Formulas from Trigonometry:
sin2 A + cos2 A = 1
cos(A ± B) = cos A cos B ∓ sin A sin B
sin 2A = 2 sin A cos A
tan 2A =
sin(A ± B) = sin A cos B ± cos A sin B
tan A±tan B
tan(A ± B) = 1∓tan
A tan B
2
2
cos 2A = cos
A
−
sin
A
q
2 tan A
1−tan2 A
sin A2 = ±
q
A
cos A2 = ± 1+cos
2
sin2 A = 21 − 12 cos 2A
sin A + sin B = 2 sin 21 (A + B) cos 12 (A − B)
cos A + cos B = 2 cos 21 (A + B) cos 12 (A − B)
sin A sin B = 12 {cos(A − B) − cos(A + B)}
sin A cos B = 21 {sin(A − B) + sin(A + B)}
1−cos A
2
sin A
tan A2 = 1+cos
A
cos2 A = 21 + 12 cos 2A
sin A − sin B = 2 cos 21 (A + B) sin 21 (A − B)
cos A − cos B = 2 sin 21 (A + B) sin 21 (B − A)
cos A cos B = 12 {cos(A − B) + cos(A + B)}
cos(θ) = sin(θ + π/2)
Differentiation Formulas:
dv
d
(uv) = u dx
+ du
v
dx
dx
dy
dy du
Chain rule: dx
= du
dx
d
du
cos
u
=
−
sin
u
dx
dx
−1
−1
du
d
π
π
√ 1
sin
u
=
,
−
<
sin
u
<
2
dx
2
2
1−u dx
d
1 du
π
π
−1
−1
tan
u
=
,
−
<
tan
u
<
2
dx
1+u dx
2
2
1 du
d
ln
u
=
dx
u dx
v(du/dx)−u(dv/dx)
d
u
=
dx v
v2
d
du
sin
u
=
cos
u
dx
dx
d
2 du
tan
u
=
sec
u
dx
dx
d
−1
√ −1 du , (0 < cos−1
cos
u
=
dx
1−u2 dx
d u
u du
e = e dx
dx
d
loga u = logua e du
, a 6= 0, 1
dx
dx
u < π)
Integration Formulas:
Z
Z
Integration by parts:
u dv = uv −
Z
du
= ln |u|
Z u
au
, a > 0, a 6= 1
au du =
ln a
Z
v du
Z
sin u du = − cos u
Z
tan u du = − ln cos u
cos u du = sin u
Z
Z
Z
sin2 u du =
cos2 u du =
u
2
u
2
−
+
eu du = eu
Z
sin 2u
4
sin 2u
4
= 12 (u − sin u cos u)
= 12 (u + sin u cos u)
du
1
u−a
=
ln
2a
u+a
2 − a2
u
Z
√
du
√
= ln(u + u2 + a2 )
u2 + a2
Z
ax
cos bx)
eax sin bx dx = e (a sina2bx−b
+b2
Z
ax
x sin ax dx = sina2ax − x cos
a
Z
sin2 ax dx = x2 − sin4a2ax
Z
2
x
x2 cos ax dx = 2x
cos
ax
+
2
a −
a
Z
tan2 ax dx = tanaax − x
Z
ln x dx = x ln x − x
Z
tan2 u du = tan u − u
Z
du
= a1 tan−1 ua
+ a2
du
√
= sin−1 ua
a2 − u2
√
du
√
= ln(u + u2 − a2 )
u2 − a2
ax
bx+b sin bx)
eax cos bx dx = e (a cos
a2 +b2
2
x2
x2 sin ax dx = 2x
sin
ax
+
−
2
3
a cos ax
a
a
Z
Z
Z
Z
u2
Z
cos ax
a2
x cos ax dx =
2
a3
Z
sin ax
Z
cos2 ax dx =
xeax dx =
eax
a
x ln x dx =
x2
2
Z
1
x
2
+
+
x sin ax
a
sin 2ax
4a
x−
1
a
ln x −
1
2
Summation Formulas:
N2
X
αN1 − αN2 +1
k
α =
, α 6= 1
1−α
k=N1
∞
X
a
kak =
, |a| < 1
(1 − a)2
k=0
n
X
a{1 − (n + 1)an + nan+1 }
kak =
(1 − a)2
k=0
Signals:
δ(t) =
d
u(t)
dt
Z
∞
X
k=0
n
X
ak =
1
,
1−a
|a| < 1
1 − an+1
,
a =
1−a
k=0
k
a 6= 1
δ[n] = u[n] − u[n − 1]
∞
X
∗
x1 (t)x2 (t) dt hx1 [n], x2 [n]i =
x1 [n]x∗2 [n]
∞
hx1 (t), x2 (t)i =
−∞
n=−∞
Ev{x(t)} = 21 {x(t) + x(−t)}
Od{x(t)} = 12 {x(t) − x(−t)}
Complex Exponential Signals:
ejω0 t
ejω0 n
Distinct signals for distinct w0
Periodic for any choice of w0
Fundamental frequency w0
Fundamental period:
w0 = 0: undefined
w0 6= 0: 2π/w0
Identical signals for values of w0 separated by multiples of 2π
Periodic only if w0 /(2π) = m/N ∈ Q
Fundamental frequency w0 /m
Fundamental period:
w0 = 0: one
w0 6= 0: 2πm/w0
Systems:
System H is linear if H{ax1 (t) + bx2 (t)} = aH{x1 (t)} + bH{x2 (t)}.
System H is time invariant if H{x(t − t0 )} = y(t − t0 ).
System H is memoryless if the current output does not depend on future or past inputs.
System H is invertible if distinct input signals produce distinct output signals.
System H is invertible if an inverse system G exists which “undoes” the action of H.
System H is causal if the current output does not depend on future inputs.
LTI system H is causal iff h(t) = 0 ∀ t < 0.
Bounded: x(t) is bounded if ∃ B ∈ R, B > 0, such that |x(t)| ≤ B ∀ t ∈ R.
System H is BIBO stable if everyZbounded input signal produces a bounded output signal.
∞
LTI system H is BIBO stable iff
|h(t)| dt < ∞.
−∞
Z ∞
Z ∞
y(t) = x(t) ∗ h(t) =
x(τ )h(t − τ ) dτ =
x(t − τ )h(τ ) dτ
−∞
y[n] = x[n] ∗ h[n] =
−∞
∞
X
∞
X
x[k]h[n − k] =
k=−∞
x[n − k]h[k]
k=−∞
Z
t
s(t) =
h(τ ) dτ
h(t) =
−∞
s[n] =
n
X
h[k]
d
s(t)
dt
h[n] = s[n] − s[n − 1]
k=−∞
2