1.5 Sinusoidal signals

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 :

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.

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.