Signals and Systems - Electrical Engineering

250 C H A P T E R 4: Frequency Analysis: The Fourier Series

The power line spectrum|Xk|^2 versusk 0 ofx(t)displays the distribution of the power of the signal over

frequency.

4.4.2 Symmetry of Line Spectra

For a real-valued periodic signalx(t), of periodT 0 , represented in the frequency domain by the Fourier

coefficients{Xk=|Xk|ej∠Xk}at harmonic frequencies{k 0 = 2 πk/T 0 }, we have that

Xk=X∗−k (4.17)

or equivalently that

1. |Xk|=|X−k|(i.e., magnitude|Xk|is even function ofk 0 )


2. ∠Xk=−∠X−k(i.e., phase∠Xkis odd function ofk 0 ) (4.18)

Thus, for real-valued signals we only need to display fork≥ 0 the

Magnitude line spectrum: Plot of|Xk|versusk 0

Phase line spectrum: Plot of∠Xkversusk 0

For a real signalx(t), the Fourier series of its complex conjugatex∗(t)is

x∗(t)=

[

∑

`

X`ej`^0 t

]∗

=

∑

`

X`∗e−j`^0 t=

∑

k

X−∗kejk^0 t

Sincex(t)=x∗(t), the above equation is equal to

x(t)=

∑

k

Xkejk^0 t

Comparing the Fourier series coefficients in the expressions, we have thatX∗−k=Xk, which means

that ifXk=|Xk|ej∠Xk, then

|Xk|=|X−k|

∠Xk=−∠X−k

or that the magnitude is an even function ofk, while the phase is an odd function ofk. Thus, the line

spectra corresponding to real-valued signals is given for only positive harmonic frequencies, with the

understanding that for negative values of the harmonic frequencies the magnitude line spectrum is

even and the phase line spectrum is odd.