Hi, I need the answer for the last problem only (7.24).Thanks o) Click on Tools

Question

Hi, I need the answer for the last problem only (7.24).Thanks

o) Click on Tools to convert files to PDF 7.14 Three two-sided sequences are as given below: rinl)- -3, 4, 0, -2, 5, 4 .-1sns4, fvln])-1 3,-2, 0, 6, -7,-3sn s2, (wIn])- (2, 6,1, -3, -4, 8),-4sns1 The sample values of the above sequences outside the specified range of time index n are assumed to be zeros. Determine the following sequences and show explicitly the ranges of the time index n for each: 7.22 Express the sequences s[n], v[n], and x[n], of Problem 7.14 as a linear weighted combination of delayed and advanced unit sample sequences. 7.23 Express the sequences s[n], v[n], and x[n], of Problem 7.14 as a linear weighted combination of delayed and advanced unit step sequences. For Problem 7.24 only find even & odd part of [n] 7.24 Determine the even and odd parts of the sequences r[n], v[n], and w[n], of Problem

Explanation / Answer

For a discrete time signal like r[n], the even and odd parts can be found using following method.

All signals can be expressed as a sum of an even signal and an odd signal.

Let r[n] = reven[n] + rodd[n]............(1)

By definition of even function, reven[-n] = reven[n] ...........(2)

and by definition of odd function, rodd[-n] = -rodd[n] ...........(3)

Replacing n with -n in equation (1),

r[-n] = reven[-n] + rodd[-n]

=> r[-n] = reven[n] - rodd[n] ......(4)(from equations 2 and 3)

Adding eq(1) and (4)

r[n] + r[-n] = 2reven[n]

=> reven[n] = (r[n] + r[-n] )/2

Subtracting eq(4) from eq(1),

r[n] + r[-n] = 2rodd[n]

=> rodd[n] = (r[n] - r[-n] )/2

now, given in question, r[n] = {-3,4,0,-2,5,4}, with -1<=n<=4; i.e. n starting from -1

=>r[-1] = -3, r[0] = 4, r[1] = -0, r[2]= -2, r[3] = 5, r[4] = 4

now, r[-n] means r values from -4<=n<=1. IT is the flipped version of r[n] about y axis.

r[-4]=4, r[-3]=5, r[-2] = -2, r[-1] = 0, r[0]= 4, r[1] =-3

reven[n] spans from n=-4 to n=4 and has values

reven[-4]=(0+4)/2 = 2 [since r[-4] = 0]
reven[-3]=(0+5)/2 = 2.5
reven[-2]=(0-2)/2 = -1
reven[-1] = (-3+0)/2 = -1.5
reven[0] = (4+4)/2 = 4,
reven[1] = (0-3)/2=-1.5
reven[2] = (-2+0)/2 = -1
reven[3] =(5+0)/2 =2.5
reven[4] =(4+0)/2 = 2

similarly, rodd[n] spans from n=-4 to n=4 and has values

rodd[-4]=(0-4)/2 = -2 [since r[-4] = 0]
rodd[-3]=(0-5)/2 = -2.5
rodd[-2]=(0+2)/2 = 1
rodd[-1] = (-3-0)/2 = -1.5
rodd[0] = (4-4)/2 = 0,
rodd[1] = (0+3)/2 = 1.5
rodd[2] = (-2+0)/2 = -1
rodd[3] =(5+0)/2 =2.5
rodd[4] =(4+0)/2 = 2

Thus,

       reven[n] = {2, 2.5, -1, -1.5, 4, -1.5, -1, 2.5, 2}
and rodd[n] = {-2, -2.5, 1, -1.5, 0, 1.5, -1, 2.5, 2}

where -4<=n<=4

[Add reven[n] and rodd[n] to verify r[n] = reven[n] + rodd[n] ]

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v[n] = {1,3,-2,0,6,-7}, -3<=n<2

v[-n] = {-7,6,0,-2,3,1}, -2<=n<3

veven[n] = {0.5, -2, 2, 0, 2, -2, 0.5}, -3<=n<3
vodd[n] = {0.5, 5, -4, 0, 4, -5, -0.5}, -3<=n<3

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w[n] = {2,6,1,-3,-4,8}, -4<=n<1

w[-n] = {8,-4,-3,1,6,2}, -1<=n<4

weven[n] = {1, 3, 0.5, 2.5, -4, 2.5, 0.5, 3, 1}, -4<=n<4
wodd[n] = {1, 3, 0.5, -5.5, 0, 5.5, -0.5, -3, -1}, -4<=n<4

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