##### Document Text Contents

Page 34

Fourier Transform Pairs of Common Functions

Likewise, the Fourier transform for the shifted delta function is

(8.61)

We will use the notation to show the time domain to frequency domain correspon-

dence. Thus, (8.60) may also be denoted as in Figure 8.1.

TABLE 8.8 Fourier Transform Properties and Theorems

Property

Linearity

Symmetry

Time Scaling

Time Shifting

Frequency Shifting

Time Differentiation

Frequency Differentiation

Time Integration

Conjugate Functions

Time Convolution

Frequency Convolution

Area under

Area under

Parseval’s Theorem

f t( ) F ω( )

a1 f1 t( ) a2 f2 t( ) …+ + a1 F1 ω( ) a2 F2 ω( ) …+ +

F t( ) 2πf ω–( )

f at( ) 1

a

-----F ω

a

----

f t t0–( ) F ω( )e

jωt0–

e

jω0tf t( )

F ω ω0–( )

d

n

dt

n

--------- f t( ) jω( )

n

F ω( )

jt–( )

n

f t( ) d

n

dω

n

-----------F ω( )

f τ( ) τd

∞–

t

F ω( )

jω

------------ πF 0( )δ ω( )+

f∗ t( ) F∗ ω–( )

f1 t( )∗f2 t( ) F1 ω( ) F2 ω( )⋅

f1 t( ) f2 t( )⋅ 1

2π

------F1 ω( )∗F2 ω( )

f t( )

F 0( ) f t( ) td

∞–

∞

=

F ω( )

f 0( ) 1

2π

------ F ω( ) ωd

∞–

∞

=

f t( ) 2 td

∞–

∞

1

2π

------ F ω( ) 2 ωd

∞–

∞

=

δ t t0–( )

δ t t0–( ) e

jωt0–⇔

f t( ) F ω( )↔

Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fifth Edition 8−17

Copyright © Orchard Publications

Page 35

Chapter 8 The Fourier Transform

Figure 8.1. The Fourier transform of the delta function

8.4.2 The Constant Function Pair

(8.62)

Proof:

and (8.62) follows.

The correspondence is also shown in Figure 8.2.

Figure 8.2. The Fourier transform of constant A

Also, by direct application of the Inverse Fourier transform, or the frequency shifting property and

(8.62), we derive the transform

(8.63)

The transform pairs of (8.62) and (8.63) can also be derived from (8.60) and (8.61) by using the

symmetry property

8.4.3 The Cosine Function Pair

(8.64)

Proof:

This transform pair follows directly from (8.63). The correspondence is also shown in

Figure 8.3.

0 t

1

ω0

f t( )

δ t( )

F ω( )

A 2Aπδ ω( )⇔

F 1– 2Aπδ ω( ){ } 1

2π

------ 2Aπδ ω( )e jωt ωd

∞–

∞

A δ ω( )e

jωt ωd

∞–

∞

Ae

jωt

ω 0= A= = = =

f t( ) F ω( )↔

A

ω

0 0

t

f t( ) F ω( )

2Aπδ ω( )

e

jω0t 2πδ ω ω0–( )⇔

F t( ) 2πf ω–( )⇔

ω0tcos

1

2

--- e

jω0t e

j– ω0t+( ) πδ ω ω0–( ) πδ ω ω0+( )+⇔=

f t( ) F ω( )↔

8−18 Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fifth Edition

Copyright © Orchard Publications

Page 67

conformable for multiplication D-4 Nyquist frequency 10-13 recursive realization digital filter

congugate of D-8 see digital filter

definition of D-1 O region of

determinant D-9 convergence 9-3

minor of D-11 octave defined 11-11 divergence 9-3

non-singular D-19 odd functions 6-11, 7-333 relationship between state equations

singular D-19 odd symmetry - see Fourier and Laplace Transform 5-28

diagonal D-1 series - symmetry residue 3-3, 9-37

diagonal elements of D-1 orthogonal functions 7-2 residue MATLAB function 3-3, 3-12

elements of D-1 orthogonal vectors 5-19 residue theorem 9-19

Hermitian D-8 orthonormal basis 5-19 right shift in the discrete-time domain

identity D-6 see Z transform - properties of

inverse of D-220 P RLC band-elimination filter - see filter

left division in MATLAB D-23 RLC band-pass filter - see filter

multiplication in MATLAB A-17 parallel form realization - see digital filter roots of polynomials in MATLAB A-3

power series of 5-9 Parseval’s theorem - see roots(p) MATLAB function 3-5, A-3

scalar D-6 Fourier transform - properties of round(n) MATLAB function A-22

size of D-7 partial fraction expansion 3-1 row vector in MATLAB A-3

skew-Hermitian D-9 alternate method of 3-14 Runge-Kutta method 5-1

skew-symmetric D-8 method of clearing the fractions 3-14 running Simulink B-7

square D-1 phase angle 11-2

symmetric D-8 phase shift filter - see filter S

trace of D-2 picket-fence effect 10-14

transpose of D-7 plot MATLAB command A-9 sampling property of the delta function

triangular polar form of complex numbers C-5 see delta function

lower D-6 polar plot in MATLAB A-23 sampling theorem 10-13

upper D-7 polar(theta,r) MATLAB function A-22 sawtooth waveform - see Laplace

zero D-2 poles 3-1 transform of common waveforms

matrix left division in MATLAB - see matrix complex 3-5 sawtooth waveform - Fourier series of

matrix multiplication in MATLAB - see matrix distinct 3-2 see Fourier series of

matrix power series - see matrix multiple (repeated) 3-7 common waveforms

maximally flat filter - see filter poly MATLAB function A-4 scalar matrix - see matrix

mesh(x,y,z) MATLAB function A-15 polyder MATLAB function A-6 scaling property of the Laplace transform

meshgrid(x,y) MATLAB command A-16 polynomial construction from see Laplace transform - properties of

m-file in MATLAB A-1, A-224 known roots in MATLAB A-4 Scope block in Simulink B-12

minor of determinant - see matrix polyval MATLAB function A-5 script file in MATLAB A-2, A-24

MINVERSE Excel function D-25 pre-sampling filter 10-13 second harmonic - see Fourier series

MMULT Excel function D-25 pre-warping 11-52 harmonics of

modulated signals 8-11 proper rational function - semicolons in MATLAB A-7

multiple eigenvalues - see eigenvalues definition of 3-1 semilogx MATLAB command A-12

multiple poles - see poles properties of the DFT semilogy MATLAB command A-12

multiplication by a

n

in discrete-time domain see DFT - common properties of series form realization - see digital filter

see Z transform - properties of properties of the Fourier Transform Shannon’s sampling theorem

multiplication by e

-naT

in discrete-time see Fourier transform - properties of see sampling theorem

domain - see Z transform - properties of properties of the Laplace Transform shift of f[n] u0[n] in discrete-time domain

multiplication by n in discrete-time domain see Laplace transform - properties of see Z transform - properties of

see Z transform - properties of properties of the Z Transform sifting property of the delta function

multiplication by n

2

indiscrete-time domain see Z transform - properties of see delta function

see Z transform - properties of signal flow graph 10-22

multiplication of complex numbers C-2 Q signals described in math form 1-1

signum function - see Fourier transform

N quarter-wave symmetry - see of common functions

Fourier series - symmetry simout To Workspace block

NaN in MATLAB A-25 quit MATLAB command A-2 in Simulink B-13

natural input-output FFT algorithm simple MATLAB symbolic function 3-6

see FFT algorithm R Simulation drop menu in Simulink B-12

network transformation simulation start icon in Simulink B-12

resistive 4-1 radius of absolute convergence 9-3 Simulink icon B-7

capacitive 4-1 ramp function 1-9 Simulink Library Browser B-8

inductive 4-1 randn MATLAB function 11-65 sine function - Fourier transform of

non-recursive realization digital filter Random Source Simulink block 11-76 see Fourier transform of

see digital filter rationalization of the quotient C-4 common functions

non-singular determinant - see matrix RC high-pass filter - see filter singular determinant - see matrix

nonlinear system G-1 RC low-pass filter - see filter Sinks library in Simulink B-18

normalized cutoff frequency 11-14 real axis C-1 sinw0t u0(t) Fourier transform of - see

notch filter - see filter real number C-2 Fourier transform of common functions

N-point DFT - see DFT - definition of real(z) MATLAB function A-22 size of a matrix - see matrix

nth-order delta function - see delta function rectangular form C-5 skew-Hermitian matrix - see matrix

numerical evaluation of Fourier coefficients rectangular pulse expressed in terms skew-symmetric matrix - see matrix

see Fourier series coefficients of the unit step function 1-4 special forms of the Fourier transform

IN-4

Page 68

see Fourier transform Transfer Fcn block in Simulink 4-17 convolution in the discrete

spectrum analyzer 7-35 Transfer Fcn Direct Form II time domain 9-8

square matrix - see matrix Simulink block 11-68 final value theorem 9-9

square waveform with even symmetry - see transfer function of initial value theorem 9-9

Fourier series of common waveforms continuous-time systems 4-13 left shift 9-5

square waveform with odd symmetry - see discrete-time systems 9-35 linearity 9-3

Fourier series of common waveforms transformation between multiplication by a

n

9-6

ss2tf MATLAB function 5-31 s and z domains 9-20 multiplication by e

-naT

9-6

stability 11-13 transformation methods for mapping multiplication by n 9-6

start simulation in Simulink B-12 analog prototype filters to digital filters multiplication by n

2

9-6

state equations Impulse Invariant Method 11-50 right shift 9-4

for continuous-time systems 5-1 Step Invariant Method 11-50 shift of f[n] u0[n] 9-3

for discrete-time systems 9-40 Bilinear transformation 11-50 summation 9-7

state transition matrix 5-8 transpose of a matrix - see matrix Z Transform of discrete-time functions

state variables Tree Pane in Simulink B-7 cosine function cosnaT 9-15

for continuous-time systems 5-1 triangular waveform expressed in terms exponential sequence e

-naT

u0[n] 9-15

for discrete-time systems 9-40 of the unit step function 1-6 geometric sequence a

n

9-12

State-Space block in Simulink B-13 triplet - see delta function sine function sinnaT 9-15

state-space equations Tukey - see Cooley and Tukey unit ramp function nu0[n] 9-16

for continuous-time systems 5-1 unit step function u0[n] 9-14

for discrete-time systems 9-40 U zero matrix - see matrix

step function - see unit step function zeros 3-1, 3-2

step invariant method - see trans- unit eigenvectors 5-18 zp2tf MATLAB function 11-16

formation methods for mapping analog unit impulse function (d(t)) 1-8

prototype filters to digital filters unit ramp function (u1(t)) 1-9

stop-band filter - see filter unit step function (u0(t)) 1-2

string in MATLAB A-15 upper triangular matrix - see matrix

subplots in MATLAB A-16 using MATLAB for finding the Laplace

summation in the discrete-time Domain transforms of time functions 2-26

see Z transform - properties of using MATLAB for finding the Fourier

symmetric matrix - see matrix transforms of time function 8-31

symmetric rectangular pulse expressed

as sum of unit step functions 1-5 V

symmetric triangular waveform expressed

as sum of unit step functions 1-6 Vandermonde matrix 10-18

symmetry - see Fourier series - symmetry Vector Scope Simulink block 11-78

symmetry property of the Fourier transform

see Fourier transform - properties of W

system function - definition of 8-34

warping 11-52

T window functions

Blackman E-10

Taylor series 5-1 Fourier series method for approximating

text MATLAB command A-13 an FIR amplitude response E-15

tf2ss MATLAB function 5-33 Hamming E-8, E-30

theorems of the DFT 10-10 Hanning E-6, E-26

theorems of the Fourier Transform 8-9 Kaiser E-12, E-33

theorems of the Laplace transform 2-2 other used as MATLAB functions E-14

theorems of the Z Transform 9-3 rectangular E-2

third harmonic - see Fourier triangular E-4, E-22

series - harmonics of Window Visualization Tool in MATLAB E-4

time convolution in DFT

see DFT - common properties of X

time integration property of the Fourier

transform - see Fourier xlabel MATLAB command A-12

transform - properties of

time periodicity property of the Laplace Y

transform - see Laplace

transform - properties of ylabel MATLAB command A-12

time scaling property of the Fourier

transform - see Fourier Z

transform - properties of

time shift in DFT Z transform

see DFT - common properties of computation of with contour

time shift property of the Fourier transform integration 9-17

see Fourier transform - properties of definition of 9-1

time shift property of the Laplace transform Inverse of 9-1, 9-25

see Laplace transform - properties of Z transform - properties of

MATLAB command A-12 convolution in the discrete

trace of a matrix - see matrix frequency domain 9-9

IN-5

Fourier Transform Pairs of Common Functions

Likewise, the Fourier transform for the shifted delta function is

(8.61)

We will use the notation to show the time domain to frequency domain correspon-

dence. Thus, (8.60) may also be denoted as in Figure 8.1.

TABLE 8.8 Fourier Transform Properties and Theorems

Property

Linearity

Symmetry

Time Scaling

Time Shifting

Frequency Shifting

Time Differentiation

Frequency Differentiation

Time Integration

Conjugate Functions

Time Convolution

Frequency Convolution

Area under

Area under

Parseval’s Theorem

f t( ) F ω( )

a1 f1 t( ) a2 f2 t( ) …+ + a1 F1 ω( ) a2 F2 ω( ) …+ +

F t( ) 2πf ω–( )

f at( ) 1

a

-----F ω

a

----

f t t0–( ) F ω( )e

jωt0–

e

jω0tf t( )

F ω ω0–( )

d

n

dt

n

--------- f t( ) jω( )

n

F ω( )

jt–( )

n

f t( ) d

n

dω

n

-----------F ω( )

f τ( ) τd

∞–

t

F ω( )

jω

------------ πF 0( )δ ω( )+

f∗ t( ) F∗ ω–( )

f1 t( )∗f2 t( ) F1 ω( ) F2 ω( )⋅

f1 t( ) f2 t( )⋅ 1

2π

------F1 ω( )∗F2 ω( )

f t( )

F 0( ) f t( ) td

∞–

∞

=

F ω( )

f 0( ) 1

2π

------ F ω( ) ωd

∞–

∞

=

f t( ) 2 td

∞–

∞

1

2π

------ F ω( ) 2 ωd

∞–

∞

=

δ t t0–( )

δ t t0–( ) e

jωt0–⇔

f t( ) F ω( )↔

Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fifth Edition 8−17

Copyright © Orchard Publications

Page 35

Chapter 8 The Fourier Transform

Figure 8.1. The Fourier transform of the delta function

8.4.2 The Constant Function Pair

(8.62)

Proof:

and (8.62) follows.

The correspondence is also shown in Figure 8.2.

Figure 8.2. The Fourier transform of constant A

Also, by direct application of the Inverse Fourier transform, or the frequency shifting property and

(8.62), we derive the transform

(8.63)

The transform pairs of (8.62) and (8.63) can also be derived from (8.60) and (8.61) by using the

symmetry property

8.4.3 The Cosine Function Pair

(8.64)

Proof:

This transform pair follows directly from (8.63). The correspondence is also shown in

Figure 8.3.

0 t

1

ω0

f t( )

δ t( )

F ω( )

A 2Aπδ ω( )⇔

F 1– 2Aπδ ω( ){ } 1

2π

------ 2Aπδ ω( )e jωt ωd

∞–

∞

A δ ω( )e

jωt ωd

∞–

∞

Ae

jωt

ω 0= A= = = =

f t( ) F ω( )↔

A

ω

0 0

t

f t( ) F ω( )

2Aπδ ω( )

e

jω0t 2πδ ω ω0–( )⇔

F t( ) 2πf ω–( )⇔

ω0tcos

1

2

--- e

jω0t e

j– ω0t+( ) πδ ω ω0–( ) πδ ω ω0+( )+⇔=

f t( ) F ω( )↔

8−18 Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fifth Edition

Copyright © Orchard Publications

Page 67

conformable for multiplication D-4 Nyquist frequency 10-13 recursive realization digital filter

congugate of D-8 see digital filter

definition of D-1 O region of

determinant D-9 convergence 9-3

minor of D-11 octave defined 11-11 divergence 9-3

non-singular D-19 odd functions 6-11, 7-333 relationship between state equations

singular D-19 odd symmetry - see Fourier and Laplace Transform 5-28

diagonal D-1 series - symmetry residue 3-3, 9-37

diagonal elements of D-1 orthogonal functions 7-2 residue MATLAB function 3-3, 3-12

elements of D-1 orthogonal vectors 5-19 residue theorem 9-19

Hermitian D-8 orthonormal basis 5-19 right shift in the discrete-time domain

identity D-6 see Z transform - properties of

inverse of D-220 P RLC band-elimination filter - see filter

left division in MATLAB D-23 RLC band-pass filter - see filter

multiplication in MATLAB A-17 parallel form realization - see digital filter roots of polynomials in MATLAB A-3

power series of 5-9 Parseval’s theorem - see roots(p) MATLAB function 3-5, A-3

scalar D-6 Fourier transform - properties of round(n) MATLAB function A-22

size of D-7 partial fraction expansion 3-1 row vector in MATLAB A-3

skew-Hermitian D-9 alternate method of 3-14 Runge-Kutta method 5-1

skew-symmetric D-8 method of clearing the fractions 3-14 running Simulink B-7

square D-1 phase angle 11-2

symmetric D-8 phase shift filter - see filter S

trace of D-2 picket-fence effect 10-14

transpose of D-7 plot MATLAB command A-9 sampling property of the delta function

triangular polar form of complex numbers C-5 see delta function

lower D-6 polar plot in MATLAB A-23 sampling theorem 10-13

upper D-7 polar(theta,r) MATLAB function A-22 sawtooth waveform - see Laplace

zero D-2 poles 3-1 transform of common waveforms

matrix left division in MATLAB - see matrix complex 3-5 sawtooth waveform - Fourier series of

matrix multiplication in MATLAB - see matrix distinct 3-2 see Fourier series of

matrix power series - see matrix multiple (repeated) 3-7 common waveforms

maximally flat filter - see filter poly MATLAB function A-4 scalar matrix - see matrix

mesh(x,y,z) MATLAB function A-15 polyder MATLAB function A-6 scaling property of the Laplace transform

meshgrid(x,y) MATLAB command A-16 polynomial construction from see Laplace transform - properties of

m-file in MATLAB A-1, A-224 known roots in MATLAB A-4 Scope block in Simulink B-12

minor of determinant - see matrix polyval MATLAB function A-5 script file in MATLAB A-2, A-24

MINVERSE Excel function D-25 pre-sampling filter 10-13 second harmonic - see Fourier series

MMULT Excel function D-25 pre-warping 11-52 harmonics of

modulated signals 8-11 proper rational function - semicolons in MATLAB A-7

multiple eigenvalues - see eigenvalues definition of 3-1 semilogx MATLAB command A-12

multiple poles - see poles properties of the DFT semilogy MATLAB command A-12

multiplication by a

n

in discrete-time domain see DFT - common properties of series form realization - see digital filter

see Z transform - properties of properties of the Fourier Transform Shannon’s sampling theorem

multiplication by e

-naT

in discrete-time see Fourier transform - properties of see sampling theorem

domain - see Z transform - properties of properties of the Laplace Transform shift of f[n] u0[n] in discrete-time domain

multiplication by n in discrete-time domain see Laplace transform - properties of see Z transform - properties of

see Z transform - properties of properties of the Z Transform sifting property of the delta function

multiplication by n

2

indiscrete-time domain see Z transform - properties of see delta function

see Z transform - properties of signal flow graph 10-22

multiplication of complex numbers C-2 Q signals described in math form 1-1

signum function - see Fourier transform

N quarter-wave symmetry - see of common functions

Fourier series - symmetry simout To Workspace block

NaN in MATLAB A-25 quit MATLAB command A-2 in Simulink B-13

natural input-output FFT algorithm simple MATLAB symbolic function 3-6

see FFT algorithm R Simulation drop menu in Simulink B-12

network transformation simulation start icon in Simulink B-12

resistive 4-1 radius of absolute convergence 9-3 Simulink icon B-7

capacitive 4-1 ramp function 1-9 Simulink Library Browser B-8

inductive 4-1 randn MATLAB function 11-65 sine function - Fourier transform of

non-recursive realization digital filter Random Source Simulink block 11-76 see Fourier transform of

see digital filter rationalization of the quotient C-4 common functions

non-singular determinant - see matrix RC high-pass filter - see filter singular determinant - see matrix

nonlinear system G-1 RC low-pass filter - see filter Sinks library in Simulink B-18

normalized cutoff frequency 11-14 real axis C-1 sinw0t u0(t) Fourier transform of - see

notch filter - see filter real number C-2 Fourier transform of common functions

N-point DFT - see DFT - definition of real(z) MATLAB function A-22 size of a matrix - see matrix

nth-order delta function - see delta function rectangular form C-5 skew-Hermitian matrix - see matrix

numerical evaluation of Fourier coefficients rectangular pulse expressed in terms skew-symmetric matrix - see matrix

see Fourier series coefficients of the unit step function 1-4 special forms of the Fourier transform

IN-4

Page 68

see Fourier transform Transfer Fcn block in Simulink 4-17 convolution in the discrete

spectrum analyzer 7-35 Transfer Fcn Direct Form II time domain 9-8

square matrix - see matrix Simulink block 11-68 final value theorem 9-9

square waveform with even symmetry - see transfer function of initial value theorem 9-9

Fourier series of common waveforms continuous-time systems 4-13 left shift 9-5

square waveform with odd symmetry - see discrete-time systems 9-35 linearity 9-3

Fourier series of common waveforms transformation between multiplication by a

n

9-6

ss2tf MATLAB function 5-31 s and z domains 9-20 multiplication by e

-naT

9-6

stability 11-13 transformation methods for mapping multiplication by n 9-6

start simulation in Simulink B-12 analog prototype filters to digital filters multiplication by n

2

9-6

state equations Impulse Invariant Method 11-50 right shift 9-4

for continuous-time systems 5-1 Step Invariant Method 11-50 shift of f[n] u0[n] 9-3

for discrete-time systems 9-40 Bilinear transformation 11-50 summation 9-7

state transition matrix 5-8 transpose of a matrix - see matrix Z Transform of discrete-time functions

state variables Tree Pane in Simulink B-7 cosine function cosnaT 9-15

for continuous-time systems 5-1 triangular waveform expressed in terms exponential sequence e

-naT

u0[n] 9-15

for discrete-time systems 9-40 of the unit step function 1-6 geometric sequence a

n

9-12

State-Space block in Simulink B-13 triplet - see delta function sine function sinnaT 9-15

state-space equations Tukey - see Cooley and Tukey unit ramp function nu0[n] 9-16

for continuous-time systems 5-1 unit step function u0[n] 9-14

for discrete-time systems 9-40 U zero matrix - see matrix

step function - see unit step function zeros 3-1, 3-2

step invariant method - see trans- unit eigenvectors 5-18 zp2tf MATLAB function 11-16

formation methods for mapping analog unit impulse function (d(t)) 1-8

prototype filters to digital filters unit ramp function (u1(t)) 1-9

stop-band filter - see filter unit step function (u0(t)) 1-2

string in MATLAB A-15 upper triangular matrix - see matrix

subplots in MATLAB A-16 using MATLAB for finding the Laplace

summation in the discrete-time Domain transforms of time functions 2-26

see Z transform - properties of using MATLAB for finding the Fourier

symmetric matrix - see matrix transforms of time function 8-31

symmetric rectangular pulse expressed

as sum of unit step functions 1-5 V

symmetric triangular waveform expressed

as sum of unit step functions 1-6 Vandermonde matrix 10-18

symmetry - see Fourier series - symmetry Vector Scope Simulink block 11-78

symmetry property of the Fourier transform

see Fourier transform - properties of W

system function - definition of 8-34

warping 11-52

T window functions

Blackman E-10

Taylor series 5-1 Fourier series method for approximating

text MATLAB command A-13 an FIR amplitude response E-15

tf2ss MATLAB function 5-33 Hamming E-8, E-30

theorems of the DFT 10-10 Hanning E-6, E-26

theorems of the Fourier Transform 8-9 Kaiser E-12, E-33

theorems of the Laplace transform 2-2 other used as MATLAB functions E-14

theorems of the Z Transform 9-3 rectangular E-2

third harmonic - see Fourier triangular E-4, E-22

series - harmonics of Window Visualization Tool in MATLAB E-4

time convolution in DFT

see DFT - common properties of X

time integration property of the Fourier

transform - see Fourier xlabel MATLAB command A-12

transform - properties of

time periodicity property of the Laplace Y

transform - see Laplace

transform - properties of ylabel MATLAB command A-12

time scaling property of the Fourier

transform - see Fourier Z

transform - properties of

time shift in DFT Z transform

see DFT - common properties of computation of with contour

time shift property of the Fourier transform integration 9-17

see Fourier transform - properties of definition of 9-1

time shift property of the Laplace transform Inverse of 9-1, 9-25

see Laplace transform - properties of Z transform - properties of

MATLAB command A-12 convolution in the discrete

trace of a matrix - see matrix frequency domain 9-9

IN-5