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Mathematical Tables
Composed by Vincent Verdult
Department of Electrical EngineeringDelft University of Technology
e-mail: [email protected]
November, 1997
Contents
1 Trigonometric Identities 2
2 Trigonometric Functions 3
3 Hyperbolic Functions 3
4 Series 4
5 Inequalities 6
6 Differential and Integral Calculus 6
7 Integral Table 8
8 Standard Limits 11
9 Convolutions 12
10 Dirac Delta Function 12
11 Fourier Transform 13
12 Laplace Transform 21
13 z-Transform 25
1 Trigonometric Identities
sin(x± π
2) = ± cosx
cos(x± π
2) = ∓ sinx
sin(x± y) = sinx cos y ± cosx sin y
cos(x± y) = cosx cos y ∓ sinx sin y
tan(x± y) =tanx± tan y
1∓ tanx tan y
2 sinx sin y = cos(x− y)− cos(x+ y)
2 cosx cos y = cos(x− y) + cos(x+ y)
2 sinx cos y = sin(x− y) + sin(x+ y)
sinx+ sin y = 2 sin(x+ y)
2cos
(x− y)2
sinx− sin y = 2 cos(x+ y)
2sin
(x− y)2
cosx+ cos y = 2 cos(x+ y)
2cos
(x− y)2
cosx− cos y = −2 sin(x+ y)
2sin
(x− y)2
tanx± tan y =sin(x± y)cosx cos y
sin 2x = 2 sinx cos y
cos 2x = cos2 x− sin2 x
tan 2x =2 tanx
1− tan2 x
2 sin2 x = 1− cos 2x
2 cos2 x = 1 + cos 2x
2
4 sin3 x = 3 sinx− sin 3x
4 cos3 x = 3 cosx+ cos 3x
8 sin4 x = 3− 4 cos 2x+ cos 4x
8 cos4 x = 3 + 4 cos 2x+ cos 4x
a cosx− b sinx = r cos(x+ θ) where r =√a2 + b2
θ = arctan(ba
)a = r cos θb = r sin θ
2 Trigonometric Functions
sinx =ejx − e−jx
2j
cosx =ejx + e−jx
2
tanx =sinxcosx
cos2 x+ sin2 x = 1
e±jx = cosx± j sinx
3 Hyperbolic Functions
sinhx =ex − e−x
2
coshx =ex + e−x
2
tanhx =sinhxcoshx
cosh2 x− sinh2 x = 1
e±x = coshx± sinhx
arcsinhx = ln(x+√x2 + 1)
arccoshx = ln(x+√x2 − 1), x ≥ 1
arctanhx =12
ln(
1 + x
1− x
), |x| < 1
3
4 Series
Series Expansions
f(x+ a) =∞∑n=0
xn
n!f (n)(a) = f(a) +
x
1!f ′(a) +
x
2!f ′′(a) + . . . Taylor’s series
ex =∞∑n=0
xn
n!= 1 +
x
1!+x2
2!+x3
3!+ . . . , |x| <∞
sinx =∞∑n=0
(−1)nx2n+1
(2n+ 1)!= x− x3
3!+x5
5!− x7
7!+ . . . , |x| <∞
cosx =∞∑n=0
(−1)nx2n
(2n)!= 1− x2
2!+x4
4!− x6
6!+ . . . , |x| <∞
sinhx =∞∑n=0
x2n+1
(2n+ 1)!= x+
x3
3!+x5
5!+x7
7!+ . . . , |x| <∞
coshx =∞∑n=0
x2n
(2n)!= 1 +
x2
2!+x4
4!+x6
6!+ . . . , |x| <∞
arcsinx =∞∑n=0
(2n)!22n(n!)2
x2n+1
2n+ 1= x+
12x3
3+
38x5
5+ . . . , |x| ≤ 1
arccosx =π
2− arcsinx =
π
2−∞∑n=0
(2n)!22n(n!)2
x2n+1
2n+ 1, |x| ≤ 1
arctanx =∞∑n=0
(−1)nx2n+1
2n+ 1= x− x3
3+x5
5− x7
7+ . . . , |x| ≤ 1
arctanhx =∞∑n=0
x2n+1
2n+ 1= x+
x3
3+x5
5+x7
7+ . . . , |x| < 1
ln(1 + x) =∞∑n=1
(−1)n+1xn
n= x− x2
2+x3
3− x4
4+ . . . , |x| ≤ 1
11− x
=∞∑n=0
xn = 1 + x+ x2 + x3 + . . . , |x| < 1
(1 + x)a =∞∑n=0
(a
n
)xn = 1 +
(a
1
)x+
(a
2
)x2 +
(a
3
)x3 + . . . , |x| < 1
where
(a
k
)=a · (a− 1) · (a− 2) · · · (a− k + 1)
k!, a real
4
Infinite Series∞∑n=1
(−1)n+1
n= ln 2
∞∑n=1
(−1)n+1
n2=π2
12
∞∑n=1
1(2n− 1)2
=π2
8
∞∑n=0
(−1)n
n=
1e
∞∑n=1
n
(n+ 1)!= 1
Finite Seriesm∑n=0
(m
n
)xm−nyn = (x+ y)m where
(m
n
)=
m!n! (m− n)!
m∑n=1
n =m(m+ 1)
2
m∑n=1
n2 =m(m+ 1)(2m+ 1)
6
m∑n=1
n3 =m2(m+ 1)2
4
m∑n=0
xn =1− xm+1
1− x
m∑n=1
(2n− 1) = m2
m−1∑n=0
(x+ ny) =m
2[2x+ (m− 1)y]
m∑n=0
ej(x+2ny) =sin [(m+ 1)y]
sin yej(x+my)
5
5 Inequalities
Inequalities for series∣∣∣∣∣m∑n=1
xnyn
∣∣∣∣∣2
≤(
m∑n=1
|xn|2)·(
m∑n=1
|yn|2)
Cauchy-Schwartz inequality
∣∣∣∣∣m∑n=1
xnyn
∣∣∣∣∣ ≤(
m∑n=1
|xn|p)1/p
·(
m∑n=1
|yn|q)1/q
where 1p + 1
q = 1, p > 1, q > 1
Holder’s inequality(m∑n=1
(xn + yn)p)1/p
≤(
m∑n=1
xpn
)1/p
+
(m∑n=1
ypn
)1/p
where xn ≥ 0, yn ≥ 0, p ≥ 1
Minkowski inequality
Inequalities for integrals∣∣∣∣∫ f(x)g(x) dx∣∣∣∣2 ≤ (∫ |f(x)|2 dx
)·(∫|g(x)|2 dx
)Cauchy-Schwartz inequality
∫|f(x)g(x)| dx ≤
(∫|f(x)|p dx
)1/p
·(∫|g(x)|q dx
)1/q
where 1p + 1
q = 1, p > 1, q > 1 Holder’s inequality(∫|f(x) + g(x)|p dx
)1/p
≤(∫|f(x)|p dx
)1/p
+(∫|g(x)|q dx
)1/q
where p > 1
Minkowski inequality
6 Differential and Integral Calculus
d
dx
(f(x)g(x)
)= f(x)g′(x) + f ′(x)g(x), Product rule
d
dx
(f(x)g(x)
)=g(x)f ′(x)− f(x)g′(x)
g2(x), Quotient rule
d
dxf(g(x)
)=df
dg
dg
dx, Chain rule
d
dx
∫ g(x)
f(x)h(x, λ) dλ = h
(x, g(x)
)g′(x)− h
(x, f(x)
)f ′(x) +
∫ g(x)
f(x)
∂
∂xh(x, λ) dλ,
Leibniz’s rule
dm
dxm(f · g) = (f · g)(m) =
m∑n=0
(m
n
)f (n)g(m−n) where
(m
n
)=
m!n! (m− n)!∫
f dg = fg −∫g df, Integration by parts
6
Integral and Derivative Table
∫f(x) dx f(x) f ′(x) =
d
dxf(x) Conditions
1n+ 1
xn+1 xn nxn−1 n 6= −1
ln |x| 1x
− 1x2
x 6= 0
x lnx− x lnx1x
x > 0
ex ex ex
1ln a
ax ax ax ln a a real, a > 0
− cosx sinx cosx
sinx cosx − sinx
− ln | cosx| tanx1
cos2 xx 6= π
2± nπ, n = 0, 1, 2, . . .
x arcsinx+√
1− x2 arcsinx1√
1− x2|x| < 1
x arccosx−√
1− x2 arccosx−1√
1− x2|x| < 1
x arctanx− 12
ln (1 + x2) arctanx1
1 + x2
coshx sinhx coshx
sinhx coshx sinhx
tanhx1
cosh2 x
arcsinhx1√
x2 + 1
arccoshx1√
x2 − 1x > 1
arctanhx1
1− x2|x| < 1
7
7 Integral Table
Indefinite Integrals
a, b, c real unless otherwise stated∫x sin ax dx =
1a2
(sin ax− ax cos ax)
∫x2 sin ax dx =
1a3
(2ax sin ax+ 2 cos ax− a2x2 sin ax)
∫x cos ax dx =
1a2
(cos ax+ ax sin ax)
∫x2 cos ax dx =
1a3
(2ax cos ax− 2 sin ax+ a2x2 sin ax)
∫xeax dx = eax
(x
a− 1a2
), a complex
∫x2eax dx = eax
(x2
a− 2xa2
+2a3
), a complex
∫x3eax dx = eax
(x3
a− 3x2
a2+
6xa3− 6a4
), a complex
∫eax sin bx dx =
eax
a2 + b2(a sin bx− b cos bx)
∫eax cos bx dx =
eax
a2 + b2(a cos bx+ b sin bx)
∫xn ln ax dx =
xn+1
n+ 1ln ax− xn+1
(n+ 1)2
∫ (lnx)n
xdx =
1n+ 1
(lnx)n+1
∫dx
b+ ax=
1a
ln |b+ ax|
∫dx
(b+ ax)n=
−1(n− 1)a(b+ ax)n−1
, n > 1
∫(b+ ax)n =
(b+ ax)n+1
a(n+ 1)n > 0
∫dx√
c+ bx+ ax2=
1√a
ln∣∣∣∣b+ 2ax+ 2
√a(c+ bx+ ax2)
∣∣∣∣ , a > 0
8
∫ √c+ bx+ ax2 dx =
b+ 2ax4a
√c+ bx+ ax2 +
4ac− b2
8a
∫dx√
c+ bx+ ax2, a > 0
∫dx
c+ bx+ ax2=
2√4ac− b2
arctan(
b+ 2ax√4ac− b2
), b2 < 4ac
−2b+ 2ax
, b2 = 4ac
1√b2 − 4ac
ln
∣∣∣∣∣b+ 2ax−√b2 − 4ac
b+ 2ax+√b2 − 4ac
∣∣∣∣∣ , b2 > 4ac
∫x dx
c+ bx+ ax2=
12a
ln |c+ bx+ ax2| − b
2a
∫dx
c+ bx+ ax2
∫dx
a2 + b2x2=
1ab
arctan(bx
a
)∫
x dx
a2 + x2=
12
ln(a2 + x2)
∫x2 dx
a2 + x2= x− a arctan
(x
a
)
∫dx
(a2 + x2)2=
x
2a2(a2 + x2)+
12a3
arctan(x
a
)∫
x dx
(a2 + x2)2=
−12(a2 + x2)∫
x2 dx
(a2 + x2)2=
−x2(a2 + x2)
+12a
arctan(x
a
)
∫dx
(a2 + x2)3=
x
4a2(a2 + x2)2+
3x8a4(a2 + x2)
+3
8a5arctan
(x
a
)∫
x2 dx
(a2 + x2)3=
−x4(a2 + x2)2
+x
8a2(a2 + x2)+
18a3
arctan(x
a
)∫
x4 dx
(a2 + x2)3=
a2x
4(a2 + x2)2− 5x
8(a2 + x2)+
38a
arctan(x
a
)
∫dx
(a2 + x2)4=
x
6a2(a2 + x2)3+
5x24a4(a2 + x2)2
+5x
16a6(a2 + x2)+
516a7
arctan(x
a
)∫
x2 dx
(a2 + x2)4=
−x6(a2 + x2)3
+x
24a2(a2 + x2)2+
x
16a4(a2 + x2)+
116a5
arctan(x
a
)
9
∫x4 dx
(a2 + x2)4=
a2x
6(a2 + x2)3− 7x
24(a2 + x2)2+
x
16a2(a2 + x2)+
116a3
arctan(x
a
)
∫dx
a4 + x4=
14a3√
2ln
(x2 + ax
√2 + a2
x2 − ax√
2 + a2
)+
12a3√
2arctan
(ax√
2a2 − x2
)∫
x2 dx
a4 + x4=−1
4a√
2ln
(x2 + ax
√2 + a2
x2 − ax√
2 + a2
)+
12a√
2arctan
(ax√
2a2 − x2
)
Definite Integrals
∫ ∞0
xa−1e−x dx = Γ(a), a > 0, (by definition)
where Γ(a+ 1) = aΓ(a)Γ(n) = (n− 1)! n = 1, 2, 3, . . .
Γ(
12
)=√π
Γ(n+ 1
2
)=
1 · 3 · 5 · · · (2n− 1)2n
√π, n = 1, 2, 3, . . .
∫ ∞0
xa−1 sin bx dx =Γ(a)ba
sin(πa
2
), 0 < |a| < 1, b > 0
∫ ∞0
xa−1 cos bx dx =Γ(a)ba
cos(πa
2
), 0 < a < 1, b > 0
∫ ∞0
e−ax sin bx dx =b
a2 + b2, a > 0
∫ ∞0
e−ax cos bx dx =a
a2 + b2, a > 0
∫ ∞−∞
e−a2x2+bx dx =
√π
aeb
2/(4a2), a > 0
∫ ∞0
xe−a2x2
dx =1
2a2, a > 0
∫ ∞0
x2e−a2x2
dx =√π
4a3, a > 0
∫ ∞0
xne−a2x2
dx =Γ(n+1
2
)2an+1
, a > 0
∫ ∞0
e−a2x2
cos bx dx =√π
2ae−b
2/(4a2), a > 0
10
∫ ∞0
sinxx
dx =π
2∫ ∞0
(sinxx
)2
dx =π
2∫ ∞0
cos axb2 + x2
dx =π
2be−ab, a > 0, b > 0
∫ ∞0
x sin axb2 + x2
dx =π
2e−ab, a > 0, b > 0
∫ ∞0
xm−1
1 + xndx =
π/n
sin(mπ/n), 0 < m < n
8 Standard Limits
Limits of Series
limn→∞
(1 +a
n)n = ea, a real
limn→∞
nkan = 0, a real, |a| < 1, k = 0, 1, 2, . . .
limn→∞
n√a = 1, a real, a > 0
limn→∞
n√n = 1
limn→∞
n−a = 0, a real, a > 0
Limits of Functions
limx→0
sinxx
= 1
limx→0
ex − 1x
= 1
limx→0
ln(1 + x)x
= 1
limx→0
(1 + ax)1/x = ea, a real
limx↓0
xa lnx = 0, a real, a > 0
limx→∞
lnxxa
= 0, a real, a > 0
limx→∞
xae−x = 0, a real
11
9 Convolutions
Regular Convolution
x(t) ∗ y(t) =∫ ∞−∞
x(t− τ)y(τ) dτ
x[n] ∗ y[n] =∞∑
m=−∞x[n−m]y[m]
Cyclical Convolution
x(t)� y(t) =∫ T
0x((t− τ) mod T
)y(τ) dτ, 0 ≤ t < T
x[n]� y[n] =N−1∑m=0
x[[n−m] mod N
]y[m], n = 0, 1, 2, . . . , N − 1
10 Dirac Delta Function
The Dirac delta function δ(t) is the singular function such that δ(t) = 0 for all t 6= 0 and∫ ∞−∞
δ(t) dx = 1
Note that δ(t)δ(t) is undefined.
Properties of the Dirac Delta Function
Property Expression Conditions
Symmetry δ(−t) = δ(t)
Scaling δ(at) =1|a|δ(t) a real, a 6= 0
Derivative of unit stepd
dtu(t) = δ(t)
Fourier transform δ(t) =1
2π
∫ ∞−∞
e±jωt dω ω real
Multiplication x(t)δ(t− θ) = x(θ)δ(t− θ) θ real
Convolution x(t) ∗ δ(t− θ) = x(t− θ) θ real
Sifting∫ ∞−∞
δ(t− θ)x(t) dt = x(θ) θ real
Multiplication by derivative x(t)δ(m)(t) =m∑n=0
(−1)n(m
n
)x(n)(0)δ(m−n)(t) m = 1, 2, 3, . . .
Sifting with derivative∫ ∞−∞
δ(m)(t)x(t) dt = (−1)mx(m)(0) m = 1, 2, 3, . . .
12
11 Fourier Transform
Continuous-Time Fourier Transform
The Continuous-Time Fourier Transform (CTFT) of a non-periodic function x(t) definedfor t real, is given by
X(ω) = F[x(t)
]=∫ ∞−∞
x(t)e−jωt dt , ω real
The inverse Fourier transform of a non-periodic function X(ω) defined for ω real, is given by
F−1[X(ω)
]=
12π
∫ ∞−∞
X(ω)ejωt dω = x(t) , t real
Parseval’s theorem for the CTFT is given by∫ ∞−∞
x(t)y∗(t) dt =1
2π
∫ ∞−∞
X(ω)Y ∗(ω) dω
Properties of the Continuous-Time Fourier Transform
Property Time signal Fourier transform Conditions
Linearity ax(t) + by(t) aX(ω) + bY (ω) a, b complex
Convolution x(t) ∗ y(t) X(ω)Y (ω)
Converse convolution x(t)y(t)1
2πX(ω) ∗ Y (ω)
Shift x(t− a) e−jωaX(ω) a real
Converse shift ejatx(t) X(ω − a) a real
Differentiationdx(t)dt
jωX(ω)
Converse differentiation −jtx(t)dX(ω)dω
Scaling x(at)1|a|X
(ω
a
)a real, a 6= 0
Time-frequency symmetry X(t) 2πx(−ω)
13
Standard Continuous-Time Fourier Transforms
Time signal Fourier transform Conditions
1 2πδ(ω)
u(t) πδ(ω) +1jω
δ(t) 1
δ(t− a) e−jωa a real
ejat 2πδ(ω − a) a real
e−a|t|2a
a2 + ω2a complex, Re(a) > 0
e−atu(t)1
a+ jωa complex, Re(a) > 0
tk−1e−at
(k − 1)!u(t)
1(a+ jω)k
a complex, Re(a) > 0k integer
e−at2
√π
aeω
2/4a a real, a > 0
sin(at) jπδ(ω + a)− jπδ(ω − a) a real
cos(at) πδ(ω + a) + πδ(ω − a) a real
sin(at)u(t) jπ
2δ(ω + a)− j π
2δ(ω − a) +
a
a2 − ω2a real
cos(at)u(t)π
2δ(ω + a)− π
2δ(ω − a) +
jω
a2 − ω2a real
e−at sin(bt)u(t)b
b2 + (a+ jω)2a, b real, a > 0
e−at cos(bt)u(t)a+ jω
b2 + (a+ jω)2a, b real, a > 0
1, |t| ≤ a
0, |t| > a
2 sin(ωa)ω
a real, a ≥ 0
sin atat
π
a, |ω| ≤ a
0, |ω| > a
a real, a ≥ 0
1− |t|
a, |t| ≤ a
0, |t| > a
4 sin2(ωa/2)ω2a
a real, a ≥ 0
14
Discrete-Time Fourier Transform
The Discrete-Time Fourier Transform (DTFT) of a non-periodic function x[n] definedfor n integer, is given by
X(θ) = F[x[n]
]=
∞∑n=−∞
x[n]e−jθn , θ real
with relative frequency θ = ωT where T is the sampling interval.The inverse Fourier transform of a periodic function X(θ) defined for θ real with period 2π,is given by
F−1[X(θ)
]=
12π
∫ π
−πX(θ)ejθn dθ = x[n] , n integer
Parseval’s theorem for the DTFT is given by∞∑
n=−∞x[n]y∗[n] =
12π
∫ π
−πX(θ)Y ∗(θ) dθ
Properties of the Discrete-Time Fourier Transform
Property Time signal Fourier transform Conditions
Linearity ax[n] + by[n] aX(θ) + bY (θ) a, b complex
Convolution x[n] ∗ y[n] X(θ)Y (θ)
Converse convolution x[n]y[n]1
2πX(θ)� Y (θ)
Shift x[n−m] e−jθmX(θ) m integer
Converse shift ejanx[n] X(θ − a) a real
Converse differentiation −jnx[n]dX(θ)dθ
Scaling x[mn]1mX
(θ
m
)m = 1, 2, 3, . . .
15
Standard Discrete-Time Fourier Transforms
Time signal Fourier transform Conditions
1 2πδ(θ)
δ[n] 1
δ[n−m] e−jθm m integer
anu[n]1
1− ae−jθa complex, |a| < 1
an, n = 0, 1, . . . ,M − 1
0, otherwise
1− aMe−jθM
1− ae−jθa complex, M = 1, 2, 3, . . .
sin[an] jπδ(θ + a)− jπδ(θ − a) a real
cos[an] πδ(θ + a) + πδ(θ − a) a real
ejan 2πδ(θ − a) a real1, |n| ≤M
0, |n| > M
sin((2M + 1)θ/2
)sin(θ/2)
M = 1, 2, 3, . . .
sin[an]nπ
, n 6= 0
a
π, n = 0
1, |θ| ≤ a
0, |θ| > a
a real, 0 < a ≤ π
Observe that the Fourier transforms are only given for one period i.e. −π ≤ θ < π.
16
Fourier Series
The Fourier Series of a periodic function x(t) defined for t real with period T , are given by
X[k] = F[x(t)
]=
1T
∫ T/2
−T/2x(t)e−j
2πTkt dt , k integer
The inverse Fourier transform of a non-periodic function X[k] defined for k integer, is givenby
F−1[X[k]
]=
∞∑k=−∞
X[k]ej2πTkt = x(t) , t real
Parseval’s theorem for the Fourier series is given by∫ T/2
−T/2x(t)y∗(t) dt =
1T
∞∑k=−∞
X[k]Y ∗[k]
Properties of the Fourier Series
Property Time signal Fourier transform Conditions
Linearity ax(t) + by(t) aX[k] + bY [k] a, b complex
Convolution x(t)� y(t) X[k]Y [k]
Converse convolution x(t)y(t)1TX[k] ∗ Y [k]
Shift x((t− a) mod T
)e−j
2πTkaX[k] a real
Converse shift ej2πTtmx(t) X[k −m] m integer
Differentiationdx(t)dt
jkX[k]
Scaling x(at)1|a|X
[k
a
]a real, a 6= 0
The Trigonometric Form of the Fourier Series is given by
x(t) =a0
2+∞∑k=1
ak cos(
2πTkt
)+∞∑k=1
bk sin(
2πTkt
)where
ak =2T
∫ T/2
−T/2x(t) cos
(2πTkt
)dt k = 0, 1, 2, . . .
bk =2T
∫ T/2
−T/2x(t) sin
(2πTkt
)dt k = 1, 2, 3, . . .
17
The relations between the complex exponential Fourier series and the coefficients of thetrigonometric form are given by
a0 = 2X[0]ak = X[k] +X[−k] k = 1, 2, 3, . . .
bk = j(X[k]−X[−k]
)k = 1, 2, 3, . . .
X[0] =12a0
X[k] =12
(ak − jbk) k = 1, 2, 3, . . .
X[−k] =12
(ak + jbk) k = 1, 2, 3, . . .
Standard Fourier Series
Time signal Fourier transform Conditions
δ(t) 11, |t| ≤ a
0, |t| > a
2aT, k = 0
1πk
sin(
2πTka
), k 6= 0
a real, 0 < a ≤ T2
t
a+ 1, −a ≤ t ≤ 0
− ta
+ 1, 0 < t ≤ a
1T, k = 0
T
π2k2asin2
(π
Tka
), k 6= 0
a real, 0 < a ≤ T2
cos
(π
2at
), |t| ≤ a
0, |t| > a
4aTTπ2 − 16a2k2π
cos(
2πTka
)a real, 0 < a ≤ T
2
Observe that the time signals are only given for one period i.e. −T2 ≤ t <
T2 .
18
Discrete Fourier Transform
The Discrete Fourier Transform (DFT) of a periodic function x[n] defined for n integerwith period N , is given by
X[k] = F[x[n]
]=
N−1∑n=0
x[n]e−j2πNkn , k integer
with k = ωTN2π where T is the sampling interval.
The inverse Fourier transform of a periodic function X[k] defined for k integer with periodN , is given by
F−1[X[k]
]=
1N
N−1∑k=0
X[k]ej2πNkn = x[n] , n integer
Parseval’s theorem for the DFT is given by
N−1∑n=0
x[n]y∗[n] =1N
N−1∑k=0
X[k]Y ∗[k]
Properties of the Discrete Fourier Transform
Property Time signal Fourier transform Conditions
Linearity ax[n] + by[n] aX[k] + bY [k] a, b complex
Convolution x[n]� y[n] X[k]Y [k]
Converse convolution x[n]y[n]1NX[k]� Y [k]
Shift x[[n−m] mod N
]e−j
2πNkmX[k] m integer
Converse shift ej2πNnmx[n] X
[[k −m] mod N
]m integer
Scaling x[mn]1mX
[k
m
]m = 1, 2, 3, . . .
Time-frequency symmetry X[n] Nx[−k]
19
Standard Discrete Fourier Transforms
Time signal Fourier transform Conditions
1 Nδ[k]
δ[n] 1
δ[n−m] e−j2πNkm m integer
an1− aN
1− ae−j2πNk
a complex, |a| < 1an, n = 0, 1, . . . ,M − 1
0, otherwise
1− aMe−j2πNkM
1− ae−j2πNk
a complex, M = 1, 2, 3, . . .
sin[
2πNnm
]jN
2δ[k +m]− jN
2δ[k −m] m integer
cos[
2πNnm
]N
2δ[k +m] +
N
2δ[k −m] m integer
ej2πNnm Nδ[k −m] m integer
Observe that the time signals are only given for one period i.e. 0 ≤ n ≤ N − 1. The Fouriertransforms are also given for one period i.e. 0 ≤ k ≤M − 1.
20
12 Laplace Transform
Two-Sided Laplace Transform
The two-sided Laplace transform of a function x(t) defined for t real, is given by
X(s) = L[x(t)
]=∫ ∞−∞
x(t)e−st dt , s ∈ Ex
The existence region Ex consist of all the complex numbers s for which the integral converges.The inverse two-sided Laplace transform of a function X(s) defined for s ∈ Ex, is given by
L−1[X(s)
]= lim
R→∞
12πj
∫ σ+jR
σ−jRX(s)est ds = x(t) , t real
Where the integral is taken along a vertical line that lies entirely in the existence region Exand all singularities of X(s) are on the left side of this line.
Properties of the Two-Sided Laplace Transform
Property Time signal Laplace transform Existence region Conditions
Linearity ax(t) + by(t) aX(s) + bY (s) Ex ∩ Ey a, b complex
Convolution x(t) ∗ y(t) X(s)Y (s) Ex ∩ Ey
Shift x(t− a) e−saX(s) Ex a real
Converse shift eatx(t) X(s− a) (s− a) ∈ Ex a complex
Differentiationdx(t)dt
sX(s) Ex
Integration∫ t
−∞x(τ) dτ
X(s)s
{s ∈ Ex|Re(s) > 0}
Converse diff. −tx(t)dX(s)ds
Ex
Scaling x(at)1|a|X
(s
a
)s
|a|∈ Ex a real, a 6= 0
21
Standard Two-Sided Laplace Transforms
Time signal Laplace transform Existence region Conditions
u(t)1s
Re(s) > 0
δ(t) 1 s complex
δ(k)(t) sk s complex k = 0, 1, 2, . . .
eatu(t)1
s− aRe(s) > Re(a) a complex
−eatu(−t) 1s− a
Re(s) < Re(a) a complex
tk−1
(k − 1)!u(t)
1sk
Re(s) > 0 k = 0, 1, 2, . . .
tk−1eat
(k − 1)!u(t)
1(s− a)k
Re(s) > Re(a) a complex, k = 0, 1, 2, . . .
− tk−1eat
(k − 1)!u(−t) 1
(s− a)kRe(s) < Re(a) a complex, k = 0, 1, 2, . . .
sin (at)u(t)a
s2 + a2Re(s) > 0 a real
cos (at)u(t)s
s2 + a2Re(s) > 0 a real
eat sin (bt)u(t)b
(s− a)2 + b2Re(s) > a a, b real
eat cos (bt)u(t)s− a
(s− a)2 + b2Re(s) > a a, b real
22
One-Sided Laplace Transform
The one-sided Laplace transform of a function x(t) defined for t ∈ [0,∞), is given by
X(s) = L[x(t)
]=∫ ∞
0x(t)e−st dt , s ∈ Ex
The existence region Ex consist of all the complex numbers s for which the integral converges.The inverse one-sided Laplace transform of a function X(s) defined for s ∈ Ex, is given by
L−1[X(s)
]= lim
R→∞
12πj
∫ σ+jR
σ−jRX(s)est ds =
{x(t) for t ≥ 0
0 for t < 0
Where the integral is taken along a vertical line that lies entirely in the existence region Exand all singularities of X(s) are on the left side of this line.
Properties of the One-Sided Laplace Transform
Property Time signal Laplace transform Existence region Conditions
Linearity ax(t) + by(t) aX(s) + bY (s) Ex ∩ Ey a, b complex
Convolution x(t) ∗ y(t) X(s)Y (s) Ex ∩ Ey
Shift x(t− a)u(t− a) e−saX(s) Ex a real, θ ≥ 0
Converse shift eatx(t) X(s− a) (s− a) ∈ Ex a complex
Differentiationdx(t)dt
sX(s)− x(0+) Ex
Integration∫ t
0x(τ) dτ
X(s)s
{s ∈ Ex|Re(s) > 0}
Converse diff. −tx(t)dX(s)ds
Ex
Scaling x(at)1aX
(s
a
)s
a∈ Ex a real, a > 0
Multiple differentiation: for any integer k ≥ 0
L[dkx(t)dtk
]= skX(s)−
k−1∑i=0
sk−i−1dix(0+)dti
, s ∈ Ex
Initial value property: x(0+) = lims→∞
sX(s)Final value property: lim
t→∞x(t) = lim
s→0sX(s)
23
Standard One-Sided Laplace Transforms
Time signal Laplace transform Existence region Conditions
11s
Re(s) > 0
u(t)1s
Re(s) > 0
δ(t) 1 s complex
δ(k)(t) sk s complex k = 0, 1, 2, . . .
eat1
s− aRe(s) > Re(a) a complex
tk−1
(k − 1)!1sk
Re(s) > 0 k = 0, 1, 2, . . .
tk−1eat
(k − 1)!1
(s− a)kRe(s) > Re(a) a complex, k = 0, 1, 2, . . .
sin (at)a
s2 + a2Re(s) > 0 a real
cos (at)s
s2 + a2Re(s) > 0 a real
eat sin (bt)b
(s− a)2 + b2Re(s) > a a, b real
eat cos (bt)s− a
(s− a)2 + b2Re(s) > a a, b real
24
13 z-Transform
Two-Sided z-Transform
The two-sided z-transform of a function x[n] defined for n integer, is given by
X(z) = Z[x[n]
]=
∞∑n=−∞
x[n]z−n , z ∈ Ex
The existence region Ex consist of all the complex numbers z for which the sum converges.The inverse two-sided z-transform of a function X(z) defined for z ∈ Ex, is given by
Z−1[X(z)
]=
12πj
∮CX(z)zn−1 dz = x[n] , n integer
Where the contour integral is taken along a counterclockwise arbitrary closed path C thatencloses all the finite poles of X(z)zn−1 and lies entirely in the existence region Ex.For the actual calculation of the inverse z-transform complex function theory is used. If thefunction X(z)zn−1 has k finite poles at z = ai, i = 1, 2, . . . , k then the inversion formula isevaluated via the theorem of residues
x[n] =k∑i=1
Resz=ai
[X(z)zn−1
]For a pole of order m at z = a, the residue is evaluated as
Resz=a
[X(z)zn−1
]=
1(m− 1)!
limz→a
[dm−1
dzm−1(z − a)mX(z)zn−1
]
For a simple pole this is equal to
Resz=a
[X(z)zn−1
]= lim
z→a
[(z − a)X(z)zn−1
]
Properties of the Two-Sided z-Transform
Property Time signal Laplace transform Existence region Conditions
Linearity ax[n] + by[n] aX(z) + bY (z) Ex ∩ Ey a, b complex
Convolution x[n] ∗ y[n] X(z)Y (z) Ex ∩ Ey
Multiple shift x[n− k] z−kX(z) Ex k integer
Converse shift anx[n] X(z
a)
z
a∈ Ex a complex
Converse diff. −nx[n] zdX(z)dz
Ex
25
Standard Two-Sided z-Transforms
Time signal z-transform Existence region Conditions
u[n]z
z − 1|z| > 1
δ[n] 1 z complex
δ[n− k] z−k z 6= 0 k = 1, 2, 3, . . .
δ[n+ k] zk z complex k = 0, 1, 2, . . .
nu[n]z
(z − 1)2|z| > 1
n2u[n]z(z + 1)(z − 1)3
|z| > 1
anu[n]z
z − a|z| > |a| a complex
anu[n− 1]a
z − a|z| > |a| a complex
nanu[n− 1]az
(z − a)2|z| > |a| a complex
(n−1k−1
)anu[n− k]
ak
(z − a)k|z| > |a| a complex, k = 0, 1, 2, . . .
−anu[−n− 1]z
z − a|z| < |a| a complex
−anu[−n]a
z − a|z| < |a| a complex
(−1)k(k−n−1k−1
)anu[−n]
ak
(z − a)k|z| < |a| a complex, k = 0, 1, 2, . . .
an
n!u[n] ea/z |z| > |a| a complex
1nu[n− 1] ln
(z
z − 1
)|z| > 1
an sin(bn)u[n]az sin(b)
z2 − 2az cos(b) + a2|z| > |a| a, b real
an cos(bn)u[n]z2 − az sin(b)
z2 − 2az cos(b) + a2|z| > |a| a, b real
an sin(bn+ c)u[n]z2 sin(c) + az sin(b− c)z2 − 2az cos(b) + a2
|z| > |a| a, b, c real
26
One-Sided z-Transform
The one-sided z-transform of a function x(n) defined for n = 0, 1, 2, . . ., is given by
X(z) = Z[x[n]
]=∞∑n=0
x[n]z−n , z ∈ Ex
The existence region Ex consist of all the complex numbers z for which the sum converges.The inverse one-sided z-transform of a function X(z) defined for z ∈ Ex, is given by
Z−1[X(z)
]=
12πj
∮CX(z)zn−1 dz =
{x[n] for n ≥ 0
0 for n < 0
Where the contour integral is taken along a counterclockwise arbitrary closed path C thatencloses all the finite poles of X(z)zn−1 and lies entirely in the existence region Ex.For the evaluation of this integral, see the section on the two-sided z-transform (page 25).
Properties of the One-Sided z-Transform
Property Time signal Laplace transform Existence region Conditions
Linearity ax[n] + by[n] aX(z) + bY (z) Ex ∩ Ey a, b complex
Convolution x[n] ∗ y[n] X(z)Y (z) Ex ∩ Ey
Back shift x[n+ k] zkX(z)−k−1∑i=0
zk−ix(i) Ex k = 0, 1, 2, . . .
Forward shift x[n− k]u[n− k] z−kX(z) Ex k = 0, 1, 2, . . .
Converse shift anx[n] X(z
a)
z
a∈ Ex a complex
Converse diff. −nx[n] zdX(z)dz
Ex
Forward shift without multiplication by a unit step function requires information of the signalx[n] for n < 0. The one-sided Laplace transform for any k = 0, 1, 2, . . . is given by
Z[x[n− k]
]= z−kX(z) +
k∑i=1
z−(k−i)x(−i) , z ∈ Ex
Initial value property: x[0] = lim|z|→∞
X(z)
Final value property: limn→∞
x[n] = limz→1
(z − 1)X(z)
27
Standard One-Sided z-Transforms
Time signal z-transform Existence region Conditions
1z
z − 1|z| > 1
u[n]z
z − 1|z| > 1
δ[n] 1 z complex
δ[n− k] z−k z 6= 0 k = 1, 2, 3, . . .
nz
(z − 1)2|z| > 1
n2 z(z + 1)(z − 1)3
|z| > 1
anz
z − a|z| > |a| a complex
anu[n]z
z − a|z| > |a| a complex
anu[n− 1]a
z − a|z| > |a| a complex
nanu[n− 1]az
(z − a)2|z| > |a| a complex
(n−1k−1
)anu[n− k]
ak
(z − a)k|z| > |a| a complex, k = 0, 1, 2, . . .
−anu[−n] −1 z complex a complex
(−1)k(k−n−1k−1
)anu[−n] (−1)k z complex a complex, k = 0, 1, 2, . . .
an
n!ea/z |z| > |a| a complex
1nu[n− 1] ln
(z
z − 1
)|z| > 1
an sin(bn)az sin(b)
z2 − 2az cos(b) + a2|z| > |a| a, b real
an cos(bn)z2 − az sin(b)
z2 − 2az cos(b) + a2|z| > |a| a, b real
an sin(bn+ c)z2 sin(c) + az sin(b− c)z2 − 2az cos(b) + a2
|z| > |a| a, b, c real
28
