Signals & Systems with Linear Algebra

[ashah99]

Homework Statement

Please see problem statement below

Relevant Equations

Inner products, norms, orthogonality

Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks!

My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the span of {s1(t), s2(t)}.
I normalize s1(t) to get u1(t) = sqrt(3)*t.
To get u2(t) I apply Gram-Schmidt by finding the projection of s2(t) onto u1(t), subtracting that result from s2(t) and then normalizing it. I got u2t) = 4*sqrt(5)*[t^2 - 3t/4]
So the projection of g(t) is g_hat(t) = r1*u1(t) + r2*u2(t), where r1 = <g(t), u1(t)> and r2 = <g(t), u2(t)>, the inner products. I get r1 = sqrt(3)/2 and r2 = -sqrt(5)/6
Putting it all together and simplifying, I get g_hat(t) = 4t - (10/3)*t^2, thus a = 4 and b = -10/3

Apologies for the formatting. Is this correct?

 

[Mark44]

Homework Statement:: Please see problem statement below
Relevant Equations:: Inner products, norms, orthogonality

View attachment 296739
Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks!

My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the span of {s1(t), s2(t)}.
I normalize s1(t) to get u1(t) = sqrt(3)*t.
To get u2(t) I apply Gram-Schmidt by finding the projection of s2(t) onto u1(t), subtracting that result from s2(t) and then normalizing it. I got u2t) = 4*sqrt(5)*[t^2 - 3t/4]
So the projection of g(t) is g_hat(t) = r1*u1(t) + r2*u2(t), where r1 = <g(t), u1(t)> and r2 = <g(t), u2(t)>, the inner products. I get r1 = sqrt(3)/2 and r2 = -sqrt(5)/6
Putting it all together and simplifying, I get g_hat(t) = 4t - (10/3)*t^2, thus a = 4 and b = -10/3

Apologies for the formatting. Is this correct?

@ashah99, looks fine to me. It took me awhile to check all your work, but everything checks:
your orthonormal vectors are indeed orthogonal and have magnitude 1.

Calculating the inner product of g(t) with each of $u_1$ and $u_2$ gets you the magnitudes of the projection of g(t) onto each of these "vectors". Then scaling these unit "vectors" by the amounts you found from the inner products produces two "vectors" of the right length. Finally, adding these two together gets you the "vector" projection of g(t) onto the "plane" that $u_1$ and $u_2$ span.

Good job!

 

[ashah99]

@ashah99, looks fine to me. It took me awhile to check all your work, but everything checks:
your orthonormal vectors are indeed orthogonal and have magnitude 1.

Calculating the inner product of g(t) with each of $u_1$ and $u_2$ gets you the magnitudes of the projection of g(t) onto each of these "vectors". Then scaling these unit "vectors" by the amounts you found from the inner products produces two "vectors" of the right length. Finally, adding these two together gets you the "vector" projection of g(t) onto the "plane" that $u_1$ and $u_2$ span.

Good job!

Thank you for taking the time to review my work - I know it was a mess, but at least I'm on the right track.

I was wondering if I could ask you a side question related to the original problem statement. Not sure if you have a background in digital communications, but would you know how to approach sketching the signal space (constellation) diagram for the 4 signal vectors? Thanks for your time.

 

[Mark44]

Thank you for taking the time to review my work - I know it was a mess, but at least I'm on the right track.

I was wondering if I could ask you a side question related to the original problem statement. Not sure if you have a background in digital communications, but would you know how to approach sketching the signal space (constellation) diagram for the 4 signal vectors? Thanks for your time.

I don't have a background in digital communications, but I do know some math. I'm assuming that u(t) in your problem is the Heaviside unit step function, where u(t) = 1 for t >= 0, and u(t) = 0 for t < 0. u(t - 1) is just the same function that's been shifted to the right by 1 unit.

If so, then
$g(t) = u(t) - u(t - 1) = \begin{cases} 0 & t < 0 \\ 1 & 0 \le t < 1 \\ 0 & t > 1\end{cases}$

So in a nutshell $s_1(t), s_2(t), s_3(t),$ and $s_4(t)$ just multiply $t, t^2,$ etc. by 1 for t between 0 and 1, and 0 elsewhere.

Here's a graph of y = g(t), based on my assumption:

 

[sysprog]

@Mark44's straight right graph looked exactly right to me, and I liked his 'nutshell' remark; I do have a background in digital communications, but I think that @Mark44 is, like others here, better at mathematics than I am, even though maybe if I think things through I can get along.