Table 1: Properties of the Continuous-Time Fourier Series
+∞
X
x(t) =
jkω0 t
ak e
k=−∞
1
ak =
T
Property
Periodic Convolution
−jkω0 t
x(t)e
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 ak b k
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




ℜe{ak } = ℜe{a−k }
ℑm{ak } = −ℑm{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
|ak |2
|x(t)|2 dt =
T T
k=−∞
ℜe{ak }
jℑm{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]
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 ak bk
r=hN i
Multiplication
ak
bk
X
x[n]y[n]
al bk−l
l=hN i
First Difference
Running Sum
x[n] − x[n − 1]
n
X
finite-valued and
x[k]
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




ℜe{ak } = ℜe{a−k }
ℑm{ak } = −ℑm{a−k }


|a

k | = |a−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]
ℜe{ak }
jℑm{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(jω)ejωt dω
x(t) =
2π −∞
Z ∞
X(jω) =
x(t)e−jωt dt
−∞
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
Differentiation in Time
Integration
d
x(t)
Zdt t
x(t)dt
−∞
Differentiation in Frequency
tx(t)
Conjugate Symmetry for Real
x(t) real
Signals
Symmetry for Real and Even
x(t) real and even
Signals
Symmetry for Real and Odd
x(t) real and odd
Signals
Even-Odd Decomposition for xe (t) = Ev{x(t)} [x(t) real]
xo (t) = Od{x(t)} [x(t) real]
Real Signals
jωX(jω)
1
X(jω) + πX(0)δ(ω)
jω
d
j X(jω)
 dω
X(jω) = X ∗ (−jω)




ℜe{X(jω)} = ℜe{X(−jω)}
ℑm{X(jω)} = −ℑm{X(−jω)}


|X(jω)|
= |X(−jω)|



<
) X(jω) = −<
) X(−jω)
X(jω) real and even
X(jω) purely imaginary and odd
ℜe{X(jω)}
jℑm{X(jω)}
Parseval’s Relation for Aperiodic Signals
Z +∞
Z +∞
1
2
|X(jω)|2 dω
|x(t)| dt =
2π
−∞
−∞
Table 4: Basic Continuous-Time Fourier Transform Pairs
Signal
+∞
X
Fourier transform
ak ejkω0 t
2π
+∞
X
ak δ(ω − kω0 )
k=−∞
k=−∞
jω0 t
e
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 )
n=−∞
x(t)
1, |t| < T1
0, |t| > T1
sin W t
πt
δ(t)
u(t)
δ(t − t0 )
e−at u(t), ℜe{a} > 0
te−at u(t), ℜe{a} > 0
tn−1
−at u(t),
(n−1)! e
ℜe{a} > 0
+∞
X
2 sin kω0 T1
δ(ω − kω0 )
k
k=−∞
+∞
2π X
2πk
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
Fourier series coefficients
(if periodic)
ak
a1 = 1
ak = 0, otherwise
a1 = a−1 = 12
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 T 1
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(ejω )ejωn dω
x[n] =
2π 2π
jω
X(e ) =
+∞
X
x[n]e−jωn
n=−∞
Property
Aperiodic Signal
Fourier transform
Periodic with
X(ejω )
period 2π
Y (ejω )
jω
aX(e ) + 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
δ(ω − 2πk)
+πX(e )
k=−∞
Differentiation in Frequency
Conjugate Symmetry
Real Signals
nx[n]
for x[n] real
Symmetry for Real, Even x[n] real and even
Signals
Symmetry for Real, Odd x[n] real and odd
Signals
Even-odd Decomposition of xe [n] = Ev{x[n]} [x[n] real]
xo [n] = Od{x[n]} [x[n] real]
Real Signals
dX(ejω )
j
dω
 jω
X(e ) = X ∗ (e−jω )




ℜe{X(ejω )} = ℜe{X(e−jω )}
ℑm{X(ejω )} = −ℑm{X(e−jω )}


|X(ejω )| = |X(e−jω )|



<
) X(ejω ) = −<
) X(e−jω )
X(ejω ) real and even
X(ejω ) purely
imaginary and odd
Parseval’s Relation for Aperiodic Signals
Z
+∞
X
1
2
|X(ejω )|2 dω
|x[n]| =
2π
2π
n=−∞
ℜe{X(ejω )}
jℑm{X(ejω )}
Table 6: Basic Discrete-Time Fourier Transform Pairs
Signal
X
Fourier transform
jk(2π/N )n
2π
ak e
+∞
X
k=−∞
k=hN i
ejω0 n
2π
+∞
X
2πk
ak δ ω −
N
cos ω0 n
π
δ(ω − ω0 − 2πl)
{δ(ω − ω0 − 2πl) + δ(ω + ω0 − 2πl)}
l=−∞
+∞
π X
{δ(ω − ω0 − 2πl) − δ(ω + ω0 − 2πl)}
j
sin ω0 n
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],
x[n]
sin W n
πn
1,
0,
|a| < 1
|n| ≤ N1
|n| > N1
=W
π sinc
0<W <π
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
δ ω−
N
k=−∞
1
1 − ae−jω
sin[ω(N1 + 12 )]
sin(ω/2)
1,
X(ejω ) =
0,
X(ejω )periodic
2πk
N
2πm
=
N
1, k = m, m ± N, m ± 2N, . . .
ak =
0, otherwise
0
irrational ⇒ The signal is aperiodic
(b) ω
2π
2πm
(a) ω0 = N
1
2 , k = ±m, ±m ± N, ±m ± 2N, . . .
ak =
0, otherwise
0
irrational ⇒ The signal is aperiodic
(b) ω
2π
2πr
(a) ω0 =
 N1
 2j , k = r, r ± N, r ± 2N, . . .
− 1 , k = −r, −r ± N, −r ± 2N, . . .
ak
=
 2j
0, otherwise
0
irrational
⇒ The signal is aperiodic
(b) ω
2π
1, k = 0, ±N, ±2N, . . .
ak =
0, otherwise
1
+
1 − e−jω
ω0
ak
ak
=
=
ak =
—
0 ≤ |ω| ≤ W
W < |ω| ≤ π
with period 2π
—
—
+∞
X
πδ(ω − 2πk)
—
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
—
1
e−jωn0
ak
(a)
l=−∞
+∞
X
Fourier series coefficients
(if periodic)
—
—
—
Table 7: Properties of the Laplace Transform
Property
Linearity
Signal
Transform
ROC
x(t)
X(s)
R
x1 (t)
X1 (s)
R1
x2 (t)
X2 (s)
R2
ax1 (t) + bx2 (t) 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 ∩ {ℜe{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
9. −
e u(−t)
(n − 1)!
8.
e−sT
10. δ(t − T )
11. [cos ω0 t]u(t)
s
+ ω02
ω0
2
s + 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
ℜe{s} > 0
ℜe{s} < 0
ℜe{s} > 0
ℜe{s} < 0
ℜe{s} > −ℜe{α}
ℜe{s} < −ℜe{α}
ℜe{s} > −ℜe{α}
ℜe{s} < −ℜe{α}
All s
ℜe{s} > 0
ℜe{s} > 0
ℜe{s} > −ℜe{α}
ℜe{s} > −ℜe{α}
sn
All s
1
sn
ℜe{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]
n
a x[n]
−jω0
X(e
z)
X zz0
X(a−1 z)
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 reversal
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. [rn cos ω0 n]u[n]
1−[r cos ω0 ]z −1
1−[2r cos ω0 ]z −1 +r 2 z −2
|z| > r
12. [rn sin ω0 n]u[n]
[r sin ω0 ]z −1
1−[2r cos ω0 ]z −1 +r 2 z −2
|z| > r