Find Fourier transform and plot spectrum by hand & MATLAB

[Captain1024]

Homework Statement

Link: http://i.imgur.com/JSm3Tqt.png

Homework Equations

$\omega=2\pi t$
Fourier: $Y(f)=\int ^{\infty}_{-\infty}y(t)\mathrm{exp}(-j\omega t)dt$
Linearity Property: $ay_1(t)+by_2(t)=aY_1(f)+bY_2(f)$, where a and b are constants
Scaling Property: $y(at)=\frac{1}{|a|}G(\frac{f}{a})$, where a is constant

The Attempt at a Solution

From the hint, I think I need to find the Fourier transform of the function for each region:
$(-2\leq t\leq -1)$: $Y(f)=te^{-j\omega t}+\frac{j}{\omega}(e^{j\omega}-e^{j2\omega})$
$(-1\leq t\leq 1)$: $Y(f)=\frac{j}{\omega}(e^{-j\omega}-e^{j\omega})$
$(1\leq t\leq 2)$: $Y(f)=-te^{-j\omega t}+\frac{3j}{\omega}(e^{-j2\omega}-e^{-j\omega})$
Please let me know if you want to see my work.

Why do I need to use the scaling property and how do I use it?

 

[Delta2]

First of all, Fourier transform is an integral from (-infinity) to (+infinity) and because you function is defined in that way and it is zero in "most" of real line, the Fourier transform will be the sum of 3 different integrals (one integral from -2 to -1, one from -1 to +1, and one from +1 to +2, the integral from -infinity to -2 is zero, and zero is also the integral from +2 to +infinity). So , even when our function is defined in a segmented way, the Fourier transform is not segmented (we won't have 3 different Fourier transforms as you wrote, it is one transform (integral) that is the sum of 3 smaller integrals as I said above), so it doesn't mean that we find the Fourier transform in each segment, rather we work on the whole Fourier transform and break up the big integral as necessary.

Second the hint says to use the triangular function as basis. How is the triangular function being defined in your textbook?

 

[Captain1024]

How is the triangular function being defined in your textbook?

The book I am using is "Introduction to Analog & Digital Communications" by Simon Haykin & Michael Moher, 2nd ed. They define the Fourier transform of a triangular pulse as $AT^2\mathrm{sinc}^2(fT)$. That is all I could find in the book. However, I found the following definition of the unit triangle function online: $\mathrm{tri}(t) = \begin{cases}1-|t| & |t| <1\\0 & |t|\geq 1\end{cases}$.

 

[Delta2]

Ok i believe it is $y(t)=2tri(\frac{t}{2})-tri(t)$. Can you verify this? If it is indeed true then all you have to use is linear and scaling properties of Fourier transform, to get the Fourier transform of y(t), knowing the transform of tri(t).

 

[Captain1024]

Ok i believe it is $y(t)=2tri(\frac{t}{2})-tri(t)$. Can you verify this?

Well, $y(t)=2\mathrm{tri}(\frac{t}{2})-\mathrm{tri}(t)$ is equivalent to a constant +1. So, that matches the given function over the region $-1\leq t\leq 1$ only.

To be honest, I'm stuck. I have not found a transform that agrees with the given function.

Is the following correct for defining y(t) in terms of the triangular function? $y(t) = \begin{cases}2(\mathrm{tri}(\frac{t}{2})) & -2\leq t\leq -1\\2(\mathrm{tri}(\frac{t}{2}))-\mathrm{tri}(t) & -1\leq t\leq 1\\2(\mathrm{tri}(\frac{t}{2})) & 1\leq t\leq 2\end{cases}$

 

[Delta2]

Well, $y(t)=2\mathrm{tri}(\frac{t}{2})-\mathrm{tri}(t)$ is equivalent to a constant +1. So, that matches the given function over the region $-1\leq t\leq 1$ only.

it is +1 in [-1,1]. But look what happens for example in [-2,-1]

$-2\leq t \leq-1\implies tri(t)=0 , -1\leq \frac{t}{2}\leq\frac{-1}{2} \implies tri(t)=0 , tri(\frac{t}{2})=1-\frac{|t|}{2}\implies 2tri(\frac{t}{2})-tri(t)=2+t-0$
I believe if u take the other cases (-infinity,-2],[1,2],[2,+infinity] you ll get the desired result.
To see it more clearly make the graph for $2tri(t/2)$ and on the same graph also draw $tri(t)$.

 

[Delta2]

Is the following correct for defining y(t) in terms of the triangular function? $y(t) = \begin{cases}2(\mathrm{tri}(\frac{t}{2})) & -2\leq t\leq -1\\2(\mathrm{tri}(\frac{t}{2}))-\mathrm{tri}(t) & -1\leq t\leq 1\\2(\mathrm{tri}(\frac{t}{2})) & 1\leq t\leq 2\end{cases}$

this is correct, but then again no need to take cases because tri(t)=0 outside [-1,1].

 

[Captain1024]

tri(t)=0 outside [-1,1].

I think I realize my mistake. When defined in terms of the triangular function, the given function is indeed $y(t)=2tri(\frac{t}{2})-tri(t)$.
FT of $\mathrm{tri}(t)$: $\mathrm{sinc}^2(f)$
FT of $\mathrm{tri}(\frac{t}{2})$: $\mathrm{sinc}^2(2f)$
Therefore FT of y(t): $2\mathrm{sinc}^2(2f)-\mathrm{sinc}^2(f)$
Which produces the following graph:

Link: http://i.imgur.com/7iJhEdi.jpg
Is this right? I'm concerned that the graph has values below the x-axis. Should the graph of the transform match the graph of the given function?

 

[Captain1024]

Also, here is what I have for my MATLAB script so far:

Link: http://i.imgur.com/c5z3NwQ.jpg

 

[Delta2]

There is absolutely no problem with Y(f) being negative or complex, the energy at frequency f is proportional to $|Y(f)|^2$.