Fourier Transform Pairs Fourier Transform Theorems Linearity aX(f)+bY(f) ax(t)by(t) S(f) 1 1 Х Scale change x(аt) 8(t) 1 а 1 Time reversal X(-f) x(t) u(t) j2nf x(-f) Duality X(t) П(/т) t sinc (Tf) x(-to) Time shift X(fe j2nfo sinc(Bt) П(f/ B) В X(f-fo Frequency shift T sinc2 (Tf) A(t/T) 2 1 -at Modulation a> 0 x(t)cos 2Tf0f +xf+fo) a j2nf 1 ,-at u(t) a >0 X(f)* Y(f) te Multiplication x()у(0) (aj2nf X(f)Y(f) x() *У(0) Convolution 2а a >0 eal a2+(2f) а"x() Differentiation (j2nf)" X(f) dt" -(7f /a)2 е e (ar)2 1 а j2nf e"6(f-fo)+ef+f) cos(2fot+ 0) Integration -CO 1 X(0)8 2 ,le/6f-fo)-8(f + fo] sin(2tfot+0) where sint sinc(t)= k-too n=+oo 1 , Σ δ - Μ. Т. 8-nT) _ Tt S k=-ao N=-co I1t) 1,lt| П()- otherwise t0, otherwise 38 MATHEMATICS - Einel Lohr ier thansform Por XCE)=S (t-no) A) +ohrie anstof m 2 ? XEf)= 1M &(F-k) Ko-e
Fourier Transform Pairs Fourier Transform Theorems Linearity aX(f)+bY(f) ax(t)by(t) S(f) 1 1 Х Scale change x(аt) 8(t) 1 а 1 Time reversal X(-f) x(t) u(t) j2nf x(-f) Duality X(t) П(/т) t sinc (Tf) x(-to) Time shift X(fe j2nfo sinc(Bt) П(f/ B) В X(f-fo Frequency shift T sinc2 (Tf) A(t/T) 2 1 -at Modulation a> 0 x(t)cos 2Tf0f +xf+fo) a j2nf 1 ,-at u(t) a >0 X(f)* Y(f) te Multiplication x()у(0) (aj2nf X(f)Y(f) x() *У(0) Convolution 2а a >0 eal a2+(2f) а"x() Differentiation (j2nf)" X(f) dt" -(7f /a)2 е e (ar)2 1 а j2nf e"6(f-fo)+ef+f) cos(2fot+ 0) Integration -CO 1 X(0)8 2 ,le/6f-fo)-8(f + fo] sin(2tfot+0) where sint sinc(t)= k-too n=+oo 1 , Σ δ - Μ. Т. 8-nT) _ Tt S k=-ao N=-co I1t) 1,lt| П()- otherwise t0, otherwise 38 MATHEMATICS - Einel Lohr ier thansform Por XCE)=S (t-no) A) +ohrie anstof m 2 ? XEf)= 1M &(F-k) Ko-e
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
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![Fourier Transform Pairs
Fourier Transform Theorems
Linearity
aX(f)+bY(f)
ax(t)by(t)
S(f)
1
1
Х
Scale change
x(аt)
8(t)
1
а
1
Time reversal
X(-f)
x(t)
u(t)
j2nf
x(-f)
Duality
X(t)
П(/т)
t sinc (Tf)
x(-to)
Time shift
X(fe j2nfo
sinc(Bt)
П(f/ B)
В
X(f-fo
Frequency shift
T sinc2 (Tf)
A(t/T)
2
1
-at
Modulation
a> 0
x(t)cos 2Tf0f
+xf+fo)
a j2nf
1
,-at u(t)
a >0
X(f)* Y(f)
te
Multiplication
x()у(0)
(aj2nf
X(f)Y(f)
x() *У(0)
Convolution
2а
a >0
eal
a2+(2f)
а"x()
Differentiation
(j2nf)" X(f)
dt"
-(7f /a)2
е
e (ar)2
1
а
j2nf
e"6(f-fo)+ef+f)
cos(2fot+ 0)
Integration
-CO
1
X(0)8
2
,le/6f-fo)-8(f + fo]
sin(2tfot+0)
where
sint
sinc(t)=
k-too
n=+oo
1
, Σ δ - Μ.
Т.
8-nT)
_
Tt
S
k=-ao
N=-co
I1t) 1,lt|
П()-
otherwise
t0, otherwise
38 MATHEMATICS
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8b662eb-b55f-4405-acee-a8a96eb1304c%2F6444d17e-eea3-468f-ad5f-c6d0734019cd%2Ftrydbxl.png&w=3840&q=75)
Transcribed Image Text:Fourier Transform Pairs
Fourier Transform Theorems
Linearity
aX(f)+bY(f)
ax(t)by(t)
S(f)
1
1
Х
Scale change
x(аt)
8(t)
1
а
1
Time reversal
X(-f)
x(t)
u(t)
j2nf
x(-f)
Duality
X(t)
П(/т)
t sinc (Tf)
x(-to)
Time shift
X(fe j2nfo
sinc(Bt)
П(f/ B)
В
X(f-fo
Frequency shift
T sinc2 (Tf)
A(t/T)
2
1
-at
Modulation
a> 0
x(t)cos 2Tf0f
+xf+fo)
a j2nf
1
,-at u(t)
a >0
X(f)* Y(f)
te
Multiplication
x()у(0)
(aj2nf
X(f)Y(f)
x() *У(0)
Convolution
2а
a >0
eal
a2+(2f)
а"x()
Differentiation
(j2nf)" X(f)
dt"
-(7f /a)2
е
e (ar)2
1
а
j2nf
e"6(f-fo)+ef+f)
cos(2fot+ 0)
Integration
-CO
1
X(0)8
2
,le/6f-fo)-8(f + fo]
sin(2tfot+0)
where
sint
sinc(t)=
k-too
n=+oo
1
, Σ δ - Μ.
Т.
8-nT)
_
Tt
S
k=-ao
N=-co
I1t) 1,lt|
П()-
otherwise
t0, otherwise
38 MATHEMATICS
-
Transcribed Image Text:Einel Lohr ier thansform
Por XCE)=S (t-no)
A)
+ohrie
anstof m
2
?
XEf)= 1M &(F-k)
Ko-e
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