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# Fourier series and filtering of a square wave signal ## 1. Definition of the Fourier series

The Fourier coefficients of a square wave signal are: Cn = -jnπ (1 – (- 1) n)

The Fourier series is written: u (t) = ∑n = 1∞2ReCnexpjn2πTt

The Fourier coefficients are defined by a function:

`cnCreneau[n_]:=-I/(n*Pi)*(1-(-1)^n);`

The following function displays the spectrum (modulus of cn) in decibel:

```plotSpectre[cn_, nmax_] := Module[{spectre},
spectre = Table[Line[{{n, -80}, {n,20*Log[10,Abs[cn[n]]+10^(-4)]}}], {n, 1, nmax}];
Return[Graphics[{RGBColor[1, 0, 0],spectre},Frame->True,AspectRatio->0.7,
PlotRange -> {{0, nmax},{-80,0}}, FrameLabel -> {"n", "20*log(|cn|)"}]];
]
```
`Show[plotSpectre[cnCreneau,50]]`

Calculation function of the partial sum of the Fourier series (period 1):

`serie[cn_,nmax_,t_]:=Sum[2*Re[cn[n]*Exp[I*2*Pi*n*t]],{n,1,nmax}];`

Partial sum at N = 20:

`Plot[serie[cnCreneau,20,t],{t,0,1},AxesLabel->{"t","u"}]`

Partial sum at N = 100:

`Plot[serie[cnCreneau,100,t],{t,0,1},AxesLabel->{"t","u"}]`

## 2. First order low pass filter

The transfer function of a first order low pass filter is: H (f) = 11 + jffc

where fc is the cutoff frequency.

Consider the case where fc = 10. We define the transfer function then we calculate the new Fourier coefficients, noting that the index n corresponds to the frequency:

```h[f_]:=1/(1+I*f/10);
cnSortie[n_]:=cnCreneau[n]*h[n];
```

Here is the new spectrum

`Show[plotSpectre[cnSortie,50]]`

and the partial sum for N = 100

`Plot[serie[cnSortie,100,t],{t,0,1},AxesLabel->{"t","u"}]`
```h[f_]:=1/(1+I*f/0.1);
`Plot[serie[cnSortie,100,t],{t,0,1},AxesLabel->{"t","u"}]`