Appendix B
Mathematical derivations
B.1
Fourier transform of frequency sweep signal
The known Fourier transforms are:
r
π
δ(ω − ω0 ) + δ(ω + ω0 )
2
r
π
F sin(ω0 t) = ı
δ(ω − ω0 ) − δ(ω + ω0 )
2
(B.1)
ω2 ω 2 1
cos
F cos(βt2 ) = √
+ sin
4β
4β
4β
2
ω
ω 2 1
cos
− sin
.
F sin(βt2 ) = √
4β
4β
4β
(B.2)
F cos(ω0 t) =
Using the trigonometric expansion for formula 3.12 yields:
F cos (ω0 + βt)t =
F cos(ω0 t + βt2 ) =
F cos(ω0 t) cos(βt2 − sin(ω0 t) sin(βt2 =
F cos(ω0 t) cos(βt2 − F sin(ω0 t) sin(βt2 =
F cos(ω0 t) ∗ F cos(βt2 − F sin(ω0 t) ∗ F sin(βt2 .
(B.3)
Replacing the separate terms in equation B.3 with those from equations B.1 and B.2,
and working out the convolutions, results in:
F cos (ω0 + βt)t =
(ω − ω0 )2
(ω − ω0 )2
1
√ (1 − ı) cos
+ (1 + ı) sin
+
(B.4)
4β
4β
4 β
2
2
(ω + ω0 )
(ω + ω0 )
(1 + ı) cos
+ (1 − ı) sin
.
4β
4β
Appendix B
Taking the Fourier transform of equation B.4 convolved with a rectangular window
between Tc − T and Tc + T leads, according to Mathematica 5.0 (Wolfram Research,
Inc.), to:
TZ
c +T
cos (ωc + βt)t e−ıωt dt =
Tc −T
r 1
1
ı
π
(ω − ωc )2
(ω − ωc )2
√ ( + )
ı cos
+ sin
×
2
4β
4β
β 4 4
1
( + ı ) 2β(Tc − T ) − (ω − ωc )
√
erfi 2 2
−
2β
1
( 2 + 2ı ) 2β(Tc + T ) − (ω − ωc )
√
erfi
+
2β
ω 2 + ωc2 ω 2 + ωc2 cos
+ ı sin
×
2β
2β
1
( 2 + 2ı ) 2β(Tc − T ) + (ω + ωc )
√
erf
−
2β
1
( 2 + 2ı ) 2β(Tc + T ) + (ω + ωc )
√
,
erf
2β
(B.5)
where ωc is the frequency at Tc . This is graphically depicted in figure 4.2, panels A
and B. The results of multiplying the time signal with a Kaiser-Bessel window are
in panels C through F. These graphs are obtained with Digital Fourier Transforms in
Matlab (The MathWorks, Inc.).
B.2
Noise cancellation
The addition of two sinusoidal signals in opposite phase may lead to cancellation.
This principle is used in active noise cancellation (section 5.4.2). The remaining
signal s is:
s = a1 sin(ωt) − (a1 + δa) sin(ωt + δφ)
= −(a1 + δa) sin(ωt) cos(δφ) + cos(ωt) sin(δφ) + a1 sin(ωt).
(B.6)
With perfect phase matching (δφ = 0), the remaining signal s = −d sin(ωt), and for
perfect amplitude matching (δa = 0),
s = −a1 sin(ωt) cos(δφ) + cos(ωt) sin(δφ) + a1 sin(ωt).
(B.7)
This can be approximated by
s = −a1 δφ cos(ωt),
when δφ is small.
90
(B.8)