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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 :
and the sine :
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\pi /\Omega $ . The frequency of the signals is $\Omega =2\pi /T$ which has units of rad/sec . Equivalently, the frequency can be expressed as $1/T$ , which has units of $se{c}^{-1}$ , cycles/sec , or Hz . The quantity $\Omega t$ 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.
and since $T=2\pi /\Omega $ , we have
Similarly, we have
Using Euler's Identity, we can also write:
and
The quantity ${e}^{j\Omega t}$ is called a complex sinusoid and can be expressed as
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.
$sin\left(\theta \right)=cos(\theta -\pi /2)$ |
$cos\left(\theta \right)=sin(\theta +\pi /2)$ |
$sin\left({\theta}_{1}\right)sin\left({\theta}_{2}\right)=\frac{1}{2}\left[cos,({\theta}_{1}-{\theta}_{2}),-,cos,({\theta}_{1}+{\theta}_{2})\right]$ |
$sin\left({\theta}_{1}\right)cos\left({\theta}_{2}\right)=\frac{1}{2}\left[sin,({\theta}_{1}-{\theta}_{2}),-,sin,({\theta}_{1}+{\theta}_{2})\right]$ |
$cos\left({\theta}_{1}\right)cos\left({\theta}_{2}\right)=\frac{1}{2}\left[cos,({\theta}_{1}-{\theta}_{2}),+,cos,({\theta}_{1}+{\theta}_{2})\right]$ |
$acos\left(\theta \right)+bsin\left(\theta \right)=\sqrt{{a}^{2}+{b}^{2}}cos\left(\theta ,-,{tan}^{-1},\left(\frac{b}{a}\right)\right)$ |
$cos({\theta}_{1}\pm {\theta}_{2})=cos\left({\theta}_{1}\right)cos\left({\theta}_{2}\right)\mp sin\left({\theta}_{1}\right)sin\left({\theta}_{2}\right)$ |
$sin({\theta}_{1}\pm {\theta}_{2})=sin\left({\theta}_{1}\right)cos\left({\theta}_{2}\right)\pm sin\left({\theta}_{1}\right)cos\left({\theta}_{2}\right)$ |
$\frac{d}{dt}cos\left(\Omega t\right)=-\Omega sin\left(\Omega t\right)$ |
$\frac{d}{dt}sin\left(\Omega t\right)=\Omega cos\left(\Omega t\right)$ |
$\int cos\left(\Omega t\right)dt=\frac{1}{\Omega}sin\left(\Omega t\right)$ |
$\int sin\left(\Omega t\right)dt=-\frac{1}{\Omega}cos\left(\Omega t\right)$ |
${\int}_{0}^{T}sin\left(k{\Omega}_{o}t\right)cos\left(n{\Omega}_{o}t\right)dt=0$ |
${\int}_{0}^{T}sin\left(k{\Omega}_{o}t\right)sin\left(n{\Omega}_{o}t\right)dt=0,k\ne n$ |
${\int}_{0}^{T}cos\left(k{\Omega}_{o}t\right)cos\left(n{\Omega}_{o}t\right)dt=0,k\ne n$ |
${\int}_{0}^{T}{sin}^{2}\left(n{\Omega}_{o}t\right)dt=T/2$ |
${\int}_{0}^{T}{cos}^{2}\left(n{\Omega}_{o}t\right)dt=T/2$ |
Now suppose that we have a sum of two sinusoids, say
It is of interest to know what the period $T$ of the sum of 2 sinusoids is. We must have
It follows that ${\Omega}_{1}T=2\pi k$ and ${\Omega}_{2}T=2\pi l$ , where $k$ and $l$ are integers. Solving these two equations for $T$ gives $T=2\pi k/{\Omega}_{1}=2\pi l/{\Omega}_{2}$ . 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 $2\pi k/{\Omega}_{1}=2\pi l/{\Omega}_{2}$ , we can write
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 ${\Omega}_{1}=10\pi $ and ${\Omega}_{2}=15\pi $ , we have $k=2$ and $l=3$ , after dividing out 5, the greatest common divisor between 10 and 15. So the period is $T=2\pi k/{\Omega}_{1}=0.4$ sec. On the other hand, if ${\Omega}_{1}=10\pi $ and ${\Omega}_{2}=10.1\pi $ , we find that $k=100$ and $l=101$ and the period increases to $T=2\pi k/{\Omega}_{1}=20$ sec. Notice also that if the ratio of ${\Omega}_{1}$ and ${\Omega}_{2}$ is not a rational number, then $x\left(t\right)$ 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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