6.1
Orthogonality:
Two vectors are orthogonal iff the dot product between them is zero.
c = f ⋅ x = f x cos(θ ) = 0
Where f denotes the magnitude or length of f (equal to the square root of the sum
of squares of its elements) and θ is the angle between the vectors. This can also be
computed by multiplying the two vectors element by element and summing the
products. Example: Determine whether or not the vectors are orthogonal,
a) x = [1,−1,0 ], f = [−1,−1,10 ]
b) x = [1,−2,1], f = [ −1,−1,3]
c) x = [1,0], f = [ 0,1]
Two continuous-time signals are orthogonal over an interval [t1, t2] iff the
correlation between them is zero on that interval:
t2
C = ∫ f ( t ) x ( t ) dt = 0
t1
⎡ 2π ⎤
Determine whether or not the signals below are orthogonal for t ∈ ⎢0, ⎥
⎣ ω ⎦
a) f (t ) = cos(ω t ), x ( t ) = sin(ω t )
b) f (t ) = cos(mω t ), x ( t ) = cos( nω t ) for m, n ∈[ Integers]
0
0
0
0
0
6.2
Orthogonality Relationships for Sin and Cos
a) ∫ cos(ω t ) sin(ω t ) dt = 0
2π
for T = integer multiples of
ω
2π
, n ≠ m, n, m ∈[ integers]
b) ∫ cos(mω t ) cos(nω t )dt = 0 for T = integer multiples of
ω
0
0
0
T0
0
0
0
0
T0
0
c) ∫ sin(mω t ) sin(nω t ) dt = 0 for T = integer multiples of
2π
, n ≠ m, n, m ∈[ integers]
ω
T
2π
, n ∈[ integers]
d) ∫ cos( nω t ) cos( nω t ) dt = for T = integer multiples of
2
ω
T
2π
, n ∈[ integers]
e) ∫ sin(nω t ) sin( nω t ) dt = for T = integer multiples of
2
ω
0
0
0
T0
0
0
0
0
0
T0
0
0
0
0
0
T0
0
Use the above relationships to derive the formula for the Fourier Series coefficients,
such that they minimize the mean square difference between a signal x ( t ) and its
Fourier Series expansion a + ∑ a cos(nω t ) + b sin( nω t ) . In other words find a0, an,
∞
[
0
n =1
(
n
0
n
0
∞
and bn that minimize: ∫ x(t ) − a + ∑ a cos(nω t ) + b sin(nω t )
T0
0
n =1
n
0
n
0
)] dt
2
6.3
Existence of Fourier Series and Dirichlet Conditions
In order for the Fourier Series coefficients to exist for some periodic function f(t),
the following condition must hold: ∫ f ( t ) dt < ∞
T0
In order for the Fourier Series to converge to a periodic function f(t), the coefficients
must exist and f(t) must have a finite number of maxima, minima, and
discontinuities in one period.
Distortion Example
Consider a pure sine wave at f=1/.06 Hz passing through an amplifier circuit. The
output is sin( 2π ft ) clipped when the absolute value exceeds sin(π / 3) ≈ 0.866 .
Determine the Fourier Series coefficients, plot the waveform using different
numbers of components, and compute the ratio of harmonic energy not located at
f=1/.06. This is often used as a measure of distortion for amplifier circuits.