Formulas from Trigonometry
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
sin(A ? B) = sin A cos B ? cos A sin B
tan(A ? B)
=
tan A?tan B 1tan A tan B
cos 2A = cos2 A - sin2 A
tan 2A
=
2 tan A 1-tan2 A
cos
A 2
=
?
1+cos A 2
sin2 A
=
1 2
-
1 2
cos 2A
sin
A
+
sin
B
=
2
sin
1 2
(A
+
B)
cos
1 2
(A
-
B)
cos
A
+
cos
B
=
2
cos
1 2
(A
+
B)
cos
1 2
(A
-
B)
sin A sin B
=
1 2
{cos(A
-
B)
-
cos(A
+
B)}
sin A cos B
=
1 2
{sin(A
-
B)
+
sin(A
+
B)}
sin
A 2
=
?
1-cos A 2
tan
A 2
=
sin A 1+cos A
cos2
A
=
1 2
+
1 2
cos 2A
sin
A
-
sin
B
=
2
cos
1 2
(A
+
B)
sin
1 2
(A
-
B)
cos
A
-
cos
B
=
2
sin
1 2
(A
+
B)
sin
1 2
(B
-
A)
cos A cos B
=
1 2
{cos(A
-
B)
+
cos(A
+
B)}
cos() = sin( + /2)
Differentiation Formulas:
d dx
(uv)
=
u
dv dx
+
du dx
v
Chain rule:
dy dx
=
dy du du dx
d dx
cos
u
=
-
sin
u
du dx
d dx
sin-1
u
=
1 1-u2
du dx
,
-
2
<
sin-1 u
<
2
d dx
tan-1 u
=
1 1+u2
du dx
,
-
2
<
tan-1 u
<
2
d dx
ln u
=
1 du u dx
d dx
u v
=
v(du/dx)-u(dv/dx) v2
d dx
sin
u
=
cos
u
du dx
d dx
tan
u
=
sec2
u
du dx
d dx
cos-1 u
=
-1 1-u2
du dx
,
(0
<
cos-1 u
<
)
d dx
eu
=
eu
du dx
d dx
loga u
=
loga u
e
du dx
,
a
=
0, 1
Integration Formulas:
Integration by parts: u dv = uv - v du
du
= ln |u|
u
au du =
au ,
ln a
a > 0, a = 1
cos u du = sin u
sin2 u du
=
u 2
-
sin 2u 4
=
1 2
(u
-
sin
u
cos
u)
cos2 u du
=
u 2
+
sin 2u 4
=
1 2
(u
+
sin
u
cos
u)
du u2 - a2
=
1 2a
ln
u-a u+a
du
= ln(u + u2 + a2)
u2 + a2
eax sin bx dx
=
eax(a sin bx-b cos bx) a2+b2
x sin ax dx
=
sin ax a2
-
x cos ax a
sin2 ax dx
=
x 2
-
sin 2ax 4a
x2
cos ax dx
=
2x a2
cos ax
+
x2 a
-
2 a3
sin ax
tan2 ax dx
=
tan ax a
-x
ln x dx = x ln x - x
eu du = eu
sin u du = - cos u
tan u du = - ln cos u
tan2 u du = tan u - u
du u2 + a2
=
1 a
tan-1
u a
du
a2 - u2
= sin-1
u a
du
= ln(u + u2 - a2)
u2 - a2
eax cos bx dx
=
eax(a cos bx+b sin bx) a2+b2
x2
sin ax dx
=
2x a2
sin ax
+
2 a3
-
x2 a
cos ax
x cos ax dx
=
cos ax a2
+
x sin ax a
cos2 ax dx
=
x 2
+
sin 2ax 4a
xeax dx
=
eax a
x
-
1 a
x ln x dx
=
x2 2
ln
x
-
1 2
1
Rules for Exponents: ab+c = abac
(ab)c = abc
(ab)c = acbc
a-b =
1b 1 a = ab
Taylor Series: Euler's Formula:
ex = 1 + x + x2 + x3 + x4 + ? ? ? = xn
2! 3! 4!
n!
n=0
x2 x4 x6
(-1)nx2n
cos x = 1 - + - + ? ? ? =
2! 4! 6!
(2n)!
n=0
x3 x5 x7
(-1)nx2n+1
sin x = 1 - + - + ? ? ? =
3! 5! 7!
(2n + 1)!
n=0
ej = cos + j sin
ej + e-j cos =
2
ej - e-j sin =
2j
Rectangular and Polar Form of a Complex Number:
z = a + jb = rej
r = |z| = a2 + b2
a = Re{z} = r cos
r2 = zz b
= arctan a
z + z a = Re{z} =
2 b = Im{z} = r sin
z - z b = Im{z} =
2j
Phasors:
Complex Signal:
z(t) = Aej(0t+) = Aejej0t
Real Signal:
x(t) = Re{z(t)} = A cos(0t + )
Phasor Representation: X = Aej
Phasor Addition:
Let x1(t) = A1 cos(w0t + 1), x2(t) = A2 cos(w0t + 2), and x(t) = x1(t) + x2(t). Then x(t) = A cos(w0t + ) and:
the phasor representation for x(t) is X = Aej = A1ej1 + A2ej2.
Continuous-Time Unit Impulse and Unit Step:
(t) dt = 1
-
x(t)(t) dt = x(0)
- t
u(t) = (t) dt
-
x(t)(t - t0) dt = x(t0)
-
2
Discrete-Time Unit Impulse and Unit Step:
[n] =
1, n = 0, 0, otherwise.
u[n] =
1, n 0, 0, n < 0.
Complex Exponential Signals:
ej0t
ej0n
Distinct signals for distinct w0 Identical signals for values of w0 separated by multiples of 2
Periodic for any choice of w0
Periodic only if w0/(2) = m/N Q
Fundamental frequency w0
Fundamental frequency w0/m
Fundamental period: w0 = 0: undefined
w0 = 0: 2/w0
Fundamental period: w0 = 0: one
w0 = 0: N = 2m/w0
Periodicity of Discrete-Time Sinusoids:
cos(0n), sin(0n), and If periodic, then write
ej0n are periodic in reduced form:
if and only if w0 is a ratio of
w0
=
m
(no
2 common
factors
two integers. between m and
N)
2 N
N : Fundamental Period
m: In each period of the discrete-time signal, the graph "goes around" m times.
Summation Formulas:
N2
k
=
N1
-
N2+1 ,
1-
k=N1
=1
ak =
1 ,
1-a
k=0
|a| < 1
kak =
a ,
(1 - a)2
k=0
|a| < 1
n
ak
=
1
-
an+1 ,
1-a
k=0
a=1
n
kak
=
a{1
-
(n + 1)an + (1 - a)2
nan+1}
k=0
Time Domain Representation of Discrete-Time Signals:
x[n] = ? ? ? + x[-2][n + 2] + x[-1][n + 1] + x[0][n] + x[1][n - 1] + x[2][n - 2] + ? ? ?
=
x[k][n - k].
k=
Systems:
System H is linear if H{ax1[n] + bx2[n]} = aH{x1[n]} + bH{x2[n]}.
System H is time invariant if H{x[n - n0]} = y[n - n0].
Impulse response: for LTI system H, h[n] = H{[n]}.
3
Convolution:
y[n] = x[n] h[n] =
x[k]h[n - k] =
x[n - k]h[k]
k=-
k=-
Convolution with [n]:
x[n] [n] = x[n]
x[n] [n - n0] = x[n - n0].
4
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