# Amplitude and phase spectrum

I want to find phase and amplitude spectrum of this signal:

Here is what I have done:

$$f(t)=\sum_{-\infty}^{+\infty}Fne^{jnwt}, Fn=\frac{1}{T}\int_{0}^{T}f(t)e^{-jnwt}dt=\frac{1}{T}\int_{t1}^{t1+\tau}Ee^{-jnwt}dt=...=\frac{E}{T}\frac{1}{jnw}e^{-jnwt1}(1-e^{-jnw\tau})=\frac{E}{T}\frac{1}{jnw}e^{-jnwt1}e^{jnw\frac{\tau}{2}}(\frac{e^{jnw\frac{\tau}{2}}-e^{-jnw\frac{\tau}{2}}}{2})*2=\frac{2E}{Tjnw}e^{-jnw(t1+\frac{\tau}{2})}\sin{(nw\frac{\tau}{2})}$$

I stuck there :( What should I do next?

The amplitude of the spectrum of this signal is related, as all pulsed signal, with the $sinc(t)$ function. You must find a trigonometric relationship between the coefficient associated with the sine function

$$\dfrac{2E}{T\,j\,n\omega}$$

and the argument of the function.

$$n\,\omega\dfrac{\tau}{2}$$

Check out the link, and see that there is a relationship between the pulse width and frequency of the $sinc(t)$ function.

• Here is solution from my book: $$|Fn|=E\frac{\tau}{T}|\frac{\sin{nw\frac{\tau}{2}}}{nw\frac{\tau}{2}}|$$ I'm trying to get that from my expression in first post but without success :( – etf Sep 30 '14 at 8:34
• Look at the Euler's expression of the sine in terms of exponentials. The denominator must be $2j$. – Martin Petrei Sep 30 '14 at 11:05
• Thanks!!! I used 2 in denominator instead of 2j. I don't know how I made such a mistake :) – etf Sep 30 '14 at 13:47