Show works Thanks A signal can be broken into an even part and an odd part. Find

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A signal can be broken into an even part and an odd part. Find the even and odd components of x(t) = (1 + t3)cos310t. Let y(t) = z(bt - a) where a and b are some constants. Describe how a and b relate signal z(t) to y(t). Your answer should be in sentance form and include the cases for b being greater then or less then 1 and a being either positive or negative. Once you're sure you have this right, memorize it. If you start with a signal f(t), then a three-times slower version is x(t) = f(t/3). A version of the signal delayed in time by 5 seconds is y(t) = f(t - 5). Let z(t) be the signal f(t) delayed by 10 seconds then sped up by a factor of three. Write z(t) in terms of f(t). You might find sketching a helpful way to double-check. Let y(t) = cos(10t). Let w(t) be y(t) slowed by five, then delayed by 2pi, then sped by four. At time t = 4pi, what is w(t)?

Explanation / Answer

20. Even = [x(t) + x(-t)] / 2 = 1/2 .cos^3(10t) [ (1+t^3) + (1-t^3) ] = cos^3(10t)

Odd = x(t) - even = t^3.cos(t)

21.y(t) = z(bt - a)

replace t by 1/b(t+a)

y( 1/b(t+a) ) = z(t)

22.x(t) = f(t) delayed by 10 s = f(t-10)

z(t) = x(t) sped up by 3 = x(3t)

So z(t) = f(3t - 10)

23.x(t) = y(t/5)

z(t) = x(t - 2pi) = y(t/5 - 2pi/5)

w(t) = z(4t) = y(4/5.t - 2/5. pi)

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