Understanding the Frequency Domain
By Mike Cowart WA5CMI
A must for audio and radio work
Before we discuss how to amplitude modulate, we must first acquaint ourselves with the
frequency domain. Modulation is much easier to understand when explained within it.
French mathematician and physicist Baron Jean Baptiste Joseph Fourier (1768-1830)
realized that any complex waveform could be decomposed into a group of sinusoids of
different frequencies and amplitudes. Wow! You say, but what the heck does that mean?
One could say that he forced us to look at complex waveforms in a different way: in the
frequency domain. Different from what? The time domain. His discovery was long before
the first radio transmission was made. We use Fourier Transforms extensively in radio, as
well as audio, work. They allow us to transform a function from the time domain to the
frequency domain.
Just why do we need this ability to see complex waveforms in the frequency domain? In
the case of radio waves we need to know the bandwidth of our emitted signal, how much
space we occupy. We need to know if our signal contains spurious emissions (spurs or
splatter). In the case of audio we need to know the frequency composition or spectral
content: how the lows, mids, and highs make up our transmitted audio and the amplitudes
of each.
Time domain
We are very entrenched in the time domain. Every time we look at a pattern on an
oscilloscope, we are viewing it in the time domain. What we see is a waveform that
varies in amplitude as time moves. Let’s look at the familiar, fundamental sine wave in
Figure 1.
Time is plotted along the horizontal axis and amplitude along the vertical. The sine
function has very distinct properties. It repeats itself between two constant extremes. That
is, it is periodic and it is circular. We normalize the sine function (a fancy term for
setting the minimum and maximum amplitudes to +/-1) so that the instantaneous value at
any point along the horizontal axis is less than +1 but greater than –1.
But what does this sine wave function look like in frequency domain? Before we answer
that, let’s make sure we understand that every sine wave has a frequency. Frequency is
defined as the inverse of the amount of time it takes to complete one cycle. Therefore, to
calculate the frequency in the time domain, determine the period of time it takes to
complete one cycle and invert it: frequency = 1 / period. In the frequency domain,
frequency is plotted as a point on the horizontal axis, and its amplitude is depicted as a
vertical line.
Frequency Domain
Let’s use an example sine wave with a period of 1 millisecond (one one-thousandth of a
second) to complete one cycle. The frequency would then be 1,000 Hz, somewhat in the
middle of the voice spectrum. This 1,000-Hz sine wave is represented in the frequency
domain as shown in Figure 2.
It is simply a vertical line with amplitude represented by its height, simple enough. But
an audio waveform is complex; it is comprised of many sine waves of different
amplitudes. So now we must see how these simple sine waves combine to make a
complex waveform.
Suppose we add another sine wave of the same amplitude with a frequency of 2000 Hz to
our 1000 Hz sine wave. Figure 3 shows how frequency 1 adds together with frequency 2
to form a complex waveform (albeit a simple one).
Study this figure carefully, and you can see how the complex wave represents the sum of
Frequencies 1 and 2 as they rise and fall at different times with different slopes. To make
the illustration simple, the two sine waves are the same amplitude. Also Notice that the
peak-to-peak amplitude of the sum is greater than either of the two individual sine waves.
This is because different points along the horizontal – or time – axis the peak amplitudes
two frequencies add together.
What does this complex waveform look like in the frequency domain? Before looking at
Figure 4 below, think what Ol’ Joe Fourier said. The complex waveform is comprised of
simple sine waves of various amplitudes. Below is the spectral display of our complex
waveform.
Surprised? Isn’t it easier to see the complex waveform in the frequency domain?
Now we are ready to look a very complex waveform: one generated by the human voice
with a full band behind it. The figure below is a snapshot of a few notes from “Hello
Dolly.” This was taken from a computer display, as is the one following.
Can you look at the waveform and determine the spectral content? At any given point
along the time axis, how many sine waves comprise the signal and what are their
individual amplitudes? Look at the frequency domain display in Figure 6 for the answer.
At a single point on the time axis, the frequency content was as is shown in Figure 6.
The spectral display gives us much more useful information than the time domain display
(oscillograph). For instance we can see that the upper low-end (between 300 and 500 Hz)
frequencies are dominant (perhaps a string bass or other lower-voiced instruments). The
midrange (600 to 1500 Hz) has some strong components, but it also has some that are
much lower. The high end falls off dramatically above 2,000 Hz With a little experience,
one can glean the same information from the oscillograph, but it is much more precise in
the frequency spectrum. It is important to remember that each vertical bar is a single sine
wave. Also, remember that this is a single point on the time axis and the spectral content
is always changing.
Important: Remember that the two figures above represent the same thing: a
complex waveform, first in the time domain and second in the frequency domain.
Do not try to relate what you see in the time-domain display to what you see in the
frequency-domain display. They are totally different domains.