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Sinusoidal signals

Sinusoidal signals are perhaps the most important type of signal that we will encounter in signal processing. There are two basic types of signals, the cosine :

x ( t ) = A cos ( Ω t )

and the sine :

x ( t ) = A sin ( Ω t )

where A is a real constant. Plots of the sine and cosine signals are shown in [link] . Sinusoidal signals are periodic signals. The period of the cosine and sine signals shown above is given by T = 2 π / Ω . The frequency of the signals is Ω = 2 π / T which has units of rad/sec . Equivalently, the frequency can be expressed as 1 / T , which has units of s e c - 1 , cycles/sec , or Hz . The quantity Ω t has units of radians and is often called the phase of the sinusoid. Recalling the effect of a time shift on the appearance of a signal, we can observe from [link] that the sine signal is obtained by shifting the cosine signal by T / 4 seconds, i.e.

Cosine and sine signals. Each signal is periodic with period T = 2 π / Ω .
sin ( Ω t ) = cos ( Ω ( t - T / 4 ) )

and since T = 2 π / Ω , we have

sin ( Ω t ) = cos ( Ω t - π / 2 ) )

Similarly, we have

cos ( Ω t ) = sin ( Ω t + π / 2 ) )

Using Euler's Identity, we can also write:

A cos ( Ω t ) = A 2 e j Ω t + e - j Ω t

and

A sin ( Ω t ) = A 2 j e j Ω t - e - j Ω t

The quantity e j Ω t is called a complex sinusoid and can be expressed as

e ± j Ω t = cos Ω t ± j sin Ω t

There are a number of trigonometric identities which are sometimes useful. These are shown in [link] . [link] shows some basic calculus operations on sine and cosine signals.

Useful trigonometric identities.
sin ( θ ) = cos ( θ - π / 2 )
cos ( θ ) = sin ( θ + π / 2 )
sin ( θ 1 ) sin ( θ 2 ) = 1 2 cos ( θ 1 - θ 2 ) - cos ( θ 1 + θ 2 )
sin ( θ 1 ) cos ( θ 2 ) = 1 2 sin ( θ 1 - θ 2 ) - sin ( θ 1 + θ 2 )
cos ( θ 1 ) cos ( θ 2 ) = 1 2 cos ( θ 1 - θ 2 ) + cos ( θ 1 + θ 2 )
a cos ( θ ) + b sin ( θ ) = a 2 + b 2 c o s θ - tan - 1 b a
cos ( θ 1 ± θ 2 ) = cos ( θ 1 ) cos ( θ 2 ) sin ( θ 1 ) sin ( θ 2 )
sin ( θ 1 ± θ 2 ) = sin ( θ 1 ) cos ( θ 2 ) ± sin ( θ 1 ) cos ( θ 2 )
Derivatives and integrals of sinusoidal signals.
d d t cos ( Ω t ) = - Ω sin ( Ω t )
d d t sin ( Ω t ) = Ω cos ( Ω t )
cos ( Ω t ) d t = 1 Ω sin ( Ω t )
sin ( Ω t ) d t = - 1 Ω cos ( Ω t )
0 T sin ( k Ω o t ) cos ( n Ω o t ) d t = 0
0 T sin ( k Ω o t ) sin ( n Ω o t ) d t = 0 , k n
0 T cos ( k Ω o t ) cos ( n Ω o t ) d t = 0 , k n
0 T sin 2 ( n Ω o t ) d t = T / 2
0 T cos 2 ( n Ω o t ) d t = T / 2

Now suppose that we have a sum of two sinusoids, say

x ( t ) = cos ( Ω 1 t ) + cos ( Ω 2 t )

It is of interest to know what the period T of the sum of 2 sinusoids is. We must have

x ( t - T ) = cos ( Ω 1 ( t - T ) ) + cos ( Ω 2 ( t - T ) ) = cos ( Ω 1 t - Ω 1 T ) + cos ( Ω 2 t - Ω 2 T )

It follows that Ω 1 T = 2 π k and Ω 2 T = 2 π l , where k and l are integers. Solving these two equations for T gives T = 2 π k / Ω 1 = 2 π l / Ω 2 . We wish to select the shortest possible period, since any integer multiple of the period is also a period. To do this we note that since 2 π k / Ω 1 = 2 π l / Ω 2 , we can write

Ω 1 Ω 2 = k l

so we seek the smallest integers k and l that satisfy [link] . This can be done by finding the greatest common divisor between k and l . For example if Ω 1 = 10 π and Ω 2 = 15 π , we have k = 2 and l = 3 , after dividing out 5, the greatest common divisor between 10 and 15. So the period is T = 2 π k / Ω 1 = 0 . 4 sec. On the other hand, if Ω 1 = 10 π and Ω 2 = 10 . 1 π , we find that k = 100 and l = 101 and the period increases to T = 2 π k / Ω 1 = 20 sec. Notice also that if the ratio of Ω 1 and Ω 2 is not a rational number, then x ( t ) is not periodic!

If there are more than two sinusoids, it is probably easiest to find the period of one pair of sinusoids at a time, using the two lowest frequencies (which will have a longer period). Once the frequency of the first two sinusoids has been found, replace them with a single sinusoid at the composite frequency corresponding to the first two sinusoids and compare it with the third sinusoid, and so on.

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Source:  OpenStax, Signals, systems, and society. OpenStax CNX. Oct 07, 2012 Download for free at http://cnx.org/content/col10965/1.15
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