Discrete-time real and complex valued sinusoidal signals are an incredibly important signal class in the study of discrete-time signals and systems. Of course, sinusoidal waves show up in all sorts of science and engineering applications, but they are particularly relevant for signal processing because they are the foundation of Fourier analysis.
Real valued sinusoids
There are two real-valued discrete-time sinusoidal wave signals: the
sine wave signal and the
cosine wave signal. They are represented mathematically as $\sin(\omega n +\phi)$ and $\cos(\omega n +\phi)$. Let's take a look a those in more detail. First, as we have seen with other discrete-time signals, $n$ is the independent variable time index, and it runs from negative infinity to infinity. The variable $\omega$ is known as the
frequency of the sinusoidal signal, and we will see how changing the value of $\omega$ impacts the rate of the signal's oscillation. The variable $\phi$ is the
phase of the signal, and changing it will shift the signal left along the time axis. Finally, the terms $\sin$ or $\cos$ return the corresponding trigonometric value to $\omega n +\phi$ for each value of the time index $n$. Here are a few examples of real sinusoidal waves:
(a) A plot of $\cos(0n)$. At every point in time, this signal takes the value $\cos(0)=1$. (b) A plot of $\sin(0n)$. At every point in time, this signal takes the value $\sin(0)=0$. (c) A plot of $\sin(\frac{\pi}{4}n+\frac{2\pi}{6})$. (d) A plot of $\cos(\pi n)$. Note how when $\omega=\pi$, as in this example, the signal is oscillating as rapidly as possible, between -1 and 1 at every single time instance. This phenomenon is the opposite of when $\omega=0$, for which the signal does not oscillate at all. So in some sense we can see that $0$ is the lowest possible frequency $\omega$, and $\pi$ is the highest. We saw in the figure above how the frequency $\omega$ influences the rate of the wave's oscillation. The other variable in the signal, the phase $\phi$, can shift the wave backwards and forwards along the time axis, without affecting the frequency. Below are plots of a cosine wave which all have the same frequency, but with a variety of phase shifts (i.e., different values of $\phi$):
$\cos\left(\frac{\pi}{6}n-0\right)$.$\cos\left(\frac{\pi}{6}n-\frac{\pi}{4}\right)$.$\cos\left(\frac{\pi}{6}n-\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{6}n\right)$.$\cos\left(\frac{\pi}{6}n-2\pi\right) = \cos\left(\frac{\pi}{6}n\right)$.Four cosine waves with the same frequency, but different phases. Note how a phase shift of $\frac{\pi}{2}$ shifts the cosine to be a sine wave, and a phase shift of $2\pi$ shifts it all the way over to where it was without any phase shift (because the cosine is periodic for this frequency).
Complex valued sinusoids
So we have reviewed the real sine waves $\sin$ and $\cos$, and perhaps seeing them in proximity brought to mind a very special relationship called
Euler's Formula : $e^{j\theta}=\cos(\theta)+j\sin(\theta)$ (you may remember this from math class with an $i$ instead, but recall engineers use that letter for current, and we call the imaginary number $j$). That formula works for any particular value of $\theta$, so of course it applies when we consider $\omega n+\phi$, as above, which gives us a complex valued sinusoid: $e^{j(\omega n +\phi)}=\cos(\omega n +\phi)+j\sin(\omega n +\phi)$. Let's look at some plots of complex sinusoids. Unlike two-dimensional real sinusoids (which have an one-dimensional independent time variable $n$ and a take a one-dimensional value at each time value), complex sinusoids are three dimensional: they have the time dimension, a real dimension, and an imaginary dimension. So they can be visualized as a three dimensional helix in space:
A three dimensional visualization of a complex sinusoid. Note that this image is continuous-valued, whereas a discrete valued version would actually appear as points along the line. If you were to look at this helix from directly above, you would see only the real portion of the helix, and it would appear to be a cosine wave. If you looked at it from the side, you would see the imaginary aspect of it, as a sine wave. The frequency variable $\omega$ controls how quickly the helix rotates across time $n$, and also the direction: positive values cause it to rotate in the counterclockwise manner shown, and negative values would result in it rotating clockwise.
While it is illuminating to visualize complex simusiods in three dimensions, in practice it is actually most common to view them in two, separately plotting either the real and the imaginary parts with respect to time, or the magnitude and phase across time:
The magnitude of a complex sinusoid.The phase, or angle, of a complex sinusoid.A complex sinusoid plotted according to its magnitude and phase. Note that the magnitude of a single complex sinusoid is trivial, as $|e^{j(\omega n +\phi)}|=1$.The real part of a complex sinusoid.The imaginary part of a complex sinusoid.A complex sinusoid plotted according to its real and imaginary parts. These are a cosine, and sine, respectively, which follows from Euler's Formula. We'll wrap up our introduction of sinusoids by briefly considering the concept of negative-valued frequencies. It is easiest to see the difference a negative frequency makes, compared to a positive frequency of the same magnitude, by expressing it all mathematically:
$e^{j (-\omega) n} ~=~ e^{-j\omega n} ~=~ \cos(-\omega n) + j \sin(-\omega n) ~=~ \cos(\omega n) - j \sin(\omega n)$So negating the frequency of a complex sinusoid has no effect on the real part of the signal (the cosine), but it flips the sign of the imaginary part (the sine). This operation (preserving the real part, but changing the sign of the imaginary part) is also known as taking the complex conjugate of the signal. So negating the frequency of a complex sinusoid is the same thing as taking the complex conjugate of it:$e^{j (-\omega) n} ~=~ e^{-j \omega n}~=~ \left( e^{j \omega n} \right)^*$.
Why use imaginary numbers?
Now perhaps you are wondering the point of using imaginary numbers. After all, aren't all real world signals, well, real-valued? They are indeed, but we can consider them as the real-part of a complex-valued signal. And why go to that trouble? There are many good reasons, but here is one to start with: exponential functions are much easier to work with than trigonometric functions. You can easily simplify $e^{a} e^{b}$ into a single term, but you very likely would be turning to a table to simplify $\sin(a)\cos(b)$, wouldn't you?
