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

ISBN 0-471-09517-6

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df parameter in FM system (frequency deviation)

i0(t) optimum MMSE estimate

ds(t, a(t)) derivative of s(t, a(t)) with respect to a{t)

dˆ(t) error in desired point estimate

d+(t) output of arbitrary nonlinear operation

8 phase of specular component (Rician channel)

Ä interval in PFM detector

Äd change in performance index

Adx desired change in d

AiV change in white noise level

Äï constraint on covariance function error

Am mean difference vector (i.e., vector denoting the dif-

ference between two mean vectors)

AQ matrix denoting difference between two inverse co-

variance matrices

E energy (no subscript when there is only one energy in

the problem)

Ea expectation over the random variable a only

Ee(N) energy in error waveform (as a function of the number

of terms in approximating series)

?/ energy in interfering signal

E{ energy on ith hypothesis

Er expected value of received energy

Et transmitted energy

Ey energy in y(t)

Glossary 675

El9 Eq energy of signals on Íã and H0 respectively

Eˆ energy in error signal (sensitivity context)

eN(t) error waveform

ˆj interval error

ˆT total error

erf (•) error function (conventional)

erf* (•) error function (as defined in text)

erfc (•) complement of error function (conventional)

erfc* (•) complement of error function (as defined in text)

ã) (eta) threshold in likelihood ratio test

?(•) expectation operation (also denoted by (•) infrequently)

F function to minimize or maximize that includes

Lagrange multiplier f(t) envelope of transmitted signal

f(t) function used in various contexts

fit: r(u),

Ò³ < è < Tf) nonlinear operation on r(u) (includes linear operation

as special case) fc oscillator frequency (toc

/ä(0 normalized difference signal

F matrix in differential equation

F(t) time-varying matrix in differential equation

Fd(t) matrix in equation describing desired signal

&+èø) factor of Sr(oj) that has all of the poles and zeros in

LHP (and \ of the zeros on jw-axis). Its transform is zero for negative time. g(t) function in colored noise correlator

g(t, A), g(t, A) function in problem of estimating A (or A) in colored

noise

g(Xi) a function of an eigenvalue

gh(t) homogeneous solution

g{(r) filter in loop

gio(r), Gl0(jw) impulse response and transfer function optimum loop

filter

gpjj) unrealizable post-loop filter

gpuo(T), Gpuo{jw) optimum unrealizable post-loop filter

g6(t) impulse solution

#ä(0 difference function in colored noise correlator

g\ a weighted sum of g(Af)

goo(0> Gao(J<*>) infinite interval solution

676 Glossary

G matrix in differential equation

G(0 time-varying matrix in differential equation

Gd linear transformation describing desired vector d

Gd(0 matrix in differential equation for desired signal

g(0 function for vector correlator

ga(A) nonlinear transformation describing desired vector d

Ã(õ) Gamma function

ó parameter (y = kV\ + A)

ó threshold for arbitrary test (frequently various constants

absorbed in y)

óa factor in nonlinear modulation problem which controls

the error variance

H0, Hl9..., Í{ hypotheses in decision problem h(t9 u) impulse response of time-varying filter (output at t due

to impulse input at u)

^ch(*> tt) channel impulse response

hL(t) low pass function (envelope of bandpass filter)

h0(t9 u) optimum linear filter

W0(r)9 H'0(jo>) optimum processor on whitened signal: impulse

response and transfer function, respectively h0Jj)9 H0U(jco) optimum unrealizable filter (impulse response and

transfer function) hw(t9 u) whitening filter

éˆ(´, è) arbitrary linear filter

h*(t9 u) linear filter in uniqueness discussion

H linear matrix transformation

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