Signals and Systems - Electrical Engineering

4.7 Convergence of the Fourier Series 269

nExample 4.10

Consider the mean-square error optimization to obtain an approximation of the periodic sig-

nalx(t)shown in Figure 4.4 from Example 4.5. We wish to obtain an approximatex 2 (t)=

α+ 2 βcos( 0 t), given that it is clear thatx(t)has an average, and that once we subtract it from the

signal the resulting signal is approximated by a cosine function. Minimize the mean-square error

E 2 =

1

T 0

∫

T 0

|x(t)−x 2 (t)|^2 dt

with respect toαandβto find these values.

Solution

To minimizeE 2 we set to zero its derivatives with respect toαandβto get

dE 2

dα

=−

1

T 0

∫

T 0

2[x(t)−α− 2 βcos( 0 t)]dt=−

1

T 0

∫

T 0

2[x(t)−α]dt= 0

dE 2

dβ

=−

1

T 0

∫

T 0

2[x(t)−α− 2 βcos( 0 t)] cos( 0 t)dt= 0

which, after getting rid ofT^20 of both sides of the above equations and applying the orthogonality

of the Fourier basis, gives

α=

1

T 0

∫

T 0

x(t)dt

β=

1

T 0

∫

T 0

x(t)cos( 0 t)dt

For the signal in Figure 4.4 we obtain

α= 1

β=

2

π

giving as approximation the signal

x 2 (t)= 1 +

4

π

cos( 2 πt)

which att=0 gives x 2 ( 0 )=2.27 instead of the expected 2;x 2 (0.25)=1 (because of the

discontinuity at this point, this value is the average of 2 and 0, the values, respectively, before

and after the discontinuity) instead of 2 andx 2 (0.5)=−0.27 instead of the expected 0. n