# Signal equation using signal waveform and Fourier series?

Part A: Obtain signal waveform from mathematical equation of the signal

Let a sinusoidal periodical signal is represented by an equation

$$y=f(t)=10+10\cos\left(\frac{2\pi f_1t}{T} +\frac{\pi}{6}\right)+10\sin\left(\frac{2\pi f_2t}{T}+ \frac{\pi}{3}\right)\tag{1}$$.

Here, let us take $f_1=10 ,f_2=5 ,T=100$

Now, with this we can get the waveform of the signal $y$ on Cartesian axis with magnitude as $Y$-axis and time as $X$-axis.

(Sorry but I don't have any software tool now so that I could show the waveform diagram)

Part B: Obtain mathematical equation of the signal from its waveform and Fourier series formula.

i.e. part B is reverse of part A.

Here, my question is that how can we obtain (probably the same) mathematical equation $(1)$ for signal $y$ from waveform of signal $y$ and Fourier series formula ?

Fourier series formula for any sinusoidal periodical signal $y$ is given as

$$y(t) = \sum\limits_{m=0}^{+\infty} a_m \cos(w_m t) + \sum\limits_{m=0}^{+\infty}b_m \sin(w_m t). \tag{2}$$

• Your question still isn't clear to me: are you asking how to find the $a_m$ and $b_m$ coefficients? Or are you asking something else? – Peter K. May 4 '16 at 12:57
• @Peter K. Sir, my final aim is to obtain $equation1$ .so you are right that I need to find $a_n$,$b_n$ . The steps I should follow are 1. Find $f(x)$ from the signal waveform 2. Compute $a_0$,$a_n$,$b_n$ 3.Put above values into the $equation2$ and see whether I get $equation 1$ or not. – signal1642 May 4 '16 at 13:17

Did you check the usual equations?

Responding to:

The steps I should follow are

1. Find $f(x)$ from the signal waveform

2. Compute $a_0$, $a_n$, $b_n$

3. Put above values into the (2) and see whether I get (1) or not.

Essentially, yes.

For your equation to relate to the $f(x)$ in the image above you may want to think about $$w_m = \frac{2\pi 5 m}{100} = \frac{\pi m}{10}$$

Also note that there needs to be an integer (or rational) ratio between $f_1$ and $f_2$ in your equation (1).

• K sir, you have given formulas for calculation of $a_0$ ,$a_n$ ,$b_n$ . But how to find $f(x)$ from the waveform of the signal?? – signal1642 May 4 '16 at 11:57
• @signal1642 : $f(x) = y(t) | _{t = x}$. – Peter K. May 4 '16 at 11:59