Table 1: Properties of the Continuous-Time Fourier Series
+∞
X
x(t) =
ak e
jkω0 t
k=−∞
1
ak =
T
Property
Periodic Convolution
x(t)e
−jkω0 t
T
ak ejk(2π/T )t
k=−∞
1
dt =
T
Z
x(t)e−jk(2π/T )t dt
T
Periodic Signal
x(t)
y(t)
Linearity
Time-Shifting
Frequency-Shifting
Conjugation
Time Reversal
Time Scaling
Z
=
+∞
X
Fourier Series Coefficients
Periodic with period T and
fundamental frequency ω0 = 2π/T
Ax(t) + By(t)
x(t − t0 )
ejM ω0 t = ejM (2π/T )t x(t)
x∗ (t)
x(−t)
x(αt),
α > 0 (periodic with period T /α)
Z
x(τ )y(t − τ )dτ
ak
bk
Aak + Bbk
ak e−jkω0 t0 = ak e−jk(2π/T )t0
ak−M
a∗−k
a−k
ak
T a k bk
T
Multiplication
+∞
X
x(t)y(t)
al bk−l
l=−∞
Differentiation
dx(t)
Z dt
t
Integration
x(t)dt
−∞
(finite-valued and
periodic only if a0 = 0)
Conjugate Symmetry
for Real Signals
x(t) real
Real and Even Signals
x(t) real and even
Real and Odd Signals
x(t) real and odd
Even-Odd Decomposition of Real Signals
2π
jkω0 ak = jk ak
T
1
1
ak =
ak
jk(2π/T )
 jkω0 ∗
ak = a−k




<e{ak } = <e{a−k }
=m{ak } = −=m{a−k }


|ak | = |a−k |



<
) ak = −<
) a−k
ak real and even
ak purely imaginary and odd
xe (t) = Ev{x(t)} [x(t) real]
xo (t) = Od{x(t)} [x(t) real]
Parseval’s Relation for Periodic Signals
Z
+∞
X
1
2
|x(t)| dt =
|ak |2
T T
k=−∞
<e{ak }
j=m{ak }
Table 2: Properties of the Discrete-Time Fourier Series
X
X
x[n] =
ak ejkω0 n =
ak ejk(2π/N )n
k=<N >
1
N
ak =
Property
Time Scaling
Periodic Convolution
x[n]e−jkω0 n =
n=<N >
1
N
X
x[n]e−jk(2π/N )n
n=<N >
Periodic signal
x[n]
y[n]
Linearity
Time shift
Frequency Shift
Conjugation
Time Reversal
X
k=<N >
Fourier series coefficients
Periodic with period N and fundamental frequency ω0 = 2π/N
Ax[n] + By[n]
x[n − n0 ]
ejM (2π/N )n x[n]
x∗ [n]
x[−n]
x[n/m] if n is a multiple of m
x(m) [n] =
0
if n is not a multiple of m
(periodic with period mN )
X
x[r]y[n − r]
x[n]y[n]
First Difference
x[n] − x[n − 1]
n
X
finite-valued and
x[k]
Periodic with
period N
Aak + Bbk
ak e−jk(2π/N )n0
ak−M
a∗−k
a−k
viewed as
1
ak periodic with
m
period mN
!
N a k bk
r=hN i
Multiplication
ak
bk
X
al bk−l
l=hN i
Running Sum
k=−∞
periodic only if a0 = 0
Conjugate Symmetry
for Real Signals
x[n] real
Real and Even Signals
x[n] real and even
Real and Odd Signals
x[n] real and odd
Even-Odd Decomposition of Real Signals
(1 − e−jk(2π/N ) )ak
1
ak
(1 − e−jk(2π/N ) )

ak = a∗−k




<e{ak } = <e{a−k }
=m{ak } = −=m{a−k }


|a | = |a−k |


 k
<
) ak = −<
) a−k
ak real and even
ak purely imaginary and odd
xe [n] = Ev{x[n]} [x[n] real]
xo [n] = Od{x[n]} [x[n] real]
<e{ak }
j=m{ak }
Parseval’s Relation for Periodic Signals
X
1 X
|x[n]|2 =
|ak |2
N
n=hN i
k=hN i
Table 3: Properties of the Continuous-Time Fourier Transform
Z ∞
1
x(t) =
X(jω)ejωt dω
2π −∞
Z ∞
x(t)e−jωt dt
X(jω) =
−∞
Property
Aperiodic Signal
Fourier transform
x(t)
y(t)
X(jω)
Y (jω)
Linearity
Time-shifting
Frequency-shifting
Conjugation
Time-Reversal
ax(t) + by(t)
x(t − t0 )
ejω0 t x(t)
x∗ (t)
x(−t)
Time- and Frequency-Scaling
x(at)
Convolution
x(t) ∗ y(t)
Multiplication
x(t)y(t)
aX(jω) + bY (jω)
e−jωt0 X(jω)
X(j(ω − ω0 ))
X ∗ (−jω)
X(−jω)
1
jω
X
|a|
a
X(jω)Y (jω)
1
X(jω) ∗ Y (jω)
2π
d
x(t)
Zdt
Differentiation in Time
t
x(t)dt
Integration
−∞
Differentiation in Frequency
tx(t)
Conjugate Symmetry for Real
Signals
x(t) real
Symmetry for Real and Even
Signals
Symmetry for Real and Odd
Signals
Even-Odd Decomposition for
Real Signals
x(t) real and even
x(t) real and odd
xe (t) = Ev{x(t)} [x(t) real]
xo (t) = Od{x(t)} [x(t) real]
jωX(jω)
1
X(jω) + πX(0)δ(ω)
jω
d
j X(jω)
 dω
X(jω) = X ∗ (−jω)




<e{X(jω)} = <e{X(−jω)}
=m{X(jω)} = −=m{X(−jω)}


|X(jω)| = |X(−jω)|



<
) X(jω) = −<
) X(−jω)
X(jω) real and even
X(jω) purely imaginary and odd
<e{X(jω)}
j=m{X(jω)}
Parseval’s Relation for Aperiodic Signals
Z +∞
Z +∞
1
2
|X(jω)|2dω
|x(t)| dt =
2π −∞
−∞
Table 4: Basic Continuous-Time Fourier Transform Pairs
Signal
+∞
X
Fourier transform
ak e
jkω0 t
k=−∞
2π
+∞
X
ak δ(ω − kω0 )
2πδ(ω − ω0 )
cos ω0 t
π[δ(ω − ω0 ) + δ(ω + ω0 )]
sin ω0 t
π
[δ(ω − ω0 ) − δ(ω + ω0 )]
j
x(t) = 1
2πδ(ω)
Periodic
square wave
1, |t| < T1
x(t) =
0, T1 < |t| ≤ T2
and
x(t + T ) = x(t)
+∞
X
δ(t − nT )
x(t)
1, |t| < T1
0, |t| > T1
sin W t
πt
δ(t)
u(t)
δ(t − t0 )
e
−at
ak
k=−∞
ejω0 t
n=−∞
Fourier series coefficients
(if periodic)
u(t), <e{a} > 0
te−at u(t), <e{a} > 0
tn−1 −at
u(t),
(n−1)! e
<e{a} > 0
+∞
X
2 sin kω0 T1
δ(ω − kω0 )
k
k=−∞
+∞
2πk
2π X
δ ω−
T
T
k=−∞
2 sin ωT1
ω
1, |ω| < W
X(jω) =
0, |ω| > W
1
1
+ πδ(ω)
jω
e−jωt0
1
a + jω
1
(a + jω)2
1
(a + jω)n
a1 = 1
ak = 0, otherwise
a1 = a−1 = 21
ak = 0, otherwise
1
a1 = −a−1 = 2j
ak = 0, otherwise
a0 = 1, ak = 0, k 6= 0
!
this is the Fourier series representation for any choice of
T >0
ω0 T1
sinc
π
ak =
kω0 T1
π
1
for all k
T
—
—
—
—
—
—
—
—
=
sin kω0 T1
kπ
Table 5: Properties of the Discrete-Time Fourier Transform
Z
1
x[n] =
X(ejω )ejωn dω
2π 2π
jω
X(e ) =
+∞
X
x[n]e−jωn
n=−∞
Property
Aperiodic Signal
Fourier transform
Periodic with
X(ejω )
jω
period 2π
Y (e )
aX(ejω ) + bY (ejω )
e−jωn0 X(ejω )
X(ej(ω−ω0 ) )
X ∗ (e−jω )
X(e−jω )
Convolution
x[n]
y[n]
ax[n] + by[n]
x[n − n0 ]
ejω0 n x[n]
x∗ [n]
x[−n]
x[n/k], if n = multiple of k
x(k) [n] =
0,
if n 6= multiple of k
x[n] ∗ y[n]
Multiplication
x[n]y[n]
Differencing in Time
x[n] − x[n − 1]
n
X
x[k]
Linearity
Time-Shifting
Frequency-Shifting
Conjugation
Time Reversal
Time Expansions
Accumulation
k=−∞
X(ejkω )
X(eZjω )Y (ejω )
1
X(ejθ )Y (ej(ω−θ) )dθ
2π 2π
(1 − e−jω )X(ejω )
1
X(ejω )
1 − e−jω
+∞
X
j0
+πX(e )
δ(ω − 2πk)
k=−∞
Differentiation in Frequency
nx[n]
Conjugate Symmetry
Real Signals
x[n] real
for
Symmetry for Real, Even
Signals
x[n] real and even
Symmetry for Real, Odd
Signals
Even-odd Decomposition of
Real Signals
x[n] real and odd
dX(ejω )
j
dω
 jω
X(e ) = X ∗ (e−jω )




<e{X(ejω )} = <e{X(e−jω )}
=m{X(ejω )} = −=m{X(e−jω )}


|X(ejω )| = |X(e−jω )|



<
) X(ejω ) = −<
) X(e−jω )
X(ejω ) real and even
X(ejω ) purely
imaginary and odd
xe [n] = Ev{x[n]} [x[n] real]
xo [n] = Od{x[n]} [x[n] real]
Parseval’s Relation for Aperiodic Signals
Z
+∞
X
1
2
|X(ejω )|2 dω
|x[n]| =
2π 2π
n=−∞
<e{X(ejω )}
j=m{X(ejω )}
Table 6: Basic Discrete-Time Fourier Transform Pairs
Signal
X
Fourier transform
ak ejk(2π/N )n
2π
+∞
X
2πk
ak δ ω −
N
+∞
X
δ(ω − ω0 − 2πl)
k=−∞
k=hN i
ejω0 n
2π
+∞
X
π
ω0
ak
ω0
2π
ω0
(b)
(a)
{δ(ω − ω0 − 2πl) + δ(ω + ω0 − 2πl)}
l=−∞
+∞
π X
{δ(ω − ω0 − 2πl) − δ(ω + ω0 − 2πl)}
j
sin ω0 n
ak
(a)
l=−∞
cos ω0 n
Fourier series coefficients
(if periodic)
ak
(b)
(a)
ω0
2π
ω0
ak
=
(b)
ω0
l=−∞
x[n] = 1
2π
+∞
X
δ(ω − 2πl)
l=−∞
Periodic
square wave
1, |n| ≤ N1
x[n] =
0, N1 < |n| ≤ N/2
and
x[n + N ] = x[n]
+∞
X
δ[n − kN ]
k=−∞
an u[n], |a| < 1
x[n]
1, |n| ≤ N1
0, |n| > N1
=W
π sinc
0<W <π
sin W n
πn
Wn
π
δ[n]
u[n]
δ[n − n0 ]
(n + 1)an u[n], |a| < 1
(n + r − 1)! n
a u[n], |a| < 1
n!(r − 1)!
2π
+∞
X
k=−∞
2πk
ak δ ω −
N
+∞
2π X
2πk
δ ω−
N
N
k=−∞
1
1 − ae−jω
sin[ω(N1 + 21 )]
sin(ω/2)
1, 0 ≤ |ω| ≤ W
X(ω) =
0, W < |ω| ≤ π
X(ω)periodic with period 2π
1
e−jωn0
2π
1, k = 0, ±N, ±2N, . . .
ak =
0, otherwise
ak
ak
=
=
ak =
—
—
—
—
k=−∞
1
(1 − ae−jω )2
1
(1 − ae−jω )r
sin[(2πk/N )(N1 + 12 )]
, k 6= 0, ±N, ±2N, . . .
N sin[2πk/2N ]
2N1 +1
,
k
=
0,
±N,
±2N, . . .
N
1
for all k
N
—
+∞
X
1
+
πδ(ω − 2πk)
1 − e−jω
2πm
=
N
1, k = m, m ± N, m ± 2N, . . .
=
0, otherwise
irrational ⇒ The signal is aperiodic
2πm
=
N
1
2 , k = ±m, ±m ± N, ±m ± 2N, . . .
=
0, otherwise
irrational ⇒ The signal is aperiodic
2πr
=
 N1
 2j , k = r, r ± N, r ± 2N, . . .
− 1 , k = −r, −r ± N, −r ± 2N, . . .
 2j
0, otherwise
irrational ⇒ The signal is aperiodic
—
—
—
Table 7: Properties of the Laplace Transform
Property
Linearity
Signal
Transform
x(t)
X(s)
R
x1 (t)
X1 (s)
R1
x2 (t)
X2 (s)
R2
ax1 (t) + bx2 (t)
ROC
aX1 (s) + bX2 (s) At least R1 ∩ R2
Time shifting
x(t − t0 )
e−st0 X(s)
R
Shifting in the s-Domain
es0 t x(t)
X(s − s0 )
Shifted version of R [i.e., s is
in the ROC if (s − s0 ) is in
R]
Time scaling
x(at)
1 s
X
|a|
a
Conjugation
x∗ (t)
X ∗ (s∗ )
Convolution
x1 (t) ∗ x2 (t)
X1 (s)X2 (s)
Differentiation in the Time Domain
d
x(t)
dt
sX(s)
Differentiation in the s-Domain
−tx(t)
Integration in the Time Domain
Z
t
x(τ )d(τ )
−∞
d
X(s)
ds
1
X(s)
s
“Scaled” ROC (i.e., s is in
the ROC if (s/a) is in R)
R
At least R1 ∩ R2
At least R
R
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) = lims→0 sX(s)
Table 8: Laplace Transforms of Elementary Functions
Signal
Transform
ROC
1. δ(t)
1
All s
2. u(t)
3. −u(−t)
tn−1
u(t)
(n − 1)!
tn−1
5. −
u(−t)
(n − 1)!
4.
6. e−αt u(t)
7. −e−αt u(−t)
tn−1 −αt
e u(t)
(n − 1)!
tn−1 −αt
e u(−t)
9. −
(n − 1)!
8.
e−sT
10. δ(t − T )
11. [cos ω0 t]u(t)
s
+ ω02
ω0
s2 + ω02
s+α
(s + α)2 + ω02
ω0
(s + α)2 + ω02
s2
12. [sin ω0 t]u(t)
13. [e−αt cos ω0 t]u(t)
14. [e−αt sin ω0 t]u(t)
15. un (t) =
1
s
1
s
1
sn
1
sn
1
s+α
1
s+α
1
(s + α)n
1
(s + α)n
dn δ(t)
dtn
16. u−n (t) = u(t) ∗ · · · ∗ u(t)
|
{z
}
n times
<e{s} > 0
<e{s} < 0
<e{s} > 0
<e{s} < 0
<e{s} > −α
<e{s} < −α
<e{s} > −α
<e{s} < −α
All s
<e{s} > 0
<e{s} > 0
<e{s} > −α
<e{s} > −α
sn
All s
1
sn
<e{s} > 0
Table 9: Properties of the z-Transform
Property
Sequence
Transform
ROC
x[n]
x1 [n]
x2 [n]
X(z)
X1 (z)
X2 (z)
R
R1
R2
Linearity
ax1 [n] + bx2 [n]
aX1 (z) + bX2 (z)
At least the intersection
of R1 and R2
Time shifting
x[n − n0 ]
z −n0 X(z)
R except for the
possible addition or
deletion of the origin
Scaling in the
ejω0 n x[n]
R
z-Domain
z0n x[n]
an x[n]
−jω0
X(e
z)
X zz0
X(a−1 z)
Time reversal
x[−n]
X(z −1 )
Inverted R (i.e., R−1
= the set of points
z −1 where z is in R)
X(z k )
R1/k
Time expansion
x(k) [n] =
x[r], n = rk
0,
n 6= rk
for some integer r
z0 R
Scaled version of R
(i.e., |a|R = the
set of points {|a|z}
for z in R)
(i.e., the set of points z 1/k
where z is in R)
Conjugation
x∗ [n]
X ∗ (z ∗ )
R
Convolution
x1 [n] ∗ x2 [n]
X1 (z)X2 (z)
At least the intersection
of R1 and R2
First difference
x[n] − x[n − 1]
(1 − z −1 )X(z)
At least the
intersection of R and |z| > 0
Accumulation
Pn
1
X(z)
1−z −1
At least the
intersection of R and |z| > 1
Differentiation
in the z-Domain
nx[n]
−z dX(z)
dz
R
k=−∞ x[k]
Initial Value Theorem
If x[n] = 0 for n < 0, then
x[0] = limz→∞ X(z)
Table 10: Some Common z-Transform Pairs
Signal
Transform
ROC
1. δ[n]
1
All z
2. u[n]
1
1−z −1
|z| > 1
3. −u[−n − 1]
1
1−z −1
|z| < 1
4. δ[n − m]
z −m
All z except
0 (if m > 0) or
∞ (if m < 0)
5. αn u[n]
1
1−αz −1
|z| > |α|
6. −αn u[−n − 1]
1
1−αz −1
|z| < |α|
7. nαn u[n]
αz −1
(1−αz −1 )2
|z| > |α|
8. −nαn u[−n − 1]
αz −1
(1−αz −1 )2
|z| < |α|
9. [cos ω0 n]u[n]
1−[cos ω0 ]z −1
1−[2 cos ω0 ]z −1 +z −2
|z| > 1
10. [sin ω0 n]u[n]
[sin ω0 ]z −1
1−[2 cos ω0 ]z −1 +z −2
|z| > 1
11. [r n cos ω0 n]u[n]
1−[r cos ω0 ]z −1
1−[2r cos ω0 ]z −1 +r 2 z −2
|z| > r
12. [r n sin ω0 n]u[n]
[r sin ω0 ]z −1
1−[2r cos ω0 ]z −1 +r 2 z −2
|z| > r