Sine and Cosine transforms for finite range
Fourier sine transform
Sn
=
f (x)
=
Z
2 L
nπ
f (t) Sin(
x)dx,
L 0
L
∞
X
nπ
Sn Sin(
x).
L
n=1
Fourier cosine transform
Cn
f (x)
=
=
2
L
Z
L
f (t) Cos(
0
nπ
x)dx,
L
∞
X
C0
nπ
+
Cn Cos(
x).
2
L
n=1
EE-2020, Spring 2009 – p. 1/18
Sine and Cosine transforms of derivatives
Finite Sine and Cosine transforms:
Fs (f )
≡
Fc (f )
≡
Z
2 ∞
Fs (ω) =
f (t) Sin(ωt)dt,
π 0
Z
2 ∞
Fc (ω) =
f (t) Cos(ωt)dt,
π 0
Sine Transforms of Derivatives
Fs (f ′ )
=
Fs (f ′′ )
=
−ωFc (f ),
2
ω f (0) − ω 2 Fs (f ),
π
Cosine Transforms of Derivatives
Fc (f ′ )
=
Fc (f ′′ )
=
2
f (0) + ωFs (f ),
π
2
− ω f ′ (0) − ω 2 Fc (f ).
π
−
EE-2020, Spring 2009 – p. 2/18
Diffusion problem in a semi-infinite medium
PDE:
ut = α2 uxx ,
BCs:
u(0, t) = A,
IC:
u(x, 0) = 0,
0 < x < ∞,
0<t<∞
0<t<∞
0≤x<∞
Transform x-variable via the Fourier sine transform, i.e. Fs [u(x, t)] = U (ω, t),
Fs [ut ] = α2 Fs [uxx ],
⇒
d
2
U (ω, t) = α2 [ ω u(0, t) − ω 2 U (ω, t)],
dt
π
For the ODE,
d
2Aωα2
U (t) =
− α2 ω 2 U (t)
dt
π
we have the solution for the IC, U (0) = 0,
U (t) =
2 2
2A
[1 − e−ω α t ],
πω
EE-2020, Spring 2009 – p. 3/18
Diffusion problem in a semi-infinite medium
By the inverse transform of Fs−1 [U (ω, t)] = u(x, t), we have
√
u(x, t) = A erfc(x/2α t),
i.e.
Z
∞
0
2
2 1 − e−aω
x
[
]Sin(ωx) dω ≡ erfc( √ ),
π
ω
2 a
where erfc(x), 0 < x < ∞, is called the complementary-error function and is given by
2
erfc(x) = √
π
Z
∞
x
e
−t2
dt,
and
2
erf(x) = √
π
Z
x
2
e−t dt,
0
EE-2020, Spring 2009 – p. 4/18
Fourier transforms
A periodic function of time, f(t), of period T can be represented by a Fourier
transform,
F(n)
f (t)
=
=
1
T
Z
T /2
f (t)e−jnω0 t dt,
−T /2
∞
X
F(n)e+jnω0 t ,
n=−∞
where F(n) is the corresponding Fourier series, where F (n) =
F (−n) =
1
(Cn
2
+ i Sn ), and F (0) =
C0
.
2
1
(Cn
2
− i Sn ),
For an aperiodic function,
F(ω)
=
f (t)
=
Z ∞
1
f (t)e−jωt dt,
2π −∞
Z ∞
F(ω)e+jωt dω.
−∞
EE-2020, Spring 2009 – p. 5/18
Integral transforms
Fourier sine transform
F (ω)
f (t)
=
=
2
π
Z
Z
∞
0
∞
f (t) Sin(ωt)dt,
F (ω) Sin(ωt)dω.
0
Fourier cosine transform
F (ω)
f (t)
=
=
2
π
Z
Z
0
∞
∞
f (t) Cos(ωt)dt,
F (ω) Cos(ωt)dω.
0
Fourier transform
F (ω)
=
f (t)
=
Z ∞
1
√
f (t) e−iωt dt,
2π −∞
Z ∞
1
√
F (ω) eiωt dω,
2π −∞
EE-2020, Spring 2009 – p. 6/18
Gaussian function
t2
exp(− 2 )
2τp
↔
where we use the identity
τp2 ω 2
τp exp(−
),
2
R∞
2
−x dx =
−∞ e
∆t · ∆ω = τp ·
√
π. And
1
=1
τp
EE-2020, Spring 2009 – p. 7/18
Rectangle function

 1,
f (t) =
 0,
when|t| ≤ τp


when|t| > τp 
↔
r
2 sinωτp
τp
≡
π
ωτp
r
2
τp Sinc(ωτp ),
π
EE-2020, Spring 2009 – p. 8/18
Some common Fourier transforms
Fourier transform
F(ω)
=
f (t)
=
Z ∞
1
√
f (t) e−iωt dt,
2π −∞
Z ∞
1
√
F(ω) eiωt dω,
2π −∞
Common Fourier transforms
2
f (x) = e−x

 1 ; for − 1 ≤ x ≤ 1
f (x) =
 0 ; elsewhere

 e−x ; for x > 0
f (x) =
 0
; for x ≤ 0
↔
2
1
F(ω) = √ e−(ω/2) ,
2
r
2 Sin(ω)
F(ω) =
,
π ω
↔
1 1 − iω
,
F(ω) = √
2
2π 1 + ω
↔
Gaussian
rectangle
EE-2020, Spring 2009 – p. 9/18
Properties of Fourier transform
if f (t) is real, then F(−ω) = F∗ (ω).
if f (t) is even, then F(−ω) = F(ω), i.e., F(ω) is even.
if f (t) is odd, then F(−ω) = −F(ω), i.e., F(ω) is odd.
Time scaling, f (a t) ↔
Frequency scaling,
1
F( ω
).
|a|
a
1
t
f
(
)
|b|
b
↔ F(b ω).
Time shifting, f (t − t0 ) ↔ F(ω)e−jωt0 .
Frequency shifting, f (t)ejω0 t ↔ F(ω − ω0 ).
Convolution theorem: A convolution of two time functions, f (t) and g(t), is defined,
1
g⊗f ≡ √
2π
Z
∞
−∞
g(t − t′ )f (t′ ) d t′ ,
the Fourier transform of the convolution is
1
√
2π
Z
∞
−∞
g ⊗ f e−jωt d t = G(ω)F(ω) = F[g]F[f ].
EE-2020, Spring 2009 – p. 10/18
Transformation of Partial Derivatives
Fourier transform
F[f ] = F(ω)
=
F −1 {F[f ]} = f (t)
=
Z ∞
1
√
f (t) e−iωt dt,
2π −∞
Z ∞
1
√
F(ω) eiωt dω,
2π −∞
Transformation of partial derivatives
F[ux ]
=
iωF[u],
F[uxx ]
=
−ω 2 F[u],
Useful identities
Z
∞
2
e−x dx =
√
π
−∞
EE-2020, Spring 2009 – p. 11/18
Diffusion problem in a infinite medium
PDE:
ut = α2 uxx ,
BCs:
no boundary condition
u(x, 0) = φ(x),
IC:
−∞ < x < ∞,
0<t<∞
−∞ ≤ x < ∞
Transform x-variable via the Fourier sine transform, i.e. F[u(x, t)] = U (ω, t),
F[ut ] = α2 F[uxx ],
⇒
d
U (ω, t) = −α2 ω 2 U (ω, t),
dt
With the IC, we can solve the ODE to have,
2 ω2 t
U (t) = U0 (ω) e−α
,
where U0 (ω) ≡ F[φ(x)]
EE-2020, Spring 2009 – p. 12/18
Diffusion problem in a infinite medium
By the inverse Fourier transformation,
2 ω2 t
u(x, t) = F −1 [U (t)] = F −1 [U0 (ω) e−α
]
With the convolution theorem,
u(x, t)
=
=
=
2 ω2 t
F −1 [U0 (ω)] ⊗ F −1 [e−α
]
2
1
− x2
φ(x) ⊗ [ √ e 4α t ]
α 2t
Z ∞
2
2
1
φ(ξ)e−(x−ξ) /4α t dξ
√
2α πt −∞
The integrand is made up of two terms:
1. The initial temperature φ(x).
x2
2. The Green function or impulse-response function G(x, t) =
−
1
[ √
e 4α2 t ]
2α πt
The initial temperature u(x, 0) = φ(x) is decomposed into a continuum of impulses.
EE-2020, Spring 2009 – p. 13/18
Integral transforms, cont.
Laplace transform
F (s)
f (t)
=
Z
∞
f (t) e−s t dt,
0
c+i∞
1
2πi
Z
=
Z
∞
=
Z
=
F (s) est ds,
c−i∞
Hankel transform (Fourier-Bessel)
Fn (ξ)
f (r)
r Jn (ξr) f (r)dr,
0
∞
Jn (ξr) Fn (ξ)dξ,
0
Mellin transform
F (s)
f (t)
=
=
Z
∞
f (t) ts−1 dt,
0
1
2πi
Z
c+i∞
F (s) x−s ds,
c−i∞
EE-2020, Spring 2009 – p. 14/18
Some common Laplace transforms
Laplace transform
F (s)
f (t)
Z
=
∞
0
1
2πi
=
f (t) e−s t dt,
Z
c+i∞
F (s) est ds,
c−i∞
Common Laplace transforms, for 0 < t < ∞,
f (t) = 1
↔
f (t) = e2t
↔
f (t) = Sin(ωt)
↔
2
f (t) = et
↔
1
,
s
1
F(s) =
s>2
s−2
ω
,
F(s) = 2
s + ω2
F(s) =
F(s) don’t exist.
EE-2020, Spring 2009 – p. 15/18
Transformation of Partial Derivatives
Laplace Transformation of partial derivatives, i.e. L[u(x, t)] ≡ U (x, s)
L[ut ]
=
L[utt ]
=
s U (x, s) − u(x, 0),
s2 U (x, s) − s u(x, 0) − ut (x, 0),
Definition of finite convolution
[g ⊗ f ](t) ≡
Z
t
0
′
′
′
g(t )f (t − t ) dt =
Z
t
0
g(t − t′ )f (t′ ) dt′ ,
Convolution theorem for Laplace transform
L[g ⊗ f ]
L−1 {L[g] L[f ]}
=
L[g] L[f ],
=
g ⊗ f.
EE-2020, Spring 2009 – p. 16/18
Heat conduction in a semi-infinite medium
PDE:
BC:
IC:
ut = uxx ,
0 < x < ∞,
ux (0, t) − u(0, t) = 0,
u(x, 0) = u0 ,
0<t<∞
0<t<∞
0≤x<∞
Transform t-variable via the Laplace transform, i.e. L[u(x, t)] = U (x),
L[ut ] = s U (x, s) − u(x, 0),
For the ODE,
d2
s U (x) − u0 =
U (x),
dx2
we have the general solution (homogeneous + a particular solution),
U (x) = c1 e
√
sx
+ c2 e
√
− sx
+
u0
,
s
EE-2020, Spring 2009 – p. 17/18
Heat conduction in a semi-infinite medium, cont.
For the BC,
d
U (0) − U (0) = 0,
dx
we have the coefficients,
√
e− sx
u0
U (x) = −u0 √
+
,
s( s + 1)
s
By the inverse Laplace transform,
u(x, t)
=
=
L−1 [U (x, s)],
√
√
√
u0 − u0 [erfc(x/2 t) − erfc( t + x/2 t)ex+t ],
where the complementary-error function is
2
erfc(x) = √
π
Z
∞
2
e−ξ dξ.
x
EE-2020, Spring 2009 – p. 18/18