Find the fourier transform of the triangle function give below.f( t)=( a+ t)/a² −a< t<0,( a-t)/a² 0< t< a,0 | t|> a please ca you give me the solving to human understand method

To find the Fourier Transform of the given triangle function, we use the definition of the Fourier Transform:

$$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$$

The function $f(t)$ is defined as:

$$f(t) = \begin{cases} (a+t)/a^2 & -a < t < 0 \\ (a-t)/a^2 & 0 < t < a \\ 0 & |t| \ge a \end{cases}$$

We can split the integral into two parts based on the definition of $f(t)$:

$$F(\omega) = \int_{-a}^{0} \frac{a+t}{a^2} e^{-j\omega t} dt + \int_{0}^{a} \frac{a-t}{a^2} e^{-j\omega t} dt$$

Step 1: Evaluate the first integral
Let $I_1 = \int_{-a}^{0} \frac{a+t}{a^2} e^{-j\omega t} dt$. We can factor out $\frac{1}{a^2}$:

$$I_1 = \frac{1}{a^2} \int_{-a}^{0} (a+t) e^{-j\omega t} dt$$

We use integration by parts, $\int u dv = uv - \int v du$. Let $u = (a+t)$ and $dv = e^{-j\omega t} dt$.
Then $du = dt$ and $v = \frac{e^{-j\omega t}}{-j\omega}$.

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