Associate Professor Ivan KOTLIAROV, PhD National Research ...
8 pages
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Associate Professor Ivan KOTLIAROV, PhD National Research ...

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Associate Professor Ivan KOTLIAROV, PhD National Research University Higher School of Economics, St. Petersburg Branch Russia E-mail: HOW MUCH SHOULD A FRANCHISEE PAY? A NEW MODEL OF CALCULATION OF ROYALTIES Abstracts: Existing algorithm of payment for external intellectual property is analyzed. It is demonstrated that this algorithm cannot be universal. Different models of royalty rate calculation in case of franchising are proposed and discussed.
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  • correct sharing of additional income between franchisor
  • franchisee
  • uslic lic lic
  • karpova n. n.
  • royalty rate
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  • calculation
  • g.g.
  • g. g.
  • income

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Publié par
Nombre de lectures 24
Langue English
Cuff (Lecture 9)
Lecture 9 ELE 301: Signals and Systems
Prof. Paul Cuff
Princeton University
Fall 2011-12
ELE 301: Signals and Systems
Discrete-time Fourier Transform
Represent a Discrete-time signal usingδfunctions
Properties of the Discrete-time Fourier Transform Periodicity I Time Scaling Property I Multiplication Property I
Periodic Discrete Duality
DFT
Constant-Coefficient Difference Equations
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
Fall 2011-12
1 / 16
2 / 16
Fourier Transform for Discrete-time Signals
x[n]
X(f)
=
=
Z i2πfn X(f)e df, 1 X i2πfn x[n]e. n=−∞
X(f) is always periodic with period 1.
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
Continuous-representation of a discrete-time signal
Notice that
F[x(t)]
Cuff (Lecture 9)
x(t)
=
=
=
,
X x[k]δ(tk). k=−∞
 ! ZX i2πft x[k]δ(tk)e dt −∞ k=−∞ Z X i2πft x[k]δ(tk)e dt −∞ k=−∞ X i2πfk x[k]e dt. k=−∞
ELE 301: Signals and Systems
Fall 2011-12
3 / 16
4 / 16
Properties of the Discrete-time Fourier Transform
Inherits properties from continuous-time. Easy Properties:
Linearity Conjugation Convolution = Multiplication in frequency domain Parseval’s Theorem (integrate over one period) Time shift
Properties that require care:
Time-scaling Multiplication (circular convolution in frequency)
Cuff (Lecture 9)
Time-scaling
ELE 301: Signals and Systems
Fall 2011-12
In continuous time we can scale by an arbitrary real number. In discrete-time we scale only by integers. For an integerk, define x[n/k] ifnis a multiple ofk, xk[n] = 0 ifnis not a multiple ofk.
Cuff (Lecture 9)
xk[n]X(kf).
ELE 301: Signals and Systems
Fall 2011-12
5 / 16
6 / 16
Compressing in time
Compressing in time requires decimation.
Cuff (Lecture 9)
Discrete Difference
ELE 301: Signals and Systems
What is the Fourier transform ofy[n] =x[n]x[n1]?
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
Fall 2011-12
7 / 16
8 / 16
Dual Derivative Formula
The dual to the continuous-time differentiation formula still holds.
Cuff (Lecture 9)
Accumulation
nx[n]
i 0 X(f). 2π
ELE 301: Signals and Systems
P n What is the Fourier transform ofy[n] =x[m]? m=−∞ (Hint: Inverse of discrete difference)
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
Fall 2011-12
9 / 16
10 / 16
Parseval’s Theorem
Theorem:
Cuff (Lecture 9)
Z 1/2 X 2 2 Ex=|x[n]|=|X(f)|df. 1/2 n=−∞
ELE 301: Signals and Systems
Multiplication Property
Fall 2011-12
Multiplication in time equates tocircular convolution in frequency.
Notice that multiplyingδfunctions is not well defined.
This is dual to what we saw in the Fourier series.
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
11 / 16
12 / 16
Discrete Periodic Duality
periodic signal = discrete transform (though not integerf) discrete signal = periodic transform
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
13 / 16
Discrete Fourier Transform Notice that a discrete and periodic signal will have a discrete and periodic transform. This is convenient for numerical computation (computers and digital systems).
TheDFT is (almost) equivalent to the discrete-time Fourier series of the periodic extension. For periodN, let   x[0] x[1] x=  . x[N1]
Then
Cuff (Lecture 9)
  a0 a1 DFT[x] =    . aN1 ELE 301: Signals and Systems
Fall 2011-12
14 / 16
The DFT Matrix
Since the Fourier transform is linear, the DFT can be encompassed in a matrix.
DFT[x] =Fx.
Matlab uses the fast-Fourier-transform algorithm to compute the DFT (using thefftcommand).
Cuff (Lecture 9)
ELE 301: Signals and Systems
Constant-Coefficient Difference Equations
n M X X aky[nk] =bkx[nk]. k=0k=0
Fall 2011-12
15 / 16
Find the Fourier Transform of the impulse response (the transfer function of the system,H(f)) in the frequency domain.
Cuff (Lecture 9)
ELE 301: Signals and Systems
Fall 2011-12
16 / 16