PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS in .NET framework

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PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
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Integration in the Time Domain:
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Initial value theorem: ~ ( 0 ' = lirn sX,(s) )
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Final value theorem: lim x ( t ) = lim sX,(s)
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B.3 THE FOURIER TRANSFORM
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DeJinition:
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Properties of the Fourier Transform: Linearity: a l x l ( t ) a 2 x 2 ( t ) * a l X , ( o ) a 2 X 2 ( o ) c Time shifting: x ( t - t o )c* e - ~ " ' o ~ ( w ) Frequency shifting: e J w ~ ~ ' x ( tX ( o - oo) c* ) Time scaling: x ( a t )
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Time reversal: x( - t ) c* X ( - o ) Duality: X ( t ) c* 2lrx( - o ) W t ) Time differentiation: -c*jwX(w) dt
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dX(4 Frequency differentiation: ( -jt ) x ( t ) c* do
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Convolution: x , ( t ) * x 2 ( t )- X , ( w ) X 2 ( w )
- X , ( o ) * X2(W ) 2lr Real signal: x ( t ) = x e ( t )+ x o ( t )- X ( o ) = A ( o ) + j B ( o ) X( -0) =X * ( o ) Even component: x e ( t ) R e { X ( o ) )=A ( w ) Odd component: x o ( t )c*j I m ( X ( o ) )=j B ( o )
Multiplication: x,(t ) x 2 ( t )
PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
[APP. B
Parseval's Relations:
Common Fourier Transforms Pairs:
p,(O =
sin w a 2a - wa Iwl < a
sin at
-pa(w)
2 sgn t o 1w
APP. B]
PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
B 4 DISCRETE-TIME LTI SYSTEMS .
Unit sample response: h [ n ]
Convolution: y [ n ] = x [ n ]* h [ n ]=
x [ k ] h [ n- k ]
Causality: h [ n ]= 0, n < 0
Stability:
n= -m
( h [ n ] ( d< a t :
B.5 THE Z-TRANSFORM
The Bilateral (or Two-sided) z-Transform:
Dejnition:
Transform: Properties of the z-
Linearity: a l x l [ n ] a 2 x 2 [ nt]- , a , X 1 ( z )+ a 2 X 2 ( z ) ,R' 3 R , nR 2 + Time shifting: x [ n - n o ]-2-"oX(z), R' 3 R n (0 < lzl < w) Multiplication by z:: z:x[n]
Multiplication by ejR1tN: ~ ~ o " ~ [ ~ ( ] - j n l ) z ) , e n e R' Time reversal: x[ - n ] t-,X
- -1,
R = lzdR '
dX(z ) Multiplication by n: nx[n]o - z , R' = R dz n 1 Accumulation: x[n] X ( z ) , R' 3 R
k = - OC
1-2-'
n {lzl > 1)
Some Common z-Transforms Pairs:
6[n]
1 , all z
PROPERTIES O F LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
[APP. B
6[n - m] -zPm, all z except 0 if m > 0, or 03 if m < 0 1 Z anu[n] =, IzI > la1 1-az-' z-a 1 Z -anu[-n - 11=, Izl < lal 1-az-' z-a
(COS
fl,n)u[n
(sin n,n)u[n]
( r ncos R,n)u[n] ( r n sin fl,n)u[n]
z 2 - (cos 0,)z , lzl > 1 z 2 - (2cos n o ) z + 1
(sin 0 , ) z , 121 > 1 z 2 - (2cos Ro)z + 1 z 2 - ( r cos n o ) z Izl > r z 2 - (2r cos n,)z + r 2 ' ( r sin 0 , ) z Izl > r ~~-(2rcos~~)z+r~' 1 -aNz-N IzI>O 1 - az-I
OsnsN-1otherwise
The Unilateral (or One-sided) z-Transform:
Some Special Properties:
Time-Shifting Property:
Initial value theorem: x[O] = lim X( z)
z-rm
Final value theorem: lim x[N] = lim (1 - z- ')X( z)
z- 1
APP. B]
PROPERTIES O F LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
45 1
B.6 THE DISCRETE-TIME FOURIER TRANSFORM Definition:
X ( R )=
x [n] eeJf'"
Properties of the Discrete-Time Fourier Transform:
Periodicity: x [ n ] X ( R ) = X ( R + 2rr) ] Linearity: a , x , [ n ]+ a 2 x 2 [ n++ a , X J R ) + a 2 X 2 ( R ) Time shifting: x [ n - n o ]++ e - ~ ~ " l l X ( R ) Frequency shifting: e ~ " ~ " x [ n ]X ( R - R,) Conjugation: x * [ n ] X*( - R ) Time Reversal: x[ - n ] X( - R ) ifn=km -X(mR) Time Scaling: x(,,,[n] = ifnzkm dX(R) Frequency differentiation: m [ n ] j-----dR First difference: x [ n ]- x [ n - 11 ++ ( 1 - e - j o ) x ( f l ) Accumulation:
x[k]
rrX(0)S ( R )+ 1 - e -jn x(n)
Convolution: x , [ n ]* x 2 [ n ] + X , ( R ) X , ( R ) + Multiplication: x , [ n ] x 2 [ n ] - X , ( R ) @ X 2 ( R ) 2rr Real sequence: x [ n ]= x , [ n ] + x,[n] X ( R ) = A ( R ) + j B ( R ) X(-R) =X*(R) Even component: x,[n] R e { X ( R ) )= A ( R ) Odd component: x,[n] j 1 m { X ( R ) )= j B ( R )
Parseval's Relations:
45 2
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