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Signal Processing Tutorial: The Basic Sinusoids

One common class of signals are sinusoids…that is they are either cosines or sines.  The video shown below briefly reviews the definition of a sine, cosine, and tangent.

[tubepress  mode="tag" tagValue= 'soh cah toa drjctu']

Mathematically, the cosine signal is generally given as

 x(t)=Acos({omega}t+{phi})= Acos(2{pi}f t+{phi})        (Equation 1)

where  {omega}=2{pi}f t is the angular frequency (radians/sec), f is the frequency (cycles/sec or Hertz) t is the independent variable time, and  {phi} is the phase shift, and A is the amplitude.  With f as the frequency, then T=1/f is the time period of the sinusoid.

Two sinusoidal waves offset from each other by...

Image via Wikipedia

The interactive graph below allows you to experiment with these parameters.   The default function shown below is a combination of a sine and cosine function.   It can be shown that the combination of sin(x)+cos(x) can be reduced to  Equation 1.  In fact, sin(x)+cos(x) can be viewed what is known as a rectangular form of asin(x)+bcos(x) or as a time function of

 x(t)= asin({omega}t)+bcos({omega}t)      (Equation 2)

Equation 1 can be viewed as the polar representation of a sinusoid since the function has amplitude and phase while Equation 2 is the rectagular representation of a general sinusoid with a an b as the components of the function x(t).   You can use conversion formulas similar used to going from one representation to another found in complex numbers.

In this case, the conversion formulas are

 a= Acos({omega}t)

 b=-Asin({omega}t)

A=sqrt{a^2+b^2}

 {phi}=tan(frac{-b}{a})

Again, using the interactive flash graph below, try changing the frequency, amplitude and phase to see how these parameters change.

[swf src="http://www.freedomuniversity.tv/resources/SimpleGraphExample/SimpleGraphExample.swf" width=671 height=452 ]

One reason why sinusoids are important has to do with physical systems since many can be modeled mathematically as sine or cosine functions of time.  For example, we can hear audio signals, including sinusoidal ones.  Tones from instruments have different pitches as a result of the different frequencies they produce.

With today’s technology, you can use a computer with an analog-to-digital converter (A-to-D) and a microphone, we can record digitally produced by an instrument.  The mi9crophone is a transducer which converts mechanical sound into an electrical signal.  This in turn takes the electrical signal coverts into a sequence of numbers that can now be stored in the computer.    A typical plot is similar to the ones shown above.

The bottom line is that common physical systems produce signals whose graphical representations look like sinusoidal signals.   You’ll learn that the sinusoidal functions (sine and cosine functions) are actually solutions to differential equations derived from the laws of physics to describe physical phenomena such as a tuning fork.

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