# Linear system - calculate output signal

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1. Oct 7, 2014

### etf

Hi!
1. The problem statement, all variables and given/known data

a) Calculate and sketch amplitude spectrum of u(t),
b) u(t) is input signal for linear time invariant system whose transfer function H(jw) is shown. Calculate output signal uo(t)

2. Relevant equations

3. The attempt at a solution

I completed task a), I got $$u(t)=\frac{3}{2}+\sum_{n=-\infty,n\neq 0}^{n=\infty}\frac{1}{2n\pi }(\sin{\frac{3n\pi}{2}}-\sin{\frac{n\pi}{2}})e^{jn\frac{2\pi}{0.5*10^{-3}}t},$$ where amplitude spectrum is $$F_n=|F_n|=\frac{1}{2n\pi }(\sin{\frac{3n\pi}{2}}-\sin{\frac{n\pi}{2}}).$$ Any suggestion about task b) ?

Last edited: Oct 7, 2014
2. Oct 8, 2014

### rude man

Use the Fourier transform.

3. Oct 10, 2014

### etf

Why should I use Fourier transform?

4. Oct 10, 2014

### rude man

Only way I can think of.

5. Oct 11, 2014

Frequency components with w<wc/2 gets multiplied with 1 and and the frequency components between wc/2 and wc gets multiplied 1/2. Since, output is convolution of input and impulse response of the system, in frequency domain it becomes the multiplication of frequency spectrum of both the signals.

6. Oct 11, 2014

### etf

Thanks for reply, but I'm still confused :( Here is "step by step" tutorial from my book:
1. Define excitation in frequency domain: Xn or X(jw)
I did it already, $$u(t)=\frac{3}{2}+\sum_{n=-\infty,n\neq 0}^{n=\infty}\frac{1}{2n\pi }(\sin{\frac{3n\pi}{2}}-\sin{\frac{n\pi}{2}})e^{jn\frac{2\pi}{0.5*10^{-3}}t},$$ where amplitude spectrum is $$F_n=|F_n|=\frac{1}{2n\pi }(\sin{\frac{3n\pi}{2}}-\sin{\frac{n\pi}{2}}).$$ (Xn from book is Fn here).
2. Multiply transfer function of system and spectrum of excitation (H(jw)Xn or H(jw)X(jw) to get response in frequency domain: Yn or Y(jw)
But my transfer function is piecewise defined, it's 1 in interval w<wc/2 and 1/2 between wc/2 and wc.
3. Apply inverse Fourier transform to find analytical expression for output signal

7. Oct 11, 2014

### rude man

Looks like your input is a pulse train. I thought it was just a pulse.
In which case you can do as post # 5 suggests.
So you take each of your frequency components (harmonics"), starting with the zero'th (dc), multiply by the corrseponding |H(jw)| of your system amplitude spectrum, and form the corresponding new sum.
A Fourier inversion is not done.