Digital Signal Processing Reference
In-Depth Information
transform may seem like a waste of eort, but not if it greatly simplies the steps
in between.
If you have had a course in dierential equations, you may have seen the Laplace
transform used for just this purpose: to provide an easy way to solve these equations.
After the equation is set up, you apply the Laplace transform (often by looking it
up in a table), then the transformed equation can be manipulated and solved using
algebra, and nally the result is inverse-transformed back to the time-domain (again,
often by looking at a table). The Laplace transform also has a use in determining
the stability of systems, and this is something we will return to in a later chapter.
1.6
Why Do We Study Sinusoids?
We look at sinusoidal functions (sine and cosine) because they are interesting func-
tions that often appear in the \analog" world. Examining a single cosine function is
easy to do, and what applies to one cosine function also applies to a signal composed
of several sinusoids. Since sinusoidal functions occur frequently, it is nice to be able
to use a few pieces of data to represent a sinusoid. Almost every sinusoid in this
text will be of the following format:
amplitudecos(2frequencyt + phase).
That is all! When the amplitude, frequency, and phase are known, we have all the
information we need to nd the value of this sinusoid for any value of time (t). This
is a very compact way of representing a signal. If the signal is more complex, say
it is composed of two sinusoids, we just need the amplitude, frequency, and phase
information of the two sinusoids. In this way, we can represent the signal, then later
remake it from this information.
Think of how the earth rotates around the sun in a year's time. The earth gets
the same amount of sunlight every day, but the earth's tilt means that one hemi-
sphere or the other receives more sun. Starting at the December solstice (around
December 22), the earth's Northern hemisphere receives the least amount of sunlight
[8]. As the earth rotates around the sun, this hemisphere gets an increasing amount
of sunlight (spring time), until it reaches the maximum (summer), then the days get
shorter (fall), and it nally returns to the low point again in winter. The seasons
vary with time. In two dimensions, one could diagram this with the sun in the
center, and the earth following a circular path around it (approximately). Another
way of viewing this information is to graph the distance with time as the x-axis.
For the y-axis, consider the horizontal distance between the two. This graph looks
like a sine wave. The seasons change with time, but they always repeat. Therefore,
this signal is periodic.
