-
Interval Sum Condition Interval Sum Condition
Infinite
|a|
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Table 1: Properties of the Continuous-Time Fourier Series
x(t) =+
k=
akejk0t =
+k=
akejk(2pi/T )t
ak =1
T
T
x(t)ejk0tdt =1
T
T
x(t)ejk(2pi/T )tdt
Property Periodic Signal Fourier Series Coefficients
x(t)y(t)
}Periodic with period T andfundamental frequency 0 = 2pi/T
akbk
Linearity Ax(t) +By(t) Aak +BbkTime-Shifting x(t t0) ake
jk0t0 = akejk(2pi/T )t0
Frequency-Shifting ejM0t = ejM(2pi/T )tx(t) akMConjugation x(t)
akTime Reversal x(t) akTime Scaling x(t), > 0 (periodic with
period T/) ak
Periodic Convolution
T
x()y(t )d Takbk
Multiplication x(t)y(t)
+l=
albkl
Differentiationdx(t)
dtjk0ak = jk
2pi
Tak
Integration
t
x(t)dt(finite-valued andperiodic only if a0 = 0)
(1
jk0
)ak =
(1
jk(2pi/T )
)ak
Conjugate Symmetryfor Real Signals
x(t) real
ak = ak
e{ak} = e{ak}m{ak} = m{ak}|ak| = |ak|
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Table 2: Properties of the Discrete-Time Fourier Series
x[n] =
k=
akejk0n =
k=
akejk(2pi/N)n
ak =1
N
n=
x[n]ejk0n =1
N
n=
x[n]ejk(2pi/N)n
Property Periodic signal Fourier series coefficients
x[n]y[n]
}Periodic with period N and fun-damental frequency 0 = 2pi/N
akbk
}Periodic withperiod N
Linearity Ax[n] +By[n] Aak +BbkTime shift x[n n0] ake
jk(2pi/N)n0
Frequency Shift ejM(2pi/N)nx[n] akMConjugation x[n] akTime
Reversal x[n] ak
Time Scaling x(m)[n] =
{x[n/m] if n is a multiple of m0 if n is not a multiple of m
1
mak
(viewed asperiodic withperiod mN
)(periodic with period mN)
Periodic Convolutionr=N
x[r]y[n r] Nakbk
Multiplication x[n]y[n]l=N
albkl
First Difference x[n] x[n 1] (1 ejk(2pi/N))ak
Running Sum
nk=
x[k](finite-valued andperiodic only if a0 = 0
) ( 1(1 ejk(2pi/N))
)ak
Conjugate Symmetryfor Real Signals
x[n] real
ak = ak
e{ak} = e{ak}m{ak} = m{ak}|ak| = |ak|
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Table 3: Properties of the Continuous-Time Fourier Transform
x(t) =1
2pi
X(j)ejtd
X(j) =
x(t)ejtdt
Property Aperiodic Signal Fourier transform
x(t) X(j)y(t) Y (j)
Linearity ax(t) + by(t) aX(j) + bY (j)Time-shifting x(t t0)
e
jt0X(j)Frequency-shifting ej0tx(t) X(j( 0))Conjugation x(t)
X(j)Time-Reversal x(t) X(j)
Time- and Frequency-Scaling x(at)1
|a|X
(j
a
)Convolution x(t) y(t) X(j)Y (j)
Multiplication x(t)y(t)1
2piX(j) Y (j)
Differentiation in Timed
dtx(t) jX(j)
Integration
t
x(t)dt1
jX(j) + piX(0)()
Differentiation in Frequency tx(t) jd
dX(j)
Conjugate Symmetry for RealSignals
x(t) real
X(j) = X(j)e{X(j)} = e{X(j)}m{X(j)} = m{X(j)}|X(j)| = |X(j)|
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Table 4: Basic Continuous-Time Fourier Transform Pairs
Fourier series coefficientsSignal Fourier transform (if
periodic)
+k=
akejk0t 2pi
+k=
ak( k0) ak
ej0t 2pi( 0)a1 = 1ak = 0, otherwise
cos0t pi[( 0) + ( + 0)]a1 = a1 =
12
ak = 0, otherwise
sin0tpi
j[( 0) ( + 0)]
a1 = a1 =12j
ak = 0, otherwise
x(t) = 1 2pi()
a0 = 1, ak = 0, k 6= 0(this is the Fourier series
rep-resentation for any choice ofT > 0
)Periodic square wave
x(t) =
{1, |t| < T10, T1 < |t|
T2
andx(t+ T ) = x(t)
+k=
2 sin k0T1k
( k0)0T1pi
sinc
(k0T1pi
)=
sin k0T1kpi
+n=
(t nT )2pi
T
+k=
(
2pik
T
)ak =
1
Tfor all k
x(t)
{1, |t| < T10, |t| > T1
2 sinT1
sinWt
pitX(j) =
{1, || < W0, || > W
(t) 1
u(t)1
j+ pi()
(t t0) ejt0
eatu(t),e{a} > 01
a+ j
teatu(t),e{a} > 01
(a+ j)2
tn1
(n1)!eatu(t),
e{a} > 0
1
(a+ j)n
Basic Continuous-Time Fourier Transform Pairs
Signal Fourier TransformFourier Series Coefficients(if
periodic)
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Table 5: Properties of the Discrete-Time Fourier Transform
x[n] =1
2pi
2pi
X(ej)ejnd
X(ej) =
+n=
x[n]ejn
Property Aperiodic Signal Fourier transform
x[n]y[n]
X(ej)Y (ej)
}Periodic withperiod 2pi
Linearity ax[n] + by[n] aX(ej) + bY (ej)Time-Shifting x[n n0]
e
jn0X(ej)
Frequency-Shifting ej0nx[n] X(ej(0))Conjugation x[n] X(ej)Time
Reversal x[n] X(ej)
Time Expansions x(k)[n] =
{x[n/k], if n = multiple of k0, if n 6= multiple of k
X(ejk)
Convolution x[n] y[n] X(ej)Y (ej)
Multiplication x[n]y[n]1
2pi
2pi
X(ej)Y (ej())d
Differencing in Time x[n] x[n 1] (1 ej)X(ej)
Accumulationn
k=
x[k]1
1 ejX(ej)
+piX(ej0)+
k=
( 2pik)
Differentiation in Frequency nx[n] jdX(ej)
d
Conjugate Symmetry forReal Signals
x[n] real
X(ej) = X(ej)e{X(ej)} = e{X(ej)}m{X(ej)} = m{X(ej)}|X(ej)| =
|X(ej)|
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Table 6: Basic Discrete-Time Fourier Transform Pairs
Fourier series coefficientsSignal Fourier transform (if
periodic)
k=N
akejk(2pi/N)n 2pi
+k=
ak
(
2pik
N
)ak
ej0n 2pi+l=
( 0 2pil)
(a) 0 =2pimN
ak =
{1, k = m,mN,m 2N, . . .0, otherwise
(b) 02pi irrational The signal is aperiodic
cos0n pi+l=
{( 0 2pil) + ( + 0 2pil)}
(a) 0 =2pimN
ak =
{12 , k = m,mN,m 2N, . . .0, otherwise
(b) 02pi irrational The signal is aperiodic
sin0npi
j
+l=
{( 0 2pil) ( + 0 2pil)}
(a) 0 =2pirN
ak =
12j , k = r, r N, r 2N, . . .
12j , k = r,r N,r 2N, . . .
0, otherwise(b) 02pi irrational The signal is aperiodic
x[n] = 1 2pi+l=
( 2pil) ak =
{1, k = 0,N,2N, . . .0, otherwise
Periodic square wave
x[n] =
{1, |n| N10, N1 < |n| N/2
andx[n+N ] = x[n]
2pi+
k=
ak
(
2pik
N
)ak =
sin[(2pik/N)(N1+1
2)]
N sin[2pik/2N ] , k 6= 0,N,2N, . . .
ak =2N1+1N , k = 0,N,2N, . . .
+k=
[n kN ]2pi
N
+k=
(
2pik
N
)ak =
1
Nfor all k
anu[n], |a| < 11
1 aej
x[n]
{1, |n| N10, |n| > N1
sin[(N1 +12 )]
sin(/2)
sinWnpin =
Wpi sinc
(Wnpi
)0 < W < pi
X() =
{1, 0 || W0, W < || pi
X()periodic with period 2pi
[n] 1
u[n]1
1 ej+
+k=
pi( 2pik)
[n n0] ejn0
(n + 1)anu[n], |a| < 11
(1 aej)2
(n+ r 1)!
n!(r 1)!anu[n], |a| < 1
1
(1 aej)r
Basic Discrete-Time Fourier Transform Pairs
Signal Fourier TransformFourier Series Co-effecient (if
periodic)
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Table 7: Properties of the Laplace Transform
Property Signal Transform ROC
x(t) X(s) R
x1(t) X1(s) R1
x2(t) X2(s) R2
Linearity ax1(t) + bx2(t) aX1(s) + bX2(s) At least R1 R2
Time shifting x(t t0) est0X(s) R
Shifting in the s-Domain es0tx(t) X(s s0) Shifted version of R
[i.e., s isin the ROC if (s s0) is inR]
Time scaling x(at)1
|a|X(sa
)Scaled ROC (i.e., s is inthe ROC if (s/a) is in R)
Conjugation x(t) X(s) R
Convolution x1(t) x2(t) X1(s)X2(s) At least R1 R2
Differentiation in the Time Domaind
dtx(t) sX(s) At least R
Differentiation in the s-Domain tx(t)d
dsX(s) R
Integration in the Time Domain
t
x()d()1
sX(s) At least R {e{s} > 0}
Initial- and Final Value Theorems
If x(t) = 0 for t < 0 and x(t) contains no impulses or
higher-order singularities at t = 0, then
x(0+) = lims sX(s)
If x(t) = 0 for t < 0 and x(t) has a finite limit as t,
then
limt x(t) = lims0 sX(s)
Properties of Laplace Transform
Property Signal Transform ROC
-
Table 8: Laplace Transforms of Elementary Functions
Signal Transform ROC
1. (t) 1 All s
2. u(t)1
se{s} > 0
3. u(t)1
se{s} < 0
4.tn1
(n 1)!u(t)
1
sne{s} > 0
5. tn1
(n 1)!u(t)
1
sne{s} < 0
6. etu(t)1
s+ e{s} >
7. etu(t)1
s+ e{s} <
8.tn1
(n 1)!etu(t)
1
(s+ )ne{s} >
9. tn1
(n 1)!etu(t)
1
(s+ )ne{s} <
10. (t T ) esT All s
11. [cos0t]u(t)s
s2 + 20e{s} > 0
12. [sin0t]u(t)0
s2 + 20e{s} > 0
13. [et cos0t]u(t)s+
(s+ )2 + 20e{s} >
14. [et sin0t]u(t)0
(s+ )2 + 20e{s} >
15. un(t) =dn(t)
dtnsn All s
16. un(t) = u(t) u(t) n times
1
sne{s} > 0
Laplace Transform of Elementary Functions
Signal Transform Roc
Haris H.
