Associate Professor Ivan KOTLIAROV, PhD National Research ...

Prof. Paul Cuff

Description

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.

• supsup sup
• correct sharing of additional income between franchisor
• franchisee
• uslic lic lic
• karpova n. n.
• royalty rate
• supsupsupsup supsupsupsup pwpwpwpw pdwpcwpbwpaw
• calculation
• g.g.
• g. g.
• income

Sujets

• Journal
• McDonald's
• Royalty
• Income
• Franchise (réseau commercial)
• Calculation
• Sharing
• Independent

Informations

• Publié par: phawyir
• Nombre de lectures: 24
• Langue: English

Extrait

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(t−k). k=−∞

 ! Z ∞ X ∞ −i2πft x[k]δ(t−k)e dt −∞ k=−∞ ∞ Z ∞ X −i2πft x[k]δ(t−k)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, x k [n] = 0 ifnis not a multiple ofk.

Cuff (Lecture 9)

x k [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[n−1]?

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 E x =|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[N−1]

Then

Cuff (Lecture 9)

  a 0 a 1 DFT[x] =     . a N−1 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 a k y[n−k] =b k x[n−k]. 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