The Fourier series expansion of sgn(cos(t)) has _________. Where sgn represents the signum function

Concept:

Even Symmetricities:

⇒ FS expansion of an even signal does not contents sin terms

ODD symmetricities:

⇒ FS expansion of an odd signal contains only sin term.

Half wave symmetricities:

\(\begin{array}{*{20}{c}} {x\left( t \right) = - x\left( {t + \frac{{{T_o}}}{2}} \right)}\\ {\begin{array}{*{20}{l}} \downarrow \\ {{c_n} = - {c_n}.{e^{\frac{{Jn{\omega _0}{T_0}}}{2}}}} \end{array}} \end{array}\)

Time shifting

\(\begin{array}{*{20}{c}} \;\\ {x\left( {t - {t_0}} \right)\rightarrow{c_n}e} \end{array} - jn{\omega _0}{t_0}\)

\(1 = - {e^{\frac{{Jn{\omega _0}{T_o}}}{2}}},\frac{{{\omega _o}{T_o}}}{2} = \pi \)

1 = -eJnπ

1 + (ejπ)n = 0

∴ eπj = -1

1 + (-1)n = 0      -----(1)

Equation (1) will be satisfied for n = odd integer

nω0  = ODD harmonic

∴ F.S expansion of an H.W.S contains only ODD harmonic

Even + H.W.S:

⇒ F.S expansion of even H.W.S signal contains cos terms with ODD harmonic.

Cos terms with ODD harmonic

Odd + H.W.S

⇒ F.S expansion of odd H.W.S. signal contains sin terms with ODD harmonics.

Analysis:

\(\begin{array}{l} \;sgn\left( {cos{\rm{\;}}t} \right){\rm{\;}} = {\rm{\;}}1;{\rm{\;}}cost{\rm{\;}} > {\rm{\;}}0\\ = {\rm{\;}} - 1;{\rm{\;}}cost{\rm{\;}} < {\rm{\;}}0 \end{array}\)

It represents square wave, which is an even and half-wave symmetry function, it contains cosine terms for all odd harmonics.