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74
Signals and Spectra Chap. 2
Polar Fourier Series
The quadrature Fourier series, Eq. (2–95), may be rearranged and written in a polar (amplitudephase)
form. The polar form is
where w(t) is real and
w(t) = D 0 +
n=q
a
n=1
D n cos(nv 0 t + w n )
(2–103)
a n = e D 0, n = 0
(2–104)
D n cos w n , n Ú 0
b n = -D n sin w n , n Ú 1
(2–105)
The latter two equations may be inverted, and we obtain
D n = e a 0, n = 0
(2–106)
2a 2 n + b 2 n, n Ú 1 f = e c 0, n = 0
2ƒc n ƒ, n Ú 1
and
w n = -tan -1 b n
£ ≥ = lc (2–107)
a n , n Ú 1
n
where the angle operator is defined by
l[# ] = tan -1 a Im[ # ]
b
(2–108)
Re[# ]
It should be clear from the context whether l denotes the angle operator or the angle
itself. For example, l90° denotes an angle of 90°, but l[1 + j2] denotes the angle operator
and is equal to 63.4° when evaluated.
The equivalence between the Fourier series coefficients is demonstrated geometrically
in Fig. 2–11. It is seen that, in general, when a physical (real) waveform w(t) is represented by
a Fourier series, c n is a complex number with a real part x n and an imaginary part y n (which
are both real numbers), and consequently, a n , b n , D n , and wn are real numbers. In addition, D n
is a nonnegative number for n Ú 1. Furthermore, all of these coefficients describe the amount
of frequency component contained in the signal at the frequency of nf 0 Hz.
In practice, the Fourier series (FS) is often truncated to a finite number of terms. For
example, 5 or 10 harmonics might be used to approximate the FS series for a square wave.
Thus, an important question arises: For the finite series, are the optimum values for the series
coefficients the same as those for the corresponding terms in the infinite series, or should the
coefficients for the finite series be adjusted to some new values to give the best finite-series
approximation? The answer is that the optimum values for the coefficients of the truncated
FS are the same as those for the corresponding terms in the nontruncated FS. †
As we have seen, the complex, quadrature, and polar forms of the Fourier series are
all equivalent, but the question is, Which is the best form to use? The answer is that it
† For a proof of this statement, see [Couch, 1995].