Signal processing involves techniques that improve our understanding
of information contained in received ultrasonic data. Normally,
when a signal is measured with an oscilloscope, it is viewed in
the time domain (vertical axis is amplitude or voltage and the
horizontal axis is time). For many signals, this is the most logical
and intuitive way to view them. Simple signal processing often
involves the use of gates to isolate the signal of interest or
frequency filters to smooth or reject unwanted frequencies.
When the frequency content of the signal is of interest, it makes
sense to view the signal graph in the frequency domain. In the
frequency domain, the vertical axis is still voltage but the horizontal
axis is frequency.
-----------Time Domain --------------------Frequency
The frequency domain display shows how much of the
signal's energy is present as a function of frequency. For a simple
signal such as a sine wave, the frequency domain representation
does not usually show us much additional information. However,
with more complex signals, such as the response of a broad bandwidth
transducer, the frequency domain gives a more useful view of the
Fourier theory says that any complex periodic waveform can be
decomposed into a set of sinusoids with different amplitudes,
frequencies and phases. The process of doing this is called Fourier
Analysis, and the result is a set of amplitudes, phases, and frequencies
for each of the sinusoids that makes up the complex waveform.
Adding these sinusoids together again will reproduce exactly the
original waveform. A plot of the frequency or phase of a sinusoid
against amplitude is called a spectrum.
The following Fourier Java applet, adapted with permission of
Stanford University, allows the user to manipulate discrete time
domain or frequency domain components and see the relationships
between signals in time and frequency domains.
The top row (light blue color) represents the real and imaginary
parts of the time domain. Normally the imaginary part of the time
domain signal is identically zero.
The middle row (peach color) represents the the real and imaginary
parts of the frequency domain.
The bottom row (light green color) represents the magnitude (amplitude)
and phase of the frequency domain signal. Magnitude is the square
root of the sum of the squares of the real and imaginary components.
Phase is the angular relationship of the real and imaginary components.
Ultrasonic transducer manufactures often provide plots of both
time domain and frequency domain (magnitude) signals characteristic
of each transducer. Use this applet to explore the relationship
between time and frequency domains.
Try replicating time domain signal in the upper left box with
a pattern similar to the image on the right. Note the resulting
bandwidth in the frequency domain (magnitude) in the lower left
box. Next try changing the magnitude, perhaps more of a "mountain"
shape tapering to zero. Note that "narrowing" the magnitude,
results in more cycles in the time domain signal.