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amboleos

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Table 1: Properties of the Continuous-Time Fourier Series
x(t) =
+∞Xk=−∞
akejk!0t =
+∞Xk=−∞
akejk(2/T)t
ak =
1
T ZT
x(t)e−jk!0tdt =
1
T ZT
x(t)e−jk(2/T)tdt
Property Periodic Signal Fourier Series Coefficients
x(t)
y(t) Periodic with period T and
fundamental frequency !0 = 2/T
ak
bk
Linearity Ax(t) + By(t) Aak + Bbk
Time-Shifting x(t − t0) ake−jk!0t0 = ake−jk(2/T)t0
Frequency-Shifting ejM!0t = ejM(2/T)tx(t) ak−M
Conjugation x∗(t) a∗
−k
Time Reversal x(−t) a−k
Time Scaling x(t), > 0 (periodic with period T/) ak
Periodic Convolution ZT
x( )y(t − )d Takbk
Multiplication x(t)y(t)
+∞X l=−∞
albk−l
Differentiation
dx(t)
dt
jk!0ak = jk
2
T
ak
Integration Z t
−∞
x(t)dt
(finite-valued and
periodic only if a0 = 0) 1
jk!0ak = 1
jk(2/T)ak
Conjugate Symmetry
for Real Signals
x(t) real



ak = a∗
−k
ℜe{ak} = ℜe{a−k}
ℑm{ak} = −ℑm{a−k}
|ak| = |a−k|
<) ak = −<) a−k
Real and Even Sig-
nals
x(t) real and even ak real and even
Real and Odd Signals x(t) real and odd ak purely imaginary and odd
Even-Odd Decompo-
sition of Real Signals xe(t) = Ev{x(t)} [x(t) real]
xo(t) = Od{x(t)} [x(t) real]
ℜe{ak}
jℑm{ak}
Parseval’s Relation for Periodic Signals
1
T ZT
|x(t)|2dt =
+∞Xk=−∞
|ak|2
Table 2: Properties of the Discrete-Time Fourier Series
x[n] = X k=<N>
akejk!0n = X k=<N>
akejk(2/N)n
ak =
1
N X n=<N>
x[n]e−jk!0n =
1
N X n=<N>
x[n]e−jk(2/N)n
Property Periodic signal Fourier series coefficients
x[n]
y[n] Periodic with period N and fun-
damental frequency !0 = 2/N
ak
bk Periodic with
period N
Linearity Ax[n] + By[n] Aak + Bbk
Time shift x[n − n0] ake−jk(2/N)n0
Frequency Shift ejM(2/N)nx[n] ak−M
Conjugation x∗[n] a∗
−k
Time Reversal x[−n] a−k
Time Scaling x(m)[n] = x[n/m] if n is a multiple of m
0 if n is not a multiple of m
1
m
ak viewed as
periodic with
period mN !
(periodic with period mN)
Periodic Convolution Xr=hNi
x[r]y[n − r] Nakbk
Multiplication x[n]y[n] Xl=hNi
albk−l
First Difference x[n] − x[n − 1] (1 − e−jk(2/N))ak
Running Sum
nX
k=−∞
x[k] finite-valued and
periodic only if a0 = 0 1
(1 − e−jk(2/N))ak
Conjugate Symmetry
for Real Signals
x[n] real



ak = a∗
−k
ℜe{ak} = ℜe{a−k}
ℑm{ak} = −ℑm{a−k}
|ak| = |a−k|
<) ak = −<) a−k
Real and Even Signals x[n] real and even ak real and even
Real and Odd Signals x[n] real and odd ak purely imaginary and odd
Even-Odd Decomposi-
tion of Real Signals
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
1
N Xn=hNi
|x[n]|2 = Xk=hNi
|ak|2
Table 3: Properties of the Continuous-Time Fourier Transform
x(t) =
1
2 Z ∞
−∞
X(j!)ej!td!
X(j!) = Z ∞
−∞
x(t)e−j!tdt
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−j!t0X(j!)
Frequency-shifting ej!0tx(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
2
X(j!) ∗ Y (j!)
Differentiation in Time
d
dt
x(t) j!X(j!)
Integration Z t
−∞
x(t)dt
1
j!
X(j!) + X(0)(!)
Differentiation in Frequency tx(t) j
d
d!
X(j!)
Conjugate Symmetry for Real
Signals
x(t) real



X(j!) = X∗(−j!)
ℜe{X(j!)} = ℜe{X(−j!)}
ℑm{X(j!)} = −ℑm{X(−j!)}
|X(j!)| = |X(−j!)|
<)X(j!) = −<)X(−j!)
Symmetry for Real and Even
Signals
x(t) real and even X(j!) real and even
Symmetry for Real and Odd
Signals
x(t) real and odd X(j!) purely imaginary and odd
Even-Odd Decomposition for
Real Signals
xe(t) = Ev{x(t)} [x(t) real]
xo(t) = Od{x(t)} [x(t) real]
ℜe{X(j!)}
jℑm{X(j!)}
Parseval’s Relation for Aperiodic Signals
Z +∞
−∞
|x(t)|2dt =
1
2 Z +∞
−∞
|X(j!)|2d!
Table 4: Basic Continuous-Time Fourier Transform Pairs
Fourier series coefficients
Signal Fourier transform (if periodic)
+∞X k=−∞
akejk!0t 2
+∞X k=−∞
ak(! − k!0) ak
ej!0t 2(! − !0)
a1 = 1
ak = 0, otherwise
cos !0t [(! − !0) + (! + !0)]
a1 = a−1 = 1
2
ak = 0, otherwise
sin !0t

j
[(! − !0) − (! + !0)]
a1 = −a−1 = 1
2j
ak = 0, otherwise
x(t) = 1 2(!)
a0 = 1, ak = 0, k 6= 0
this is the Fourier series rep-
resentation for any choice of
T > 0 !
Periodic square wave
x(t) = 1, |t| < T1
0, T1 < |t| ≤ T
2
and
x(t + T) = x(t)
+∞X k=−∞
2 sin k!0T1
k
(! − k!0)
!0T1

sinc k!0T1
=
sin k!0T1
k
+∞Xn=−∞
(t − nT)
2
T
+∞X k=−∞
! −
2k
T ak =
1
T
for all k
x(t) 1, |t| < T1
0, |t| > T1
2 sin !T1
!
—
sinWt
t
X(j!) = 1, |!| < W
0, |!| > W
—
(t) 1 —
u(t)
1
j!
+ (!) —
(t − t0) e−j!t0 —
e−atu(t),ℜe{a} > 0
1
a + j!
—
te−atu(t),ℜe{a} > 0
1
(a + j!)2 —
tn−1
(n−1)!e−atu(t),
ℜe{a} > 0
1
(a + j!)n —
Table 5: Properties of the Discrete-Time Fourier Transform
x[n] =
1
2 Z2
X(ej!)ej!nd!
X(ej!) =
+∞ X n=−∞
x[n]e−j!n
Property Aperiodic Signal Fourier transform
x[n]
y[n]
X(ej!)
Y (ej!) Periodic with
period 2
Linearity ax[n] + by[n] aX(ej!) + bY (ej!)
Time-Shifting x[n − n0] e−j!n0X(ej!)
Frequency-Shifting ej!0nx[n] X(ej(!−!0))
Conjugation x∗[n] X∗(e−j!)
Time Reversal x[−n] X(e−j!)
Time Expansions x(k)[n] = x[n/k], if n = multiple of k
0, if n 6= multiple of k
X(ejk!)
Convolution x[n] ∗ y[n] X(ej!)Y (ej!)
Multiplication x[n]y[n]
1
2 Z2
X(ej)Y (ej(!−))d
Differencing in Time x[n] − x[n − 1] (1 − e−j!)X(ej!)
Accumulation
nX
k=−∞
x[k]
1
1 − e−j!X(ej!)
+X(ej0)
+∞Xk=−∞
(! − 2k)
Differentiation in Frequency nx[n] j
dX(ej!)
d!
Conjugate Symmetry for
Real Signals
x[n] real



X(ej!) = X∗(e−j!)
ℜe{X(ej!)} = ℜe{X(e−j!)}
ℑm{X(ej!)} = −ℑm{X(e−j!)}
|X(ej!)| = |X(e−j!)|
<)X(ej!) = −<)X(e−j!)
Symmetry for Real, Even
Signals
x[n] real and even X(ej!) real and even
Symmetry for Real, Odd
Signals
x[n] real and odd
X(ej!) purely
imaginary and odd
Even-odd Decomposition of
Real Signals
xe[n] = Ev{x[n]} [x[n] real]
xo[n] = Od{x[n]} [x[n] real]
ℜe{X(ej!)}
jℑm{X(ej!)}
Parseval’s Relation for Aperiodic Signals
+∞ X n=−∞
|x[n]|2 =
1
2 Z2
|X(ej!)|2d!
Table 6: Basic Discrete-Time Fourier Transform Pairs
Fourier series coefficients
Signal Fourier transform (if periodic)
X k=hNi
akejk(2/N)n 2
+∞X k=−∞
ak ! −
2k
N ak
ej!0n 2
+∞X l=−∞
(! − !0 − 2l)
(a) !0 = 2m
N
ak = 1, k = m,m ± N,m ± 2N, . . .
0, otherwise
(b) !0
2 irrational ⇒ The signal is aperiodic
cos !0n
+∞X l=−∞
{(! − !0 − 2l) + (! + !0 − 2l)}
(a) !0 = 2m
N
ak = 1
2 , k = ±m,±m ± N,±m ± 2N, . . .
0, otherwise
(b) !0
2 irrational ⇒ The signal is aperiodic
sin !0n

j
+∞X l=−∞
{(! − !0 − 2l) − (! + !0 − 2l)}
(a) !0 = 2r
N
ak = 

1
2j , k = r, r ± N, r ± 2N, . . .
− 1
2j , k = −r,−r ± N,−r ± 2N, . . .
0, otherwise
(b) !0
2 irrational ⇒ The signal is aperiodic
x[n] = 1 2
+∞X l=−∞
(! − 2l) ak = 1, k = 0,±N,±2N, . . .
0, otherwise
Periodic square wave
x[n] = 1, |n| ≤ N1
0, N1 < |n| ≤ N/2
and
x[n + N] = x[n]
2
+∞X k=−∞
ak ! −
2k
N ak = sin[(2k/N)(N1+1
2 )]
N sin[2k/2N] , k 6= 0,±N,±2N, . . .
ak = 2N1+1
N , k = 0,±N,±2N, . . .
+∞X k=−∞
[n − kN]
2
N
+∞X k=−∞
! −
2k
N ak =
1
N
for all k
anu[n], |a| < 1
1
1 − ae−j! —
x[n] 1, |n| ≤ N1
0, |n| > N1
sin[!(N1 + 1
2 )]
sin(!/2)
—
sinWn
n = W
sinc 􀀀Wn
0 < W <
X(ej!) = 1, 0 ≤ |!| ≤ W
0, W < |!| ≤
X(ej!)periodic with period 2
—
[n] 1 —
u[n]
1
1 − e−j! +
+∞X k=−∞
(! − 2k) —
[n − n0] e−j!n0 —
(n + 1)anu[n], |a| < 1
1
(1 − ae−j!)2 —
(n + r − 1)!
n!(r − 1)!
anu[n], |a| < 1
1
(1 − ae−j!)r —
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) e−st0X(s) R
Shifting in the s-Domain es0tx(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
|a|
X s
a “Scaled” ROC (i.e., s is in
the 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 Domain
d
dt
x(t) sX(s) At least R
Differentiation in the s-Domain −tx(t)
d
ds
X(s) R
Integration in the Time Domain Z t
−∞
x( )d( )
1
s
X(s) 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. (t) 1 All s
2. u(t)
1
s
ℜe{s} > 0
3. −u(−t)
1
s
ℜe{s} < 0
4.
tn−1
(n − 1)!
u(t)
1
sn ℜe{s} > 0
5. −
tn−1
(n − 1)!
u(−t)
1
sn ℜe{s} < 0
6. e−tu(t)
1
s +
ℜe{s} > −ℜe{}
7. −e−tu(−t)
1
s +
ℜe{s} < −ℜe{}
8.
tn−1
(n − 1)!
e−tu(t)
1
(s + )n ℜe{s} > −ℜe{}
9. −
tn−1
(n − 1)!
e−tu(−t)
1
(s + )n ℜe{s} < −ℜe{}
10. (t − T) e−sT All s
11. [cos !0t]u(t)
s
s2 + !2
0
ℜe{s} > 0
12. [sin !0t]u(t)
!0
s2 + !2
0
ℜe{s} > 0
13. [e−t cos !0t]u(t)
s +
(s + )2 + !2
0
ℜe{s} > −ℜe{}
14. [e−t sin !0t]u(t)
!0
(s + )2 + !2
0
ℜe{s} > −ℜe{}
15. un(t) =
dn(t)
dtn sn All s
16. u−n(t) = u(t) ∗ · · · ∗ u(t)
| {z } n times
1
sn ℜe{s} > 0
Table 9: Properties of the z-Transform
Property Sequence Transform ROC
x[n] X(z) R
x1[n] X1(z) R1
x2[n] X2(z) R2
Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) At least the intersection
of R1 and R2
Time shifting x[n − n0] z−n0X(z) R except for the
possible addition or
deletion of the origin
Scaling in the ej!0nx[n] X(e−j!0z) R
z-Domain zn
0 x[n] X z
z0 z0R
anx[n] X(a−1z) Scaled version of R
(i.e., |a|R = the
set of points {|a|z}
for z in R)
Time reversal x[−n] X(z−1) Inverted R (i.e., R−1
= the set of points
z−1 where z is in R)
Time expansion x(k)[n] = x[r], n = rk
0, n 6= rk
X(zk) R1/k
for some integer r (i.e., the set of points z1/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
k=−∞ x[k] 1
1−z−1X(z) At least the
intersection of R and |z| > 1
Differentiation nx[n] −z dX(z)
dz R
in the z-Domain
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. [n] 1 All z
2. u[n] 1
1−z−1 |z| > 1
3. u[−n − 1] 1
1−z−1 |z| < 1
4. [n − m] z−m All z except
0 (if m > 0) or
∞ (if m < 0)
5. nu[n] 1
1−z−1 |z| > ||
6. −nu[−n − 1] 1
1−z−1 |z| < ||
7. nnu[n] z−1
(1−z−1)2 |z| > ||
8. −nnu[−n − 1] z−1
(1−z−1)2 |z| < ||
9. [cos !0n]u[n] 1−[cos !0]z−1
1−[2 cos !0]z−1+z−2 |z| > 1
10. [sin !0n]u[n] [sin !0]z−1
1−[2 cos !0]z−1+z−2 |z| > 1
11. [rn cos !0n]u[n] 1−[r cos !0]z−1
1−[2r cos !0]z−1+r2z−2 |z| > r
12. [rn sin !0n]u[n] [r sin !0]z−1
1−[2r cos !0]z−1+r2z−2 |z| > r

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