... decomposed.
Chapter 8- The Discrete FourierTransform 145
Type of Transform Example Signal
Fourier Transform
Fourier Series
Discrete Time Fourier Transform
Discrete Fourier Transform
signals that are ... term: transform, is extensively used in Digital Signal
Processing, such as: Fourier transform, Laplace transform, Z transform,
Hilbert transform, Discrete Cosine transform, etc. Just what is a transform?
To ... Fourier decomposition, the heart of the
DFT.
The Family of FourierTransform
Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830),
a French mathematician and physicist. (Fourier...
... four members of the Fourier
transform family (Fourier transform, Fourier Series, DFT, and DTFT).
Figure 10-1 provides an example of how homogeneity is a property of the
Fourier transform. Figure ... properties of the Fourier Transform, how a mathematical change in one
domain results in a mathematical change in the other domain.
Linearity of the Fourier Transform
The FourierTransform is linear, ... frequency
domain multiplexing.
The Discrete Time Fourier Transform
The Discrete Time FourierTransform (DTFT) is the member of the Fourier
transform family that operates on aperiodic, discrete...
... called Fouriertransform pairs. Several common
pairs are presented in this chapter.
Delta Function Pairs
For discrete signals, the delta function is a simple waveform, and has an
equally simple Fourier ... cycles,
the result will be zero.
The sinc function is widely used in DSP because it is the Fouriertransform pair
of a very simple waveform, the rectangular pulse. For example, the sinc
function ... sinc function. Forsin(x)/x
continuous signals, the rectangular pulse and the sinc function are Fourier
transform pairs. For discrete signals this is only an approximation, with the
error being due...
... with zeros to make it an
Chapter 12- The Fast FourierTransform 235
TABLE 12-4
The Fast FourierTransform in BASIC.
1000 'THE FAST FOURIER TRANSFORM
1010 'Upon entry, N% contains the ... the far right
column in Fig. 12-3).
225
CHAPTER
12
The Fast Fourier Transform
There are several ways to calculate the Discrete FourierTransform (DFT), such as solving
simultaneous linear equations ... the
improvements are only about 20-40%. In one of these methods, the time
Chapter 12- The Fast FourierTransform 233
5000 'COMPLEX DFT BY CORRELATION
5010 'Upon entry, N% contains the number...
... the complex Fourier transform, they are
introduced by the real Fourier transform. In the world of mathematics, the
complex Fouriertransform is a greater truth than the real Fourier transform.
This ... types of Fourier transforms: the
Discrete FourierTransform (discrete, periodic), the Discrete Time
Fourier Transform (discrete, aperiodic), the Fourier Series (continuous,
periodic), and the Fourier ... Laplace and z-transforms. These complex transforms
are the foundation of theoretical DSP.
The Real DFT
All four members of the Fouriertransform family (DFT, DTFT, Fourier
Transform & Fourier Series)...
... Trong miền tần số: phương pháp Fourier Transform.
- Trong miền thời gian và tần số: STFT( Short Time Fourier Transform) .
2.4.1. Phương pháp Fourier:
2.4.1.1. Biến đổi Fourier:
Cho một hàm f(t) ... kết thúc của sự
kiện thì Fourier không phát hiện được.
2.4.2. Phương pháp STFT:
Để đạt được một biến đổi Fourier cục bộ, chúng ta có thể định nghĩa một biến đổi
Fourier cửa sổ. Tín hiệu đầu ... đổi Fourier của nó được định nghĩa:
() () ()
tfedtetfF
titj
,
ωω
ω ==
∫
∞
∞−
−
(2.14)
Biến đổi Fourier ngược là:
() ()
∫
∞
∞−
= ωω
π
ω
deFtf
tj
2
1
(2.15)
Giả sử rằng biến đổi Fourier...
... Circular Convolution
The Fourier transform, the Laplace transform, and the z -transform of the linear con-
volution of two time functions are simply the products of the transforms of the
individual ... frequencies
f
k
kf
s
N
, k 0, 1, , N À 1: 7:3:1
322
FAST FOURIERTRANSFORM AND ITS APPLICATIONS
DFT and z -transform
Consider a sequence x(n) having the z -transform X(z) with an ROC that includes
the unit ... discrete-time Fouriertransform X! at the N equally
spaced frequencies !
k
2pk=N, k 0, 1, , N À 1. If the sequence x(n) has a finite
duration of length N, the DFT of a sequence yields its z-transform...
... waw
w
j++
t
e
a-
22
2
wa
a
+
)2/(
22
s
t
e
-
2/
22
2
ws
ps
-
e
t
etu
a-
)(
wa
j
+
1
t
tetu
a-
)(
2
)(
1
wa
j
+
Ø Trigonometric Fourier Series
()
å
¥
=
++=
1
000
)sin()cos()(
n
nn
ntbntaatf
ww
where
ò
òò
=
==
T
n
T
T
n
dtnttf
T
b
dtnttf
T
adttf
T
a
0
0
0
0
0
0
)sin()(
2
... Series
()
å
¥
=
++=
1
000
)sin()cos()(
n
nn
ntbntaatf
ww
where
ò
òò
=
==
T
n
T
T
n
dtnttf
T
b
dtnttf
T
adttf
T
a
0
0
0
0
0
0
)sin()(
2
and, )cos()(
2
, )(
1
w
w
Ø Complex Exponential Fourier Series
ò
å
-
¥
-¥=
==
T
ntj
n
n
ntj
n
dtetf
T
FeFtf
0
0
)(
1
where, )(
w
w
Signals &...
... k=1:6*nt
index=mod(k,nt)+1;
set(h,’cdata’,real(pp(:,:,index)));
drawnow;
end
22
CHAPTER 2. DISCRETE FOURIER TRANSFORM
Likewise, the transform of the imaginary even function is the imaginary even
function . Finally, the transform of the imaginary odd function ... i.e.,
the function has the form . On transformation of , each of the
four individual parts transforms according to the table.
Most “industry standard” methods of Fouriertransform set the zero frequency ... then
it is useful to Fouriertransform (FT) that variable.
In seismology, the earth does not change with time (the ocean does!) so for the earth, we
can generally gain by Fourier transforming the...
... This result shows that multiplyingthe Fourier transform
of one function by the complex conjugate of the Fouriertransform of the other gives
the Fouriertransform of their correlation. The correlation ... a simple transform pair
g ∗ h ⇐⇒ G ( f ) H ( f ) “ConvolutionTheorem” (12.0.9)
In other words, the Fouriertransform of the convolution is just the product of the
individual Fourier transforms.
The ... (12.0.1) it is evident at once that Fourier transformation is a
linear operation. The transform of the sum of two functions is equal to the sum of
the transforms. The transform of a constant times...
... showed
that a discrete Fouriertransform of length N can be rewritten as the sum of two
discrete Fourier transforms, each of length N/2. One of the two is formed from the
12.1 FourierTransform of Discretely ... Fourier Transform
(Englewood Cliffs, NJ: Prentice-Hall).
Elliott, D.F., and Rao, K.R. 1982,
Fast Transforms: Algorithms, Analyses, Applications
(New
York: Academic Press).
12.2 Fast FourierTransform ... that a
routine for calculating discrete Fourier transforms can also, with slight modification,
calculate the inverse transforms.
502
Chapter 12. Fast Fourier Transform
Sample page from NUMERICAL...
... a slow
Fourier transform, of order N
2
instead of order N log
2
N. Our advice is to stay clear
of such FFT implementations, with perhaps one class of exceptions, the Winograd
Fourier transform ... in these
transforms, and the fact that the Winograd transform cannot be done “in place.”
Finally, an interesting class of transforms for doing convolutions quickly are
number theoretic transforms. ... isign)
Replaces
data[1 2*nn] by its discrete Fourier transform, if isign is input as 1; or replaces
data[1 2*nn] by nn times its inverse discrete Fourier transform, if isign is input as −1.
data...
... functionsused by the Fourier transform( a), sine transform( b), and cosine transform
(c), are plotted. The first five basis functions are shown in eachcase. (For the Fourier transform, the real
and ... stepsanalogous
to those leading up to equation (12.3.11), you will find that the Fouriertransform is
520
Chapter 12. Fast Fourier Transform
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 1982,
Fast Transforms: Algorithms, Analyses, Applications
(New
York: Academic Press).
Brigham, E.O. 1974,
The Fast Fourier Transform
(Englewood Cliffs, NJ: Prentice-Hall). [1]
Bloomfield, P. 1976,
Fourier...
... REFERENCES AND FURTHER READING:
Nussbaumer, H.J. 1982,
Fast FourierTransform and Convolution Algorithms
(New York: Springer-
Verlag).
12.5 Fourier Transforms of Real Data in Two
and Three Dimensions
Two-dimensional ... rows. If
isign is input as −1, data is replaced by its inverse
transform times the product of the lengths of all dimensions.
12.5 Fourier Transforms of Real Data in Two and Three Dimensions
525
Sample ... the full transform can be triviallyobtained
from the subset of lattice values n that have
0 ≤ n
1
≤ N
1
− 1
0 ≤ n
2
≤ N
2
− 1
···
0≤ n
L
≤
N
L
2
(12.5.3)
524
Chapter 12. Fast Fourier Transform
Sample...