Consider the trigonometric form of the Fourier series
x
(
t
)
=
a
0
+
∑
n
=
1
∞
a
n
cos
(
n
Ω
0
t
)
+
∑
n
=
1
∞
b
n
sin
(
n
Ω
0
t
)
x
(
t
)
=
a
0
+
∑
n
=
1
∞
a
n
cos
(
n
Ω
0
t
)
+
∑
n
=
1
∞
b
n
sin
(
n
Ω
0
t
)
(1)It is important to state under what conditions this series (the right-hand side of Equation 1) will actually converge to x(t)x(t). The nature of the convergence also needs to be specified. There are several ways of defining the convergence of a series.
- Uniform convergence: define the finite sum:
xN(t)=a0+∑n=1Nancos(nΩ0t)+∑n=1Nbnsin(nΩ0t)xN(t)=a0+∑n=1Nancos(nΩ0t)+∑n=1Nbnsin(nΩ0t)
(2)x(t)-xN(t)<ϵx(t)-xN(t)<ϵ
(3) - Point-wise convergence: as with uniform convergence, we require that
x(t)-xN(t)(t)<ϵx(t)-xN(t)(t)<ϵ
(4) - Mean-squared convergence: here, the series converges if for all tt:
limN→∞∫t0t0+Tx(t)-xN(t)(t)2dt=0limN→∞∫t0t0+Tx(t)-xN(t)(t)2dt=0
(5)
Dirichlet has given a series of conditions which are necessary for a periodic signal to have a Fourier series. If these conditions are met, then
- the Fourier series has point-wise convergence for all tt at which x(t)x(t) is continuous.
- where x(t)x(t) has a discontinuity, then the series converges to the midpoint between the two values on either side of the discontinuity.
The Dirichlet Conditions are:
- x(t)x(t) has to be absolutely integrable on any period:
∫t0t0+Tx(t)dt<∞∫t0t0+Tx(t)dt<∞
(6) - x(t)x(t) can have only a finite number of discontinuities on any period.
- x(t)x(t) can have only a finite number of extrema on any period.
Most periodic signals of practical interest satisfy these conditions.
References
