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Just because some of us can read and write and do a little math, that doesn't mean we deserve to conquer the Universe.

—Kurt Vonnegut, Hocus Pocus , 1990

This appendix gathers together all of the math facts used in the text. They are divided into six categories:

  • Trigonometric identities
  • Fourier transforms and properties
  • Energy and power
  • Z-transforms and properties
  • Integral and derivative formulas
  • Matrix algebra

So, with no motivation or interpretation, just labels, here they are:

Trigonometric identities

  • Euler's relation
    e ± j x = cos ( x ) ± j sin ( x )
  • Exponential definition of a cosine
    cos ( x ) = 1 2 e j x + e - j x
  • Exponential definition of a sine
    sin ( x ) = 1 2 j e j x - e - j x
  • Cosine squared
    cos 2 ( x ) = 1 2 1 + cos ( 2 x )
  • Sine squared
    sin 2 ( x ) = 1 2 1 - cos ( 2 x )
  • Sine and Cosine as phase shifts of each other
    sin ( x ) = cos π 2 - x = cos x - π 2 cos ( x ) = sin π 2 - x = - sin x - π 2
  • Sine–cosine product
    sin ( x ) cos ( y ) = 1 2 sin ( x - y ) + sin ( x + y )
  • Cosine–cosine product
    cos ( x ) cos ( y ) = 1 2 cos ( x - y ) + cos ( x + y )
  • Sine–sine product
    sin ( x ) sin ( y ) = 1 2 cos ( x - y ) - cos ( x + y )
  • Odd symmetry of the sine
    sin ( - x ) = - sin ( x )
  • Even symmetry of the cosine
    cos ( - x ) = cos ( x )
  • Cosine angle sum
    cos ( x ± y ) = cos ( x ) cos ( y ) sin ( x ) sin ( y )
  • Sine angle sum
    sin ( x ± y ) = sin ( x ) cos ( y ) ± cos ( x ) sin ( y )

Fourier transforms and properties

  • Definition of Fourier transform
    W ( f ) = - w ( t ) e - j 2 π f t d t
  • Definition of Inverse Fourier transform
    w ( t ) = - W ( f ) e j 2 π f t d f
  • Fourier transform of a sine
    F { A sin ( 2 π f 0 t + Φ ) } = j A 2 - e j Φ δ ( f - f 0 ) + e - j Φ δ ( f + f 0 )
  • Fourier transform of a cosine
    F { A cos ( 2 π f 0 t + Φ ) } = A 2 e j Φ δ ( f - f 0 ) + e - j Φ δ ( f + f 0 )
  • Fourier transform of impulse
    F { δ ( t ) } = 1
  • Fourier transform of rectangular pulse
  • With
    Π ( t ) = 1 - T / 2 t T / 2 0 otherwise ,
    F { Π ( t ) } = T sin ( π f T ) π f T T sinc ( f T ) .
  • Fourier transform of sinc function
    F { sinc ( 2 W t ) } = 1 2 W Π f 2 W
  • Fourier transform of raised cosine
  • With
    w ( t ) = 2 f 0 sin ( 2 π f 0 t ) 2 π f 0 t cos ( 2 π f Δ t ) 1 - ( 4 f Δ t ) 2 ,
    F { w ( t ) } = 1 | f | < f 1 1 2 1 + cos π ( | f | - f 1 ) 2 f Δ f 1 < | f | < B 0 | f | > B ,
    with the rolloff factor defined as β = f Δ / f 0 .
  • Fourier transform of square-root raised cosine (SRRC)
  • With w ( t ) given by
    1 T sin ( π ( 1 - β ) t / T ) + ( 4 β t / T ) cos ( π ( 1 + β ) t / T ) ( π t / T ) ( 1 - ( 4 β t / T ) 2 ) t 0 , ± T 4 β 1 T ( 1 - β + ( 4 β / π ) ) t = 0 β 2 T 1 + 2 π sin π 4 β + 1 - 2 π cos π 4 β t = ± T 4 β ,
    F { w ( t ) } = 1 | f | < f 1 1 2 1 + cos π ( | f | - f 1 ) 2 f Δ 1 / 2 f 1 < | f | < B 0 | f | > B .
  • Fourier transform of periodic impulse sampled signal
  • With
    F { w ( t ) } = W ( f ) ,
    and
    w s ( t ) = w ( t ) k = - δ ( t - k T s ) , F { w s ( t ) } = 1 T s n = - W ( f - ( n / T s ) ) .
  • Fourier transform of a step
  • With
    w ( t ) = A t > 0 0 t < 0 , F { w ( t ) } = A δ ( f ) 2 + 1 j 2 π f .
  • Fourier transform of ideal π / 2 phase shifter (Hilbert transformer) filterimpulse response
  • With
    w ( t ) = 1 π t t > 0 0 t < 0 , F { w ( t ) } = - j f > 0 j f < 0 .
  • Linearity property
  • With F { w i ( t ) } = W i ( f ) ,
    F { a w 1 ( t ) + b w 2 ( t ) } = a W 1 ( f ) + b W 2 ( f ) .
  • Duality property With F { w ( t ) } = W ( f ) ,
    F { W ( t ) } = w ( - f ) .
  • Cosine modulation frequency shift property
  • With F { w ( t ) } = W ( f ) ,
    F { w ( t ) cos ( 2 π f c t + θ ) } = 1 2 e j θ W ( f - f c ) + e - j θ W ( f + f c ) .
  • Exponential modulation frequency shift property
  • With F { w ( t ) } = W ( f ) ,
    F { w ( t ) e j 2 π f 0 t } = W ( f - f 0 ) .
  • Complex conjugation (symmetry) property If w ( t ) is real valued,
    W * ( f ) = W ( - f ) ,
    where the superscript * denotes complex conjugation (i.e.,  ( a + j b ) * = a - j b ) . In particular, | W ( f ) | is even and W ( f ) is odd.
  • Symmetry property for real signals Suppose w ( t ) is real.
    If w ( t ) = w ( - t ) , then W ( f ) is real. If w ( t ) = - w ( - t ) , W ( f ) is purely imaginary.
  • Time shift property
  • With F { w ( t ) } = W ( f ) ,
    F { w ( t - t 0 ) } = W ( f ) e - j 2 π f t 0 .
  • Frequency scale property
  • With F { w ( t ) } = W ( f ) ,
    F { w ( a t ) } = 1 a W ( f a ) .
  • Differentiation property
  • With F { w ( t ) } = W ( f ) ,
    d w ( t ) d t = j 2 π f W ( f ) .
  • Convolution multiplication property
  • With F { w i ( t ) } = W i ( f ) ,
    F { w 1 ( t ) * w 2 ( t ) } = W 1 ( f ) W 2 ( f )
    and
    F { w 1 ( t ) w 2 ( t ) } = W 1 ( f ) * W 2 ( f ) ,
    where the convolution operator “ * ” is defined via
    x ( α ) * y ( α ) - x ( λ ) y ( α - λ ) d λ .
  • Parseval's theorem
  • With F { w i ( t ) } = W i ( f ) ,
    - w 1 ( t ) w 2 * ( t ) d t = - W 1 ( f ) W 2 * ( f ) d f .
  • Final value theorem
  • With lim t - w ( t ) = 0 and w ( t ) bounded,
    lim t w ( t ) = lim f 0 j 2 π f W ( f ) ,
    where F { w ( t ) } = W ( f ) .

Energy and power

  • Energy of a continuous time signal s ( t ) is
    E ( s ) = - s 2 ( t ) d t
    if the integral is finite.
  • Power of a continuous time signal s ( t ) is
    P ( s ) = lim T 1 T - T / 2 T / 2 s 2 ( t ) d t
    if the limit exists.
  • Energy of a discrete time signal s [ k ] is
    E ( s ) = - s 2 [ k ]
    if the sum is finite.
  • Power of a discrete time signal s [ k ] is
    P ( s ) = lim N 1 2 N k = - N N s 2 [ k ]
    if the limit exists.
  • Power Spectral Density
  • With input and output transforms X ( f ) and Y ( f ) of a linear filter with impulse response transform H ( f ) (such that Y ( f ) = H ( f ) X ( f ) ),
    P y ( f ) = P x ( f ) | H ( f ) | 2 ,
    where the power spectral density (PSD) is defined as
    P x ( f ) = lim T | X T ( f ) | 2 T ( Watts / Hz ) ,
    where F { x T ( t ) } = X T ( f ) and
    x T ( t ) = x ( t ) Π t T ,
    where Π ( · ) is the rectangular pulse [link] .

Z-transforms and properties

  • Definition of the Z-transform
    X ( z ) = Z { x [ k ] } = k = - x [ k ] z - k
  • Time-shift property
  • With Z { x [ k ] } = X ( z ) ,
    Z { x [ k - Δ ] } = z - Δ X ( z ) .
  • Linearity property
  • With Z { x i [ k ] } = X i ( z ) ,
    Z { a x 1 [ k ] + b x 2 [ k ] } = a X 1 ( z ) + b X 2 ( z ) .
  • Final Value Theorem for z -transforms If X ( z ) converges for | z | > 1 and all poles of ( z - 1 ) X ( z ) are inside the unit circle, then
    lim k x [ k ] = lim z 1 ( z - 1 ) X ( z ) .

Integral and derivative formulas

  • Sifting property of impulse
    - w ( t ) δ ( t - t 0 ) d t = w ( t 0 )
  • Schwarz's inequality
    - a ( x ) b ( x ) d x 2 - | a ( x ) | 2 d x - | b ( x ) | 2 d x
    and equality occurs only when a ( x ) = k b * ( x ) , where superscript * indicates complex conjugation (i.e.,  ( a + j b ) * = a - j b ).
  • Leibniz's rule
    d a ( x ) b ( x ) f ( λ , x ) d λ d x = f ( b ( x ) , x ) d b ( x ) d x - f ( a ( x ) , x ) d a ( x ) d x + a ( x ) b ( x ) f ( λ , x ) x d λ
  • Chain rule of differentiation
    d w d x = d w d y d y d x
  • Derivative of a product
    d d x ( w y ) = w d y d x + y d w d x
  • Derivative of signal raised to a power
    d d x ( y n ) = n y n - 1 d y d x
  • Derivative of cosine
    d d x cos ( y ) = - ( sin ( y ) ) d y d x
  • Derivative of sine
    d d x sin ( y ) = ( cos ( y ) ) d y d x

Matrix algebra

  • Transpose transposed
    ( A T ) T = A
  • Transpose of a product
    ( A B ) T = B T A T
  • Transpose and inverse commutativity If A - 1 exists,
    A T - 1 = A - 1 T .
  • Inverse identity If A - 1 exists,
    A - 1 A = A A - 1 = I .

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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