G
−4
Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fifth Edition Copyright © Orchard PublicationsFigure G.2. Model for the describing function in (G.11)
The waveform displayed by the XY Graph block shows the normalized value of the describing function (Y axis) as a function of the ratio (X Axis).
Note: The XY Graph block does not display grid lines. They were added manually.
Index
Symbols Cauer filter - see elliptic filter convolution property of the Fourier Cayley-Hamilton theorem 5-10 transform - see Fourier transform % (percent) symbol in MATLAB A-2 characteristic equation 5-18 properties of
cheb1ap MATLAB function 11-35 Cooley and Tukey 10-18
A cheb2ap MATLAB function 11-38 cosine function - Fourier transform of Chebyshev filters - see filter see Fourier transform of abs(z) in MATLAB A-22 Chebyshev Type I analog low-pass common functions
active analog filter - see filter filter design - see filter design cosw0t u0(t) function - Fourier transform of
adjoint of a matrix - see matrix Chebyshev Type I filters - see filter see Fourier transform of
admittance Chebyshev Type I low-pass filter common functions
capacitive 4-2 magnitude-square function Cramer’s rule D-15
inductive 4-2 - see filter cross correlation F-1
complex input 4-11 Chebyshev Type II analog low-pass
algebraic constrain block in Simulink B-18 filter design - see filter design D
aliasing 10-13 Chebyshev Type II filters - see filter
all-pass filter - see filter circuit analysis with Laplace transforms 4-1 d2c MATLAB function 9-43 all-pole approximation 11-20 circuit analysis with state variables 5-21 data points in MATLAB A-13
all-pole low-pass filter circuit transformation from time decade - definition of 11-11
see filter - low-pass to complex frequency 4-1 decimation in frequency
alternate form of the trigonometric clc MATLAB command A-2 see FFT algorithm
Fourier series - see Fourier series clear MATLAB command A-2 decimation in time - see FFT algorithm
alternate method of partial cofactor of a matrix - see matrix deconv MATLAB function A-6
fraction expansion - see partial collect(s) MATLAB symbolic function 3-12 default color in MATLAB A-14
fraction expansion column vector in MATLAB A-17 default line in MATLAB A-14
angle(z) MATLAB function A-22 command screen in MATLAB A-1 default marker in MATLAB A-14
argument 11-2 command window in MATLAB A-1 default values in MATLAB A-11
attenuation rate 11-11 commas in MATLAB A-7 delta (impulse) function
autocorelation F-6 comment line in MATLAB A-2 definition of 1-8
autoscale icon in Simulink B-12 Commonly Used Blocks in Simulink B-7 doublet 1-13
axis MATLAB command A-15 complex admittance - see admittance Fourier transform of - see Fourier complex conjugate in MATLAB A-4 transform of common functions
B complex conjugate pairs 3-5 higher order 1-13
complex impedance - see impedance nth-order 1-13
band-elimination filter - see filter complex number C-2 sampling property of 1-11
band-elimination filter design complex numbers in MATLAB A-2 sifting property of 1-12
see filter design complex poles 3-5 triplet 1-13
band-limited signal 10-13 Complex to Magnitude-Angle demo in MATLAB A-2
band-pass filter - see filter block in Simulink C-7 DeMoivre’s theorem 11-14
band-pass filter design computation of the state derivation of the Fourier transform from
see filter design transition matrix 5-10 the Laplace Transform 8-24
band-stop filter - see filter computation of the Z Transform with describing function G-1 Bessel filter - see filter contour integration - see Z transform determinant of a matrix bilinear MATLAB function 11-59 Configuration Parameters see matrix
bilinear transformation - see in Simulink B-12 DFT - common properties of
transformation methods for mapping congugate of a matrix - see matrix even time function 10-9 analog prototype filters to digital filters conj(A) MATLAB function D-8 even frequency function 10-9 Blackman window function conjugate of a complex number C-3 frequency convolution 10-13 see window functions conjugate time and frequency functions frequency shift 10-12 bode MATLAB function 11-23 of the Fourier transform - see linearity 10-11 box MATLAB command A-12 Fourier transform - properties of odd time function 10-9 buttap MATLAB function 11-17 constant function - Fourier transform of odd frequency function 10-9 buttefly operation 10-23 see Fourier transform - properties of time convolution 10-12
Butterworth analog low-pass Contents Pane in Simulink B-7 time shift 10-11
filter design - see filter design contour integral 9-17 DFT - definition of 10-1 conv MATLAB function A-6 N-point 10-2
C convolution in the complex frequency diagonal elements of a matrix - see matrix
domain - see Laplace transform diagonal matrix - see matrix c2d MATLAB function 9-42 properties of differentiation in complex frequency capacitive admittance - see admittance convolution in the discrete-frequency domain property of the Laplace capacitive impedance - see impedance domain - see Z transform - properties of transform - see Laplace transform cascade form realization - see digital filter convolution in the discrete-time - properties of
Category I FFT algorithm - see domain - see Z transform - properties of differentiation in time domain property FFT algorithm convolution in the time domain property of the Laplace transform - see Category II FFT algorithm - see of the Laplace transform - see Laplace transform - properties of FFT algorithm Laplace transform - properties of differentiation property of the Fourier Cauchy’s residue theorem convolution integral defined 6-7 transform 8-12 - see Fourier see residue theorem graphical evaluation of 6-8 transform - properties of
digital filter 11-1, 11-49, 11-67 exponential form of complex numbers C-5 Finite Impulse Response (FIR) digital filter Finite Impulse Response (FIR) 11-49 exponential form of the Fourier series - see digital filter
Infinite Impulse Response (IIR) 11-49 see Fourier series FIR - see digital filter
realization of exponential order function - first harmonic - see Fourier series
Direct Form I 11-67 definition of 2-2 harmonics of
Direct Form II 11-68 eye(n) MATLAB function D-6 first-order circuit 5-1
cascade (series) form 11-70 eye(size(A)) MATLAB function D-7 first-order simultaneous
non-recursive 11-49 differential equations 5-1
parallel form 11-70 F Flip Block command in Simulink B-11
recursive 11-49 format in MATLAB A-29
series (cascade) form 11-70 factor(s) MATLAB symbolic function 3-4 fourier MATLAB command 8-33 Digital Filter Design Simulink block 11-70 Fast Fourier Transform (FFT) 10-1, 10-17 Fourier series
digital filter design with Simulink 11-67 FDA Tool Digital Filter Design exponential form of 7-30
dimpulse MATLAB function 9-26 Simulink block 11-78 method used in window functions E-15 Dirac MATLAB function 1-20 FFT algorithm trigonometric form of 7-2
Direct Form I realization - see digital filter Category I 10-19 alternate form of 7-24
realization of Category II 10-19 Fourier series coefficients - evaluation of
Direct Form II realization - see digital filter decimation in frequency 10-19 numerical evaluation using Excel 7-43
realization of decimation in time 10-19 numerical evaluation using MATLAB® 7-44
direct term in MATLAB 3-3 double-memory technique 10-19 Fourier series of common waveforms discontinuous function - definition of 1-2 in-place 10-19 full-wave rectifier 7-20, 7-23 Discrete Fourier Transform (DFT) 10-1 natural input-output 10-19 half-wave rectifier 7-17, 7-20
discrete impulse response 9-36 FFT definition of 10-1, 10-17 square waveform
discrete-time system transfer function 9-36 fft(x) MATLAB function 10-5, 11-65 with even symmetry 7-9, 7-13 discrete unit step function 9-3 figure window in MATLAB A-13 with odd symmetry 7-8, 7-12 discrete-time exponential sequence 9-15 filter - see also digital filter sawtooth 7-9, 7-15
discrete-time systems 9-1 active triangular 7-9, 7-16
discrete-time unit ramp function 9-16 high-pass 4-22, 4-30 Fourier series - harmonics of discrete-time unit step function 9-14 low-pass 4-22, 4-30 first 7-1, 7-10
Display block in Simulink B-18 all-pass 11-1, 11-89 second 7-1, 7-10
display formats in MATLAB A-29 all-pole 11-20 third 7-1, 7-10
distinct eigenvalues - see eigenvalues band-elimination 11-1, 11-8 Fourier series - symmetry
distinct poles - see poles band-pass 11-1, 11-6 even 7-7
division of complex numbers C-4 band-stop - see band-elimination half-wave 7-7, 7-33
dot operator in MATLAB Bessel 11-90 odd 7-7
division with A-19 Chebyshev 11-10 quarter-wave 7-7 (footnote)
exponentiation with A-19 Inverted 11-36 types of 7-7
multiplication with A-19 magnitude-square function 11-10 Fourier integral - see Fourier transform
double-memory technique prototype 11-10 Fourier transform
see FFT Algorithm Type I 11-24 definition of 8-1
doublet - see delta function Type II 11-36 inverse of 8-1
elliptic 11-37 special forms of 8-2
E high-pass 4-30, 11-1, 11-4 Fourier transform - properties of
low-pass 4-30, 11-1, 11-2 area under f(t) 8-15
Editor Window in MATLAB A-1 low-pass analog filter area under F(w) 8-15
Editor/Debugger in MATLAB A-1 prototypes 11-10 conjugate time and frequency
eig(x) MATLAB function 5-17 maximally flat 11-13 (footnote) functions 8-13
eigenvalues notch (band-elimination) 11-8 constant function 8-18
distinct 15-1 phase shift 11-1, 11-89 frequency convolution 8-14
multiple (repeated) 5-15 RC high-pass 11-4 frequency differentiation 8-12
eigenvector 5-19 RC low-pass 11-2 frequency shifting 8-11
e-jwt u0(t) Fourier transform of - see Fourier RLC band-elimination 4-31, 11-8 imaginary time functions - Fourier
transform of common functions RLC band-pass 4-31, 11-6 transform of 8-6 element-by-element operation in MATLAB filter design - see also digital filter linearity 8-9
division A-19 band-elimination 11-39 Parseval's theorem 8-15
exponentiation A-19 band-pass 11-39 real time functions - Fourier
multiplication A-19 Butterworth analog low-pass 11-13 transform of 8-3
elements of the matrix - see matrix Chebyshev symmetry 8-9
ellip MATLAB function 11-40 Type I 11-24 time convolution 8-14
elliptic filter - see filter Type II 11-36 time differentiation 8-12
elliptic filter design - see filter design elliptic 11-37 time integration 8-13 eps in MATLAB A-21 high-pass 11-39 time scaling 8-10
Euler’s identities C-4 low-pass 11-10 time shifting 8-11
even functions 6-4, 7-32 filter MATLAB function 11-60 Fourier Transform derivation from
even symmetry - see Fourier final value theorem in Laplace transform Laplace transform 8-25
series - symmetry see Laplace transform - properties of Fourier transform of common functions Excel's Analysis ToolPak 10-6 final value theorem in Z transform cosw 0t 8-18
exit MATLAB command A-2 see Z transform - properties of cosw0t u0(t) 8-23 expand MATLAB symbolic function 3-10 find MATLAB function 11-64 delta (d(t) and d(t-a)) 8-16
e-jw0t 8-19 ifourier MATLAB function 8-31 integration in
e-jw0t u0(t) 8-23 IIR - see digital filter complex frequency domain 2-7
signum (sgn(t)) 8-19 ilaplace MATLAB function 3-4 time domain 2-6
sinw0t 8-19 imag(z) MATLAB function A-22 linearity 2-3
sinw0t u0(t) 8-23 imaginary axis - definition of C-2 scaling 2-4
unit step (u0(t)) 8-21 imaginary number - definition of C-1 time periodicity 2-8
Fourier transform of common waveforms imaginary time functions 8-6 time shift 2-3
combined rectangular pulses 8-28 see Fourier transform - properties of Laplace transform of common waveforms
cosine within a rectangular pulse 8-29 impedance full-rectified 2-30
shifted rectangular pulse 8-27 capacitive 4-2 half-rectified 2-25
symmetrical rectangular pulse 8-26 inductive 4-2 linear segment 2-22
periodic time functions 8-29 complex input 4-8, 4-9 pulse 2-21
fourth-order all-pole low-pass filter improper integral - definition of 2-15 rectangular periodic 2-24
see filter, low-pass improper rational function - sawtooth 2-30
fplot in MATLAB A-25 definition of 3-1, 3-12 triangular 2-23
frequency convolution in DFT impulse function - see delta function Laplace transform of common functions see DFT - common properties of impulse invariant method - see transform of e-at coswt u
0(t) 2-20
frequency convolution in Fourier transformation methods for mapping transform of e-at sinwt u 0(t) 2-20
transform of - see Fourier transform analog prototype filters to transform of e-at u 0(t) 2-18
- properties of digital filters transform of coswt u0(t) 2-19
frequency differentiation in Fourier increments between points transform of d(t) 2-17 transform of - see Fourier transform in MATLAB A-14 transform of d(t-a) 2-17 - properties of inductive admittance - see admittance transform of sinwt u0(t) 2-19
frequency shift in DFT inductive impedance - see impedance transform of tn u
0(t) function 2-14
see DFT - common properties of infinite impulse response - see digital filter transform of tn e-at u 0(t) 2-18
frequency shift in Fourier transform initial value theorem in Laplace transform transform of u0(t) 2-12
see Fourier transform - properties of see Laplace transform - properties of transform of u1(t) 2-12
frequency shift in Laplace transform initial value theorem in Z transform leakage 10-14
see Laplace transform - properties of see Z transform - properties of left shift in discrete-time domain freqz MATLAB function 11-54 in-place FFT algorithm 10-20 see Z transform - properties of
full rectification waveform - Laplace see FFT algorithm Leibnitz’s rule 2-6
transform of - see Laplace transform integration in frequency in Laplace transform lims= MATLAB command A-25 of common waveforms see Laplace transform - properties of line spectra 7-35
full-wave rectifier - Fourier series of - see integration in time in Laplace transform linear difference equation 9-35 Fourier series of common waveforms see Laplace transform - properties of linearity in DFT
Function Block Parameters in Simulink B-10 Inverse Fourier transform see DFT - common properties of
function files in MATLAB A-25 see Fourier transform linearity in discrete-time domain
fundamental frequency 7-1 Inverse Laplace transform see Z transform - properties of
fzero MATLAB function A-25 see Laplace transform linearity property in Fourier transform inverse of a matrix - see matrix see Fourier transform - properties of
G Inverse Z transform - see Z transform linearity property in Laplace transform
inversion integral 9-29 see Laplace transform - properties of Gain block in Simulink B-9, B-18 Inverted Chebyshev filter - see filter linspace MATLAB command A-13
gamma function 2-15 ln (natural log) A-12
Gaussian elimination method D-17 J log in MATLAB A-12
generalized factorial function 2-14 log(x) MATLAB function A-12
Gibbs phenomenon 7-23 j operator C-1 log10(x) MATLAB function A-12
grid MATLAB command A-11 log2(x) MATLAB function A-12 gtext MATLAB command A-13 K loglog MATLAB command A-12
long division of polynomials 9-33
H Kaiser window - see windows lower triangular matrix - see matrix
low-pass analog filter prototypes - see filter
half-wave rectifier - Fourier series of - see L low-pass filter - see filter
Fourier series of common waveforms lp2bp MATLAB function 11-41
half-wave symmetry L’ Hôpital’s rule 2-15 lp2bs MATLAB function 11-46
see Fourier series - symmetry laplace MATLAB function 2-26 lp2hp MATLAB function 11-41 Hamming window - see windows Laplace integral - see Laplace transform lp2lp MATLAB function 11-41
Hanning window - see windows Laplace transform
Heaviside MATLAB function 1-19 definition of 2-1 M
help in MATLAB A-2 Inverse of 2-1, 3-1
Hermitian matrix - see matrix Laplace transform - properties of magnitude-squared function 11-10 higher order delta functions - see delta convolution in the complex main diagonal of a matrix - see matrix
function frequency domain 2-11 Math Operations in Simulink B-11
high-pass filter - see filter convolution in the time domain 2-11 MATLAB Demos A-2
high-pass filter design - see filter design differentiation in complex MATLAB’s Editor/Debugger A-1 frequency domain 2-6 matrix (matrices)
I differentiation in time domain 2-4 adjoint of D-19
final value theorem 2-10 cofactor of D-11
identity matrix - see matrix frequency shift 2-4 conformable for addition D-2
ifft(x) MATLAB function 10-5 initial value theorem 2-9 conformable for subtraction D-2
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 an 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 n2 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
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 an 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 n2 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 u 0[n] 9-15
for discrete-time systems 9-40 of the unit step function 1-6 geometric sequence an 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