Chapitre 6

The computation of FIR filter coefficients relies on the utilisation of the digital Fourier transform and spectral windows (rectangular, Hamming, Hanning, etc.).

A sequence of samples h(n) spaced by \(T_e=\frac{1}{F_e}\) admits, for the digital Fourier transform \(H(j\omega)\) :

A FIR filter is defined by a transfer function :

\((Z)=\sum_{k=0}^{p}{a_kZ^{-k}\ }\) (4)

Or in a harmonic regime :

\(H\left(j\omega\right)=\sum_{k=0}^{p}{a_k\left(e^{j\omega T_e}\right)^{-k}=}\sum_{k=0}^{p}{a_ke^{-kj\omega T_e}}\ \)(5)

It often happens that these coefficients exhibit symmetry properties :

\(a_k=a_{p-k}\ \ \ or a_k=-a_{p-k}\)

If we group the terms two by two :

\(a_ke^{-kj\omega T_e}+a_{p-k}e^{-\left(p-k\right)j\omega T_e}=a_k\left[e^{-kj\omega T_e}+e^{-\left(p-k\right)j\omega T_e}\right]\)

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =a_ke^{-j\frac{p}{2}\omega T_e}\left[e^{\left(\frac{p}{2}-k\right)j\omega T_e}+e^{-\left(\frac{p}{2}-k\right)j\omega T_e}\right]\)

\(\quad\) \(\quad\) \(\quad\) \(=2a_kcos\left[\left(\frac{p}{2}-k\right)\omega T_e\right]e^{-j\frac{p}{2}\omega T_e}\ \)(6)

If \(a_k=-a_{p-k}\) , we obtain \(2{ja}_ksin\left[\left(\frac{p}{2}-k\right)\omega T_e\right]e^{-j\frac{p}{2}\omega T_e}\)

We'll then have :

\(H\left(j\omega\right)\ =e^{-j\frac{p}{2}\omega T_e}{\underbrace{\sum_{k=0}^{\frac{p}{2}}{2a_kcos\left[\left(\frac{p}{2}-k\right)\omega T_e\right]}} _{ A }}\) (7)

From where :

\(Arg\left[H\left(j\omega\right)\right]\ =-\frac{p}{2}\omega T_e\omega(+\pi)\) (the \(+\pi\) term intervenes when \(A<0\))