ECE 3084 PROBLEM SET 2, Lecture notes of Signals and Systems

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I

a Given that hm F e

in

a Ip

Io (^) ip Io NEX^ N x^ L

O L J^

X b (^) Vm (^) hm (^) ip f h^ t^ i^ x^ e^ de Ty's

ices

nice so (^) Vm (^5) Ih T^ it ice (^) x de o (^) n se

r se

case 1 if X C^ L

j L t

It

7 f^

un o

no (^) common part

nice es for

this

region g

case 5 x 2L

j L (^) t

It

7 I^

un o

ice no (^) common part

for

nicx e

this

region T X L X^ T 2 a pit

T too

Oct T t T b (^) ult O^ t^ o I t 20 c ret o always

a (^) fx Apex DX^ pdf p x Sx (^) pulse function with

width DX

Fx x^ Fx^ x^

probability density

functions

W X^ T (^) Xz

Fx xz Fw W^ I^ P x txzEW

J cumulative (^) distribution I^ Iff^ fxzlxaldx.de

function

Fxitxalw (^) fIdxzFx X^ f Fx x^ dx T integrate

over all

regions

of

space x (^) TX Ew Idx

fat's

ÉÉ w x

Fw w^ s^

f

fxzCx Fx W X^ dx

Fwlw didw^ Fwcw sdldwfjfx.CAFx^ Cw^ Xz^ dxa f fxzlxz^ didw Fx (^) W X dx Fw w^ f

Fx x^ Fx CW^ Xz^ dx

Fw w^ Fxx FX x

Fw w^ AZ DX Triangular function

centered at^ w

The pdf

for the^ random variable

w (^) x t (^) Xz is a triangular function

centered at^ w^ where^ the width^ of

the (^) triangular function (^) depends on^ DX and its^ scaled^ by AZ^ x^ Dx