.49a. Assume that the channel is time-invariant with impulse response h(r) [or transfer function #(/)]. Let
#(/) = 1, I/I < W,
= 0, otherwise.
The output observation interval is infinite. The signal input is s(t), 0 < t < T and is normalized to have unity energy. The additive white Gaussian noise has spectral height N0j2.
1. Find the optimum receiver.
2. Choose s(t), 0 < t < Ă, to maximize d2.
394 4.8 Problems
Problem 4.3.20. Repeat Problem 4.3.19 for the h{t, r) given below:
hit, r) = h{t - r), 0<r<I,f <r<f,-oo<,<oo,
= 0, elsewhere.
Problem 4.3.21. The system of interest is shown in Fig. P4.6. Design an optimum binary signaling system subject to the constraints:
1. j* s2(t)dt = Et.
2. j(f) = 0, t < 0,
t < T.
3. h{r) = e~k\ r > 0,
= 0, r < 0.
Either
*«(f)
Source 1,0 Transmitter ±s(t) Channel
h(r)
r(t) 0<tST
Fig. P4.6
Section P.4.4 Signals with Unwanted Parameters
Mathematical Preliminaries
Formulas. Some of the problems in this section require the manipulation of Bessel functions and Q functions. A few convenient formulas are listed below. Other relations can be found in  and the appendices of  and 
I. Modified Bessel Functions
1 C2n
/„(z) A — exp i±jnO) exp (z cos â) dd, (F.1.1)
˛ŇŇ Jo
Ř = /-»(*), (F.1.2)
m - f^TT)’ V ^ -1, -2,..z « 1, (F.1.3)
*»'• <F-M> ~k (2” Iv(z)) = z~+ t(z), (F.1.5)
~ J; (*• Ř) = (F.1.6)
II. Marcum9s Q-function 
Q(V2a, Ó2b) = J exp {-a 4- x) 10{2ó/ax) dx, (F.2.1)
Qia, a) = i[l + /0(a2) exp (-a2)], (F.2.2)
Signals with Unwanted Parameters 395
1 + Q(a, b) - Q(b, a) Ü2 - a
?2 _ a2 Ëîî , 2äŔó \
¦ J,., “»' - *> '¦(^Tp) Ë’ ‘>«>0. (F.2.3)
+ V + Oj1 e(*/0l2 + V’ ^v+~v)’ <F'2‘4)
Q(a, 6) ~ erfc* (b - a), b» \,b» b - a. (F.2.5)
III. Rician variables 
Consider the two statistically independent Rician variables, xx and x2 with probability densities,
pXkm = exp (--~2ŕ/ę2) /o(?#)' 0 < ak < CO, (F.3.1)
k k k 0 < Xk < oo,
ę = 1, 2.
The probability of interest is
P* = Pr [*2 > *l].
Define the constants
a = g22 č _ fli2 _ Oi Previous << 1 .. 137 138 139 140 141 142 < 143 > 144 145 146 147 148 149 .. 254 >> Next 