Sinusoidal Functions, Waves, and Signals Continued

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Sinusoidal Functions, Waves, and Signals Continued
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Waves and Signals

• Recall that the general form for a sinusoidal
  wave is v(t) A sin (?t ?)
• Fourier analysis can be used to determine the
  frequency components of a signal
• Periodic signals have distinct frequency
  components a fundamental frequency and
  harmonics, found from the Fourier series
• Nonperiodic signals have a continuous frequency
  spectrum, found from the Fourier Transform
• There are other ways to analyze signals, but
  Fourier is the most basic and common, and is used
  as a first step in signal processing

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Digitizing Signals

• Signals are sampled and converted to digital when
  they are sampled at discrete time points
• The number of samples per second is known as the
  sampling rate
• The Nyquist criteria and sampling theorem tell us
  that the sampling rate should be at least twice
  the highest frequency component of the signal
• Digital signals also have discrete or digitized
  amplitudes

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Digitizing signals
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Digitizing Signals

• The amplitude of the points in the signal is also
  digitized into bits
• The resolution of the system is based on the
  number of bits used in the digitization
• The use of n bits gives 2n levels
• So, for example, a 3 bit system will resolve the
  signal into 8 distinct levels
• Increasing the number of bits increases the
  resolution and therefore the accuracy of the
  digitized signal, but also increases the memory
  storage requirements
• All computer based computations will
  automatically digitize signals, because they all
  operate on numerical values, not functions
• The figure on the next page shows the previous
  signal digitized using three bit (8 level)
  resolution

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Digitized Signal
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Signal Transmission

• Traditional transmission of signals occurs along
  conductors, either along cables or in tranmission
  lines
• The signal can be transmitted in this way as an
  analog voltage
• It can also be transmitted as a series of voltage
  pulses, representing ones and zeros in a
  digital signal

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Signal Transmission

• Standard wire cables are used for low frequency
  signals
• Higher frequency signals and signals that are
  susceptible to noise (like cable) are transmitted
  via coaxial cable an external conductor acts as
  a shield for the internal, signal carrying wire
• Fiber optic transmission can be used in either
  digital or analog signal transmission, but
  current applications of FO transmission are
  almost all digital

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Wireless Transmission

• Wireless transmission of signal occurs using the
  ambient atmosphere as a transmission medium.
• Wireless transmission requires a transmitter
  antenna and a receiver antenna
• Wireless transmission almost always requires that
  the signal be superimposed on a carrier wave of
  higher frequency

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Wireless Transmission

• Some frequencies propagate more easily through
  the atmosphere than others
• Among these frequencies, the FCC has designated
  certain bands to be set aside for different types
  of signal transmission (e.g., cell phone, TV,
  military communications, etc)
• The signal is transmitted in the form of an
  electromagnetic traveling wave

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Traveling Waves

• Traveling waves are different from sinusoidal
  currents and voltages because they propagate
  through the atmosphere
• The mathematical form of a sinusoidal traveling
  wave is given by
• E(t) A cos(?t - kx)
• Where k is 2pi divided by the wavelength and x
  is the direction of wave propagation
• This wave looks like a sine wave that is moving
  through space so if you observe the wave at an
  instant of time, it is spread out in space, if
  you look at one point in space, you will see
  sinusoidal variation in time

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Complex Number Review

• Sinusoidal functions are frequently represented
  using complex number notation this simplifies
  problems by representing them using the phase and
  amplitude only
• Sines and cosines are related to exponential
  functions through complex number theory and
  Eulers relation

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Complex Numbers

• A complex number has a real part and an imaginary
  part
• For example, consider the complex number c a
  jb in this case j is the imaginary number
  often referred to in mathematics as i, and
  equal to the square root of negative one.
• The real part of the complex number c is a and
  the imaginary part is jb
• In electrical engineering we use j because i
  is usually reserve for current
• The expression for c given above is referred to
  as the rectangular form of a complex number

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Complex Numbers

• Complex numbers can be plotted like vectors, with
  the vertical axis representing the imaginary part
  and the horizontal axis representing the real
  part
• This is called the complex plane an example is
  shown on the next page, plotting c a jb

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Complex Number Plot
imaginary
b
Complex number
real
a
The length of the vector a jb is known as the
magnitude the Angle the vector forms with the
real axis is known as the phase
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Complex Numbers

• Because of the relationship of phase and
  magnitude, often complex numbers are expressed in
  polar form, as shown in the equation to the
  right.
• Eulers Formula can be used to convert from polar
  form to rectangular form.
• The equations on the previous page can be used to
  convert from rectangular form to polar form.

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Eulers Formula
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Sinusoidal Signals

• Because of Eulers formula, a typical sinusoidal
  signal can be represented as an amplitude and
  phase
• First, be sure the sinusoidal function is given
  as a cosine function
• Next, write the cosine function as the real part
  of the exponential equivalent using Eulers
  formula
• Drop the time variation, remembering that it is
  present in the cosine function, but not a part of
  the phase and amplitude
• Finally, write the phase and amplitude in
  shorthand notation, known as phasor notation
• To go back to the time variation, simply plug
  the phase and amplitude into the cosine function,
  adding back in the frequency and time part of the
  expression