discrete fourier transform scaling property proof
The Discrete Fourier Transform
Ngày tải lên :
13/09/2012, 09:49
... decomposed.
Chapter 8- The Discrete Fourier Transform 145
Type of Transform Example Signal
Fourier Transform
Fourier Series
Discrete Time Fourier Transform
Discrete Fourier Transform
signals that ... 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 ... ]
141
CHAPTER
8
The Discrete Fourier Transform
Fourier analysis is a family of mathematical techniques, all based on decomposing signals into
sinusoids. The discrete Fourier transform (DFT) is...
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Fourier Transform Properties
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13/09/2012, 09:49
... frequency
domain multiplexing.
The Discrete Time Fourier Transform
The Discrete Time Fourier Transform (DTFT) is the member of the Fourier
transform family that operates on aperiodic, discrete signals. The ... the Fourier Transform
The Fourier Transform is linear, that is, it possesses the properties of
homogeneity and additivity. This is true for all four members of the Fourier
transform family (Fourier ... 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 (a) shows an arbitrary...
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Fourier Transform Pairs
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13/09/2012, 09:49
... called Fourier transform 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 ... one domain is discrete, the other domain must
be periodic, and vice versa. This holds for all four members of the Fourier
transform family. Since the DFT views both domains as discrete, it must ... 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 to aliasing....
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The Fast Fourier Transform
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13/09/2012, 09:50
... with zeros to make it an
Chapter 12- The Fast Fourier Transform 235
TABLE 12-4
The Fast Fourier Transform 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 Fourier Transform (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 Fourier Transform 233
5000 'COMPLEX DFT BY CORRELATION
5010 'Upon entry, N% contains the number...
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The Complex Fourier Transform
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13/09/2012, 10:20
... continuous or discrete, and it can be either
periodic or aperiodic. This defines four types of Fourier transforms: the
Discrete Fourier Transform (discrete, periodic), the Discrete Time
Fourier Transform ... the complex Fourier transform, they are
introduced by the real Fourier transform. In the world of mathematics, the
complex Fourier transform is a greater truth than the real Fourier transform.
This ... Laplace and z-transforms. These complex transforms
are the foundation of theoretical DSP.
The Real DFT
All four members of the Fourier transform family (DFT, DTFT, Fourier
Transform & Fourier Series)...
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Các phương pháp xử lý tín hiệu dùng bộ lọc, phương pháp Fourier Transform, STFT( Short Time Fourier Transform)
Ngày tải lên :
01/04/2013, 16:14
... 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...
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Real-Time Digital Signal Processing - Chapter 7: Fast Fourier Transform and Its Applications
Ngày tải lên :
28/10/2013, 05:15
... 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 FOURIER TRANSFORM 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 ... increment
f
s
=N, and is sometimes referred to as the bin spacing of the DFT outputs. The spacing
DISCRETE FOURIER TRANSFORM
307
± Segment the input sequence of length M with L À 1zeros padded at the end.
±...
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Tài liệu Table of Fourier Transform Pairs ppt
Ngày tải lên :
13/12/2013, 12:15
... 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 &...
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Tài liệu Fast Fourier Transform part 1 doc
Ngày tải lên :
21/01/2014, 18:20
... This result shows that multiplyingthe Fourier transform
of one function by the complex conjugate of the Fourier transform of the other gives
the Fourier transform of their correlation. The correlation ... a simple transform pair
g ∗ h ⇐⇒ G ( f ) H ( f ) “ConvolutionTheorem” (12.0.9)
In other words, the Fourier transform of the convolution is just the product of the
individual Fourier transforms.
The ... of the Fourier transform. (We’ll use
the “⇐⇒ ” symbol to indicate transform pairs.) If
h(t) ⇐⇒ H ( f )
is such a pair, then other transform pairs are
h(at) ⇐⇒
1
| a |
H (
f
a
) “time scaling ...
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Tài liệu Fast Fourier Transform part 2 ppt
Ngày tải lên :
21/01/2014, 18:20
... showed
that a discrete Fourier transform 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 Fourier Transform of Discretely ... generate the
required powers of W . So, the discrete Fourier transform appears to be an O(N
2
)
process. These appearances are deceiving! The discrete Fourier transform can,
in fact, be computed in ... equation (12.1.6) is called the discrete Fourier transform of the N
points h
k
. Let us denote it by H
n
,
H
n
≡
N−1
k=0
h
k
e
2πikn/N
(12.1.7)
The discrete Fourier transform maps N complex numbers...
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Tài liệu Fast Fourier Transform part 3 docx
Ngày tải lên :
21/01/2014, 18:20
... 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 ... 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. ... 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...
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Tài liệu Fast Fourier Transform part 4 pdf
Ngày tải lên :
21/01/2014, 18:20
... 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 ... the components of the
discrete Fourier transform satisfy
F
N−n
=(F
n
)* (12.3.1)
where the asterisk denotes complex conjugation. By the same token, the discrete
Fourier transform of a purely ... 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...
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Tài liệu Fast Fourier Transform part 5 pptx
Ngày tải lên :
21/01/2014, 18:20
... REFERENCES AND FURTHER READING:
Nussbaumer, H.J. 1982,
Fast Fourier Transform and Convolution Algorithms
(New York: Springer-
Verlag).
12.5 Fourier Transforms of Real Data in Two
and Three Dimensions
Two-dimensional ... data[], unsigned long nn[], int ndim, int isign)
Replaces
data by its ndim-dimensional discrete Fourier transform, if isign is input as 1.
nn[1 ndim] is an integer array containing the lengths ... 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...
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Tài liệu Fast Fourier Transform part 6 pdf
Ngày tải lên :
21/01/2014, 18:20
... 1974,
The Fast Fourier Transform
(Englewood Cliffs, NJ: Prentice-Hall).
Swartztrauber, P. N. 1986,
Mathematics of Computation
, vol. 47, pp. 323–346.
530
Chapter 12. Fast Fourier Transform
Sample ... REFERENCES AND FURTHER READING:
Nussbaumer, H.J. 1982,
Fast Fourier Transform and Convolution Algorithms
(New York: Springer-
Verlag).
12.5 Fourier Transforms of Real Data in Two
and Three Dimensions
Two-dimensional ... (12.3.5) implemented on the last transform index. The case of
i3=1 is coded separately, to account for the fact that speq is to be filled instead of
12.5 Fourier Transforms of Real Data in Two and...
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