Signals and Systems Chapter 2
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Transcript of Signals and Systems Chapter 2
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Signals and SystemsChapter 2
Biomedical EngineeringDr. Mohamed Bingabr
University of Central Oklahoma
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Outline
β’ Signalsβ’ Systemsβ’ The Fourier Transformβ’ Properties of the Fourier Transformβ’ Transfer Functionβ’ Circular Symmetry and the Hankel Transform
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Introduction
Signal Type- Continuous Signal: x-ray attenuation- Discrete Signal: times of arrival of photons in a
radioactive decay process in PET- Mixed signal: CT scan signal g(l,ΞΈk)
System Type- Continuous-continuous system
- Continuous input Continuous output- Continuous-discrete system
- Continuous input Discrete output
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Signals
function
image
(x,y) : is a pixel locationf : is pixel intensity
2-D continuous signal is defined as f(x,y)
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Point Impulse
1-D point impulse (delta, Dirac, impulse function)
πΏ (π₯ )=0 ,π₯β 0 ,
β«β β
β
π (π₯)πΏ (π₯ )ππ₯= π (0 ) .
2-D point impulseπΏ (π₯ , π¦ )=0 ,(π₯ , π¦ )β (0 , 0)
β«β β
β
β«β β
β
π (π₯ , π¦)πΏ (π₯ , π¦ )ππ₯ππ¦= π (0 , 0 ) .
Point impulse is used in the characterization of image resolution and sampling
πΏ (π₯ )
π₯
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Point Impulse Properties
1- Sifting property
β«β β
β
β«β β
β
π (π₯ , π¦)πΏ (π₯βπ , π¦βπ )ππ₯ππ¦= π (π ,π ) .
We can interpret the product of a function with a point impulse as another point impulse whose volume is equal to the value of the function at the location of the point impulse.
2- Scaling property πΏ (ππ₯ ,ππ¦ )= 1|ππ|
πΏ(π₯ , π¦ )
2- Even function πΏ (βπ₯ ,β π¦ )=πΏ(π₯ , π¦ )
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Line Impulse
This is a line whose unite normal is oriented at an angle ΞΈ relative to the x-axis and is at distance l from the origin in the direction of the unit normal.
The line impulse associated with line
πΏπ (π₯ , π¦ )=πΏπ (π₯πππ π ,+π¦ π πππβ π )
Line also used to assist image resolution
πΏ (π ,π )={(π₯ , π¦ )β¨π₯πππ π+π¦π πππ=π}
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Comb and Sampling Functions
2-D comb function
πΏπ (π₯ , π¦ ;β π₯ , β π¦ )= βπ=β β
β
βπ=β β
β
πΏ(π₯βπβ π₯ , π¦βπβ π¦ )
Used in medical imaging production (sampling CT image 1024 x 1024), manipulation, and storage.
ππππ(π₯)=βββ
β
πΏ(π₯βπ)
ππππ(π₯ , π¦)= βπ=β β
β
βπ=β β
β
πΏ(π₯βπ , π¦βπ)
Sampling function
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1-D Rect and Sinc Functions
Rect function is used in medical imaging for sectioning.
ππππ‘ (π₯ )={ 1 , for | π₯β¨ΒΏ12
0 , for |π₯β¨ΒΏ 12
π πππ (π₯ )= π πππ π₯π π₯
Sinc function is used in medical imaging for reconstruction.
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2-D Rect and Sinc Functions
ππππ‘ (π₯ , π¦ )={ 1 , for | π₯β¨ΒΏ12
andβ¨π¦β¨ΒΏ12
0 , for |π₯β¨ΒΏ 12
and|π¦|> 12
ππππ‘ (π₯ , π¦ )=ππππ‘ (π₯)ππππ‘(π¦ )
π πππ(π₯ , π¦ )={ 1 , for π₯=π¦=0sin (π π₯ ) sin (π π¦ )
π 2π₯π¦, otherwise .
π πππ (π₯ , π¦ )=π πππ (π₯ )π πππ (π¦ )
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Exponential and Sinusoidal Signals
π(π₯ , π¦ )=π π2 π (π’ 0π₯+π£0π¦ )
π (π₯ , π¦ )=πππ [ 2π (π’0 π₯+π£0 π¦ ) ]+ ππ ππ [2π (π’0π₯+π£0 π¦ ) ]
x and y have distance units.
u0 and v0 are the fundamental frequencies and their units are the inverse of the units of x and y.
πππ [2π (π’0π₯+π£0 π¦ ) ]=0.5π π2 π (π’ 0π₯+π£0 π¦ )+0.5πβ π2 π (π’0π₯+π£0 π¦ )
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Separable and Periodic Signals
β’ A signal f(x, y) is separable if f(x, y)= f1(x) f2(y)
β’ Separable signal model signal variations independently in the x and y direction.
β’ Decomposing a signal to its components f1(x) and f2(y) might simplify signal processing.
Periodicity
A signal f(x, y) is periodic if f(x, y)= f(x+X, y) = f(x, y+Y)
X and Y are the signal periods in the x and y direction, respectively.
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Systems
A continuous system is defined as a transformer Ο¨ of an input continuous signal f(x,y) to an output continuous signal g(x,y).
Linear Systems
g(x, y)= Ο¨ [f(x, y)]
Ο¨ [βπ=1
πΎ
π€π π π(π₯ , π¦)]=βπ=1
πΎ
π€πΟ¨ [ π π(π₯ , π¦) ]
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Impulse Response
If we know the system response to an impulse
then with linearity we can know the system response to any input.
h (π₯ , π¦ ; π ,π )=Ο¨ [πΏππ (π₯ , π¦ ) ]
is the system impulse response function or known as point spread function (PSF).
System output g() for any input f().
π (π₯ , π¦ )=β«β β
β
β«β β
β
π (π ,π ) h (π₯ , π¦ ;π ,π )ππ ππ
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Impulse Response
System output g() for any input f().
π (π₯ , π¦ )=β«β β
β
β«β β
β
π (π ,π ) h (π₯ , π¦ ;π ,π )ππ ππ
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Shift Invariance System
A system is shift invariant if an arbitrary translation of the input results in an identical translation in the output.Then with linearity we can know the system response to any input.
π π₯0 π¦ 0(π₯ , π¦ )= π (π₯βπ₯0 , π¦β π¦0 )Let the input
then the output
Ο¨ []= h()
System response to a shifted impulse
g()= Ο¨ []
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Linear Shift-Invariance (LSI) System
Linear shift-invariant (LSI) System Response
π (π₯ , π¦ )=β«β β
β
β«β β
β
π (π ,π ) h (π₯βπ , π¦βπ ) ππ ππ
Convolution Integral representation of system response
π (π₯ , π¦ )=h (π₯ , π¦ )β π (π₯ , π¦)
Example: Consider a continuous system with input-output equation g(x,y) = xyf(x,y). Is the system linear and shift-invariant?
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Connection of LSI Systems
π (π₯ , π¦ )=h1 (π₯ , π¦ )βh2 (π₯ , π¦ )β π (π₯ , π¦ )
π (π₯ , π¦ )=[h1 (π₯ , π¦ )+h2 (π₯ , π¦ )]β π (π₯ , π¦)
Cascade
Parallel
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Connection of LSI Systems
Example: Consider two LSI systems connected in cascade, with Gaussian PSFs of the form:
h1 (π₯ , π¦ )= 1
2π π12 π
β (π₯2+π¦2 ) /2π 12
h2 (π₯ , π¦ )= 1
2π π 22 π
β (π₯2+π¦2 ) /2π22
where Ο1 and Ο2 are two positive constants.What is the PSF of the system?
h (π₯ , π¦ )= 1
2π (π12+π2
2 )πβ (π₯2+π¦2 ) /2 (π1
2+π22 )
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Separable Systems
A 2-D LSI system with PSF h(x, y) is a separable system if there are two 1-D systems with PSFs h1(x) and h2(y), such that h(x,y) = h1(x)h2(y)
h (π₯ , π¦ )= 1
2π π 2 πβ (π₯2+π¦ 2) /2π2
h1 (π₯ )= 1
β2π ππβπ₯2 /2π 2
This PSF is separable
h2 (π¦ )= 1
β2π ππβ π¦2 /2π 2
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Separable Systems
In practice it is easier and faster to execute two consecutive 1-D operations than a single 2-D operation.
π€ (π₯ , π¦ )=β«β β
β
π (π , π¦ ) h1 (π₯βπ )ππ
g (π₯ , π¦ )=β«β β
β
π€ (π₯ ,π ) h2 (π¦βπ )ππ
For every y
For every x
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Stable Systems
A system is a bounded-input bounded-output (BIBO) stable system if
For bounded input
for every (x, y)
The output is bounded
and
β«β β
β
β«β β
β
ΒΏ h (π₯ , π¦ )β¨ππ₯ππ¦<β
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1-D Fourier Transform (time)
β«
dtetxfX ftj 2)()2(
1-N
0n
2
)()(n
N
kj
enxkX
Continuous 1-D Fourier Transform
Discrete 1-D Fourier Transform
x(n) = [125 145 148 140 110]X(k) = [668 -29.2 - j38 7.7 - j12.96 7.7 - j12.96 -29.2 - j38]
|X(k)| = [668 47.9 15.1 15.1 47.9] Phase = [0 -127.5 -59.3 59.3 127.5]
Ts = 0.25 secfs = 1/Ts = 4 Hzfmax = fs/2 = 2 Hz Sig length (T) = (N-1)*Ts=1 sec fres = 1/T = 1 Hz
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1-D Fourier Transform
1-D inverse Fourier transform
πΉ (π’ )=β±1π· ( π ) (π’)=β«β β
β
π (π₯ )πβ π2 ππ’π₯ππ₯
π (π₯ )=β± 1π·β1 (πΉ ) (π₯ )=β«
β β
β
πΉ (π’ )π π 2ππ’π₯ππ’
1-D Fourier transform
Example:
What is the Fourier transform of the ππππ‘ (π₯ )={ 1 , for | π₯β¨ΒΏ
12
0 , for |π₯β¨ΒΏ 12
u is the spatial frequency
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Fourier Transform
The 2-D Fourier transform of f(x, y)
u and v are the spatial frequencies
πΉ (π’ ,π£ )=β± 2π· ( π ) (π’ ,π£ )=β«β β
β
β«β β
β
π (π₯ , π¦ )πβ π 2π (π’π₯+π£π¦ )ππ₯ππ¦
The 2-D inverse Fourier transform of F(u, v)
π (π₯ , π¦)=β«β β
β
β«β β
β
πΉ (π’ ,π£ )π π2π (π’π₯+π£π¦ )ππ’ππ£
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Fourier Transform
Magnitude (magnitude spectrum) of FT
|πΉ (π’ ,π£ )|=βπΉπ 2 (π’ ,π£ )+πΉ πΌ
2 (π’ ,π£ )
β πΉ (π’ ,π£ )=π‘ππβ1( πΉ πΌ(π’ ,π£ )πΉπ (π’ ,π£))
Angle (phase spectrum) of the FT
πΉ (π’ ,π£)=|πΉ (π’ ,π£)|π π β πΉ (π’ ,π£ )
Example: What is the Fourier transform of the point impulse ?
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Fourier Transform Pairs
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Examples of Fourier Transform
Example: What is the Fourier transform of
Answer:β± 2π· ( π ) (π’ ,π£ )=πΏ (π’βπ’0 ,π£βπ£0 )
If the spatial frequency u0 and v0 are zero then f(x,y) =1 and the spectrum F(u,v) will be .
Slow signal variation in space produces a spectral content that is primarily concentrated at low frequencies.
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Examples of Fourier Transform
Three images of decreasing spatial variation (from left to right) and the associated magnitude spectra [depicted as log(1 + |F(u, Ο )|)].
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Examples of Fourier Transform
>> img1 = imread('\\PHYSICSSERVER\MBingabr\BiomedicalImaging\mri.tif');
>> imshow(img1)
>> size(img1)
ans = 256 256
>> FFT_img1 = fftshift(fft2(img1));
>> Abs_FFT_img1 = abs(FFT_img1)
>> surf(Abs_FFT_img1(110:140,110:140))
>> Log_Abs_FFT_img1=log10(1+Abs_FFT_img1);
>> surf(Log_Abs_FFT_img1(110:140,110:140))
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Properties of the Fourier Transform
Linearity
Properties are used in theory and application to simplify calculation.
β± 2π· (π1 π +π2π ) (π’ ,π£ )=π1πΉ (π’ ,π£ )+π2πΊ(π’ ,π£ )
Translation
If F(u,v) is the FT of a signal f(x, y) then the FT of a translated signal
β± 2π· ( π π₯0 π¦0 ) (π’ ,π£ )=πΉ (π’ ,π£ )πβ π2 π(π’ π₯0+π£ π¦0 )
π π₯0 π¦ 0(π₯ , π¦ )= π (π₯βπ₯0 , π¦β π¦0 ) is
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Properties of the Fourier Transform
Conjugation and Conjugate Symmetry
If F(u,v) is the FT of a signal f(x, y) then
πΉ (π’ ,π£)=πΉβ(βπ’ ,βπ£)
πΉπ (π’ ,π£)=πΉπ (βπ’ , βπ£ ) πΉ πΌ (π’ ,π£)=βπΉ πΌ (βπ’ ,βπ£)
ΒΏπΉ (π’ ,π£ )β¨ΒΏβ¨πΉ (βπ’ ,βπ£ )β¨ΒΏ
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Properties of the Fourier Transform
Scaling
If F(u,v) is the FT of a signal f(x, y) and if
π ππ (π₯ , π¦ )= π (ππ₯ ,ππ¦ )
β± 2π· ( π ππ ) (π’ ,π£ )= 1
ΒΏππβ¨ΒΏπΉ (π’π ,π£π )ΒΏ
Example
Detectors of many medical imaging systems can be modeled as rect functions of different sizes and locations. Compute the FT of the following
π (π₯ , π¦ )=ππππ‘( π₯βπ₯0
β π₯,π¦β π¦0
β π¦ )
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Properties of the Fourier Transform
Rotation
If F(u,v) is the FT of a signal f(x, y) and if
π π (π₯ , π¦ )= π (π₯πππ πβ π¦π πππ ,π₯π πππ+π¦πππ π )
β± 2π· ( π π ) (π’ ,π£ )=πΉ (π’πππ πβπ£ π πππ ,π’π πππ+π£ πππ π )
If f(x, y) is rotated by an angle , then its FT is rotated by the same angle.
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Properties of the Fourier Transform
ConvolutionThe Fourier transform of the convolution f(x, y) * g(x, y) is
β± 2π· ( π βπ) (π’ ,π£ )=πΉ (π’ ,π£ )πΊ (π’ ,π£ )
Example:
Convolution property simplify the difficult task of calculating the convolution in the spatial domain to multiplication in the frequency domain.
π (π₯ , π¦ )=π πππ (ππ₯ ,ππ¦ ) π (π₯ , π¦ )=π πππ (ππ₯ ,ππ¦ )Find Fourier transform of the convolution f(x, y) * g(x, y)
0 < V U
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Properties of the Fourier Transform
ProductThe Fourier transform of the product f(x, y) g(x, y) is the convolution of their Fourier transforms.
β± 2π· ( π π ) (π’ ,π£ )=πΉ (π’ ,π£ )βπΊ (π’ ,π£ )
Separable Product
If f(x, y)=f1(x)f2(y) then
where
ΒΏβ«β β
β
β«β β
β
πΊ (π ,π )πΉ (π’βπ ,π£βπ )ππ ππ
πΉ 1 (π’)=β± 1π· ( π 1 ) (π’ )
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Separability of the Fourier TransformThe Fourier transform F(u,v) of a 2-D signal f(x, y) can be calculated using two simpler 1-D Fourier transforms, as follows:
π (π’ , π¦ )=β«ββ
β
π (π₯ , π¦ )πβ π2 ππ’π₯ππ₯ For every y.1)
πΉ (π’ ,π£ )=β«ββ
β
π (π’ , π¦ )πβ π2 ππ£ π¦ππ¦ For every x.2)
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Transfer FunctionThe systemβs transfer function (frequency response) H(u, v) is the Fourier transform of the systemβs PSF h(x,y).
π» (π’ ,π£ )=β«β β
β
β«β β
β
h (π₯ , π¦ )πβ π2 π(π’π₯+π£π¦ )ππ₯ ππ¦
The inverse Fourier transform of the transfer function H(u, v) is the point spread function h(x,y).
h (π₯ , π¦ )=β«β β
β
β«β β
β
π» (π’ ,π£ )πβ π2 π(π’π₯+π£π¦ )ππ’ππ£
The output G(u, v) of a system in response to input F(u, v) is the product of the input with the transfer function H(u, v) .
πΊ (π’ ,π£ )=π» (π’ ,π£ )πΉ (π’ ,π£ )
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Transfer Function
Example: Consider an idealized system whose PSF is h(x,y) = (x-x0, y-y0). What is the transfer function H(u, v) of the system, and what is the system output g(x, y) to an input signal f(x, y).
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Low Pass Filter
π» (π’ ,π£ )={ 1 , for βπ’2+π£2β€π
0 , for βπ’2+π£2>π
πΊ (π’ ,π£)={πΉ (π’ ,π£ ) for βπ’2+π£2β€π
0 , for βπ’2+π£2>π
c1 > c2
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Circular Symmetry
Often, the performance of a medical imaging system does not depend on the orientation of the patient with respect to the system. The independence arises from the circular symmetry of the PSF.
A 2-D signal f(x, y) is circularly symmetric if
fΞΈ(x, y) = f(x, y) for every ΞΈ.
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Property of Circular Symmetry
β’ f(x, y) is even in both x and y
β’ F(u, v) is even in both u and v
β’ | F(u, v) | = F(u, v)
β’ F(u, v) = 0
β’ f(x, y) = f(r) where
β’ F(u, v) = F(q) where
π=βπ₯2+π¦2
π=βπ’2+π£2
f(r) and F(q) are one dimensional signals representing two dimensional signals
![Page 44: Signals and Systems Chapter 2](https://reader035.fdocuments.in/reader035/viewer/2022062309/56813b45550346895da422d8/html5/thumbnails/44.jpg)
Hankel Transform
The relationship between f(r) and F(q) is determined by Hankel Transform.
πΉ (π )=2πβ«0
β
π (π ) π½ π (2πππ )πππ
where J0(r) is the zero-order Bessel function of the first kind.
π½π (π )= 1πβ«0
π
cos (ππβππ πππ ) ππ
π½ 0 (π )= 1πβ«0
π
cos (ππ πππ )ππ
The nth-order Bessel function
for n = 0, 1, 2, β¦
πΉ (π )=β { π (π )}
![Page 45: Signals and Systems Chapter 2](https://reader035.fdocuments.in/reader035/viewer/2022062309/56813b45550346895da422d8/html5/thumbnails/45.jpg)
Hankel Transform
The inverse Hankel transform.
π (π )=2πβ«0
β
πΉ (π) π½ π (2πππ )πππ
unit disk jink function
![Page 46: Signals and Systems Chapter 2](https://reader035.fdocuments.in/reader035/viewer/2022062309/56813b45550346895da422d8/html5/thumbnails/46.jpg)
sinc and jink functions
ExampleIn some medical imaging systems, only spatial frequencies smaller than q0 can be imaged. What is the function having uniform spatial frequencies within the desk of radius q0 and what is its inverse Fourier transform.