The Fourier Transform
Learning Objectives
definition of Fourier
Fo rier Transform and its inverse
in erse
reasons for use of the Fourier Transform
brief introduction to Fourier transform properties
delta function and its spectrum
definition of dB scale
definition of -6dB
6dB bandwidth
bandwidth - time domain connection
Fourier Transforms for UT Systems
input voltage
vi(t)
output voltage
vR((t))
Time domain
Frequency domain
+
+∞
V (ω ) = ∫ v(t ) exp(iω t )dt
−∞
1 +∞
v(t ) =
∫ V (ω ) exp(−iω t )dω
2π −∞
or, equivalently
+∞
Vi (ω ),VR (ω )
V ( f ) = ∫ v(t ) expp(2πi f t )dt
… volts-sec
or volts-μsec
−∞
+∞
v(t ) = ∫ V ( f ) expp(−2πi f t )dff
−∞
for NDE problems t is usually in μsec, f in MHz
Fourier Transforms
V(f )=
+∞
∫ v ( t ) exp ( 2π if t ) dt
−∞
v (t ) =
+∞
∫ V ( f ) exp ( −2π i f t ) df
−∞
A few properties of Fourier Transforms
If
then
v (t ) ↔ V ( f )
v ( t − t0 ) ↔ exp ( 2π if t0 ) V ( f )
dv
↔ −2π if V ( f )
dt
Example Fourier transform
V(f )=
+∞
∫ v ( t ) exp ( 2π i f t ) dt
−∞
t0
V ( f ) = ∫ A exp ( 2π ift ) dt
v(t)
0
A exp ( 2π ift )
2π if
0
to
A
=
t0
t
=
A ⎡⎣exp ( 2π ift0 ) − 1⎤⎦
2π if
At0 exp ( iπ fto ) sin (π fto )
=
π fft0
>> u=linspace(0, 5, 100);
>> u = u + eps*(u = = 0);
>> yf = exp(i*pi*u).*sin(pi*u)./(pi*u);
>> plot(u, abs(yf))
>> xlabel('f*t
xlabel( f t_00'))
1
>> ylabel('abs(V/At_0)')
0.9
0.8
0.7
abs(V//At0 )
0.6
0.5
0.4
0.3
0.2
0.1
0
0
05
0.5
1
15
1.5
2
2.5
2
5
f*t0
3
35
3.5
4
45
4.5
5
>> plot(u, angle(yf))
3.5
3
25
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Concept of a delta function (impulse)
v(t)
Consider takingg the limit
t0 → 0
with
A
At0 = 1
t
t0
V ( f ) →1
v (t ) → δ (t )
1.0
t
f
Some properties of delta functions
⎧0 τ < a or τ > b
⎪
d = ⎨ g (τ ) a < τ < b
∫a g ( t ) δ ( t − τ ) dt
⎪
⎩ g (τ ) / 2 τ = a or τ = b
δ (t −τ ) = 0 t ≠ τ
δ (t −τ )
b
τ
t
H (t −τ )
⎧0 t < τ
1.0
⎪
δ ( u − τ ) du = ⎨1/ 2 t = τ
05
0.5
∫
u =−∞
⎪1 t > τ
⎩
= H ( t − τ ) unit step function
u =t
τ
t
A Gaussian spectrum with a center frequency fc and bandwidth bw
{
}
2
2
V ( f ) = π A exp ⎡ −4π 2 A2 ( f − f c ) ⎤ + exp ⎡ −4π 2 A2 ( f + f c ) ⎤
⎣
⎦
⎣
⎦
v ( t ) = cos ( 2π f c t ) exp ⎡⎣ −π t 2 / 4 A2 ⎤⎦
exp ⎡ −4π A
⎣
2
2
(f
− fc )
2
⎤
⎦
A=
ln 2
π bw
Vmax
Vmax / 2
⎛V ⎞
V
( dB ) = 20 log10 ⎜ ⎟
Vr
⎝ Vr ⎠
⎛1⎞
20 log10 ⎜ ⎟ = − 6.02 dB
⎝ 2⎠
bw
fc
frequency
bw = -6 dB bandwidth
fc = center frequency
In real ultrasonic systems the spectrum is not symmetric, so
how
ow do we define
de e a bandwidth
b dw d andd center
ce e frequency?
eque cy?
Vmax
Vmax /2
f1
center frequency:
fc
f2
f1 + f 2
fc =
2
-6 dB bandwidth (in % of center frequency):
f 2 − f1
bw =
× 100
×100
fc
Transducer specifications (from the manufacturer)
- 6 dB bandwidth in %
fc = 10 MHz
b = 4 MHz
bw
0.2
1
0.1
0
0
-20
bw = 2 MHz
v(t)
V(( f )
[note scale differences]
-10
0
10
20
-1
-1
04
0.4
1
0.2
0
0
-20
-10
0
10
20
-1
-1
1
1
0.5
0
-0.5
0
0.5
1
-0.5
0
0.5
1
-0.5
0
0.5
1
bw = 1 MHz
0
-20
-10
0
10
frequency, MHz
20
-1
-1
time, μ sec
% Gaussian_script
f= linspace(-20, 20, 200);
t =linspace(-1,1,
p (
500);
)
subplot(3,2,1)
bw =4;
fc = 10;
[y,z]= Gauss_funcs(f,t,fc, bw);
plot(f y)
plot(f,
subplot(3, 2, 2)
plot(t,z)
subplot(3, 2, 3)
bw =2;
fc = 10;
[y,z]= Gauss_funcs(f,t,fc, bw);
plot(f,y)
subplot(3,2,4)
plot(t z)
plot(t,z)
subplot(3, 2, 5)
bw =1;
fc = 10;
[y,z]= Gauss_funcs(f,t,fc, bw);
plot(f,y)
l (f )
subplot(3,2,6)
plot(t,z)
function [y, z] =Gauss_funcs(f,t,fc,bw)
a = sqrt(log(2))/(pi*bw);
t(l (2))/( i*b )
y = sqrt(pi)*a*(exp(-(2*a*pi*(f - fc)).^2) + exp(-(2*a*pi*(f + fc)).^2));
z = cos(2*pi*fc*t).*exp(-(1/(4*a^2))*t.^2);
Note: there are a number of forms used for the Fourier
Transforms. Some of these are different from the ones we
will use here:
+∞
V (ω ) = ∫ v(t ) exp(iω t )dt
−∞
1 +∞
v(t ) =
∫ V (ω ) exp(−iω t )dω
2π −∞
Some examples:
1
V (ω ) =
2π
1
v (t ) =
2π
+∞
∫ v ( t ) exp ( iω t ) dt
−∞
+∞
∫ V (ω ) eexpp ( −iω t ) dω
−∞
(often seen in the
math literature)
V (ω ) =
+∞
d
∫ v ( t ) exp ( − jω t ) dt
−∞
1
v (t ) =
2π
+∞
∫ V (ω ) exp ( + jω t ) dω
(often seen in the
EE literature)
−∞
All off these
h
f
forms
are acceptable.
bl In
I fact
f we could
ld write
i
Fourier transform pairs in general as:
+∞
V (ω ) = N1 ∫ v ( t ) exp ( ±iω t ) dt
−∞
+∞
v ( t ) = N 2 ∫ V (ω ) exp ( miω t ) dω
−∞
as long as N1 N 2 =
1
2π
References
Sneddon, I.N., Fourier Transforms, McGraw-Hill, New York,
1951.
1951
Bracewell, R.N., The Fourier Transform and its Applications,
3rd Ed. McGraw-Hill, New York, 2000.