Parseval’s Theorem—Power Distribution over Frequency

4.4.1 Parseval’s Theorem—Power Distribution over Frequency

Although periodic signals are infinite-energy signals, they have finite power. The Fourier series provides a way to find how much of the signal power is in a certain band of frequencies.

The power P x of a periodic signal x(t), of period T 0 , can be equivalently calculated in either the time or the frequency domain:

2 P X x = dt

|x(t)| 2 = |X

4.4 Line Spectra 249

The power of a periodic signal x(t) of period T 0 is given by

|x(t)| dt

Replacing the Fourier series of x(t) in the power equation we have that

2 1 X X ∗ j 0 kt −j 0 mt

|x(t)| dt =

after we apply the orthonormality of the Fourier exponentials. Even though x(t) is real, we let |x(t)| 2 = x(t)x ∗ ( t) in the above equations, permitting us to express them in terms of X k and its conjugate. The

above indicates that the power of x(t) can be computed in either the time or the frequency domain giving exactly the same result.

Moreover, considering the signal to be a sum of harmonically related components or

x(t) X = X

X jk 0 t

x k ( t)

the power of each of these components is given by

and the power of x(t) is the sum of the powers of the Fourier series components. This indicates that the power of the signal is distributed over the harmonic frequencies {k 0 }. A plot of |X k | 2 versus the harmonic frequencies k 0 ,k = 0, ±1, ±2, . . . , displays how the power of the signal is distributed over the harmonic frequencies. Given the discrete nature of the harmonic frequencies {k 0 } this plot consists of a line at each frequency and as such it is called the power line spectrum (that is, a periodic signal has no power in nonharmonic frequencies). Since {X k } are complex, we define two additional

spectra, one that displays the magnitude |X k | versus k 0 , called the magnitude line spectrum, and the phase line spectrum or ∠X k versus k 0 showing the phase of the coefficients {X k } for k 0 . The power line spectrum is simply the magnitude spectrum squared.

A periodic signal x(t), of period T 0 , is represented in the frequency by its Magnitude line spectrum :

|X k | vs k 0 (4.15)

Phase line spectrum :

∠X k vs k 0 (4.16)

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

The power line spectrum |X k | 2 versus k 0 of x(t) displays the distribution of the power of the signal over frequency.

4.4.2 Symmetry of Line Spectra

For a real-valued periodic signal x(t), of period T 0 , represented in the frequency domain by the Fourier coefficients {X k = |X k |e j∠X k } at harmonic frequencies {k 0 = 2πk/T 0 }, we have that

(4.17) or equivalently that 1. |X k | = |X −k | (i.e., magnitude |X k | is even function of k 0 )

X k =X ∗ −k

(4.18) Thus, for real-valued signals we only need to display for k ≥ 0 the

2. ∠X k = −∠X −k (i.e., phase ∠X k is odd function of k 0 )

Magnitude line spectrum: Plot of |X k | versus k 0 Phase line spectrum: Plot of ∠X k versus k 0

For a real signal x(t), the Fourier series of its complex conjugate x ∗ ( t) is

( t)

jℓ = t X

X ∗ −jℓ 0 t

X −k ∗ e jk 0 t

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

x(t) X

k e jk 0

Comparing the Fourier series coefficients in the expressions, we have that X ∗ −k =X k , which means that if X k = |X k |e j∠X k , then

|X k | = |X −k | ∠X k = −∠X −k

or that the magnitude is an even function of k, while the phase is an odd function of k. 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.