Digital Signal Processing

By Steven W. Smith, Ph.D.

- 1: The Breadth and Depth of DSP
- 2: Statistics, Probability and Noise
- 3: ADC and DAC
- 4: DSP Software
- 5: Linear Systems
- 6: Convolution
- 7: Properties of Convolution
- 8: The Discrete Fourier Transform
- 9: Applications of the DFT
- 10: Fourier Transform Properties
- 11: Fourier Transform Pairs
- 12: The Fast Fourier Transform
- 13: Continuous Signal Processing
- 14: Introduction to Digital Filters
- 15: Moving Average Filters
- 16: Windowed-Sinc Filters
- 17: Custom Filters
- 18: FFT Convolution
- 19: Recursive Filters
- 20: Chebyshev Filters
- 21: Filter Comparison
- 22: Audio Processing
- 23: Image Formation & Display
- 24: Linear Image Processing
- 25: Special Imaging Techniques
- 26: Neural Networks (and more!)
- 27: Data Compression
- 28: Digital Signal Processors
- 29: Getting Started with DSPs
- 30: Complex Numbers
- 31: The Complex Fourier Transform
- 32: The Laplace Transform
- 33: The z-Transform
- 34: Explaining Benford's Law

Your laser printer will thank you!

The Delta Function

Continuous signals can be decomposed into scaled and shifted *delta functions*,
just as done with discrete signals. The difference is that the continuous delta
function is much more complicated and mathematically abstract than its discrete
counterpart. Instead of defining the continuous delta function by what it *is*, we
will define it by the *characteristics it has*.

A thought experiment will show how this works. Imagine an electronic circuit
composed of linear components, such as resistors, capacitors and inductors.
Connected to the input is a signal generator that produces various shapes of
short *pulses*. The output of the circuit is connected to an oscilloscope,
displaying the waveform produced by the circuit in response to each input
pulse. The question we want to answer is: *how is the shape of the output pulse
related to the characteristics of the input pulse*? To simplify the investigation,
we will only use input pulses that are much shorter than the output. For
instance, if the system responds in milli-seconds, we might use input pulses
only a few microseconds in length.

After taking many measurement, we come to three conclusions: First, the *shape*
of the input pulse does not affect the shape of the output signal. This is
illustrated in Fig. 13-1, where various shapes of short input pulses produce
exactly the same shape of output pulse. Second, the shape of the output
waveform is totally determined by the characteristics of the system, i.e., the
value and configuration of the resistors, capacitors and inductors. Third, the
*amplitude* of the output pulse is directly proportional to the *area* of the input
pulse. For example, the output will have the same amplitude for inputs of: 1
volt for 1 microsecond, 10 volts for 0.1 microseconds, 1,000 volts for 1
nanosecond, etc. This relationship also allows for input pulses with *negative*
areas. For instance, imagine the combination of a 2 volt pulse lasting 2
microseconds being quickly followed by a -1 volt pulse lasting 4 microseconds.
The total area of the input signal is *zero*, resulting in the output doing *nothing*.

Input signals that are brief enough to have these three properties are called
impulses. In other words, an impulse is any signal that is entirely zero except
for a short *blip* of arbitrary shape. For example, an impulse to a microwave
transmitter may have to be in the *picosecond* range because the electronics
responds in *nanoseconds*. In comparison, a volcano that erupts for *years* may
be a perfectly good impulse to geological changes that take *millennia*.

Mathematicians don't like to be limited by any particular system, and commonly
use the term *impulse* to mean a signal that is short enough to be an impulse to
*any possible* system. That is, a signal that is *infinitesimally* narrow. The
continuous delta function is a normalized version of this type of impulse.
Specifically, the continuous delta function is mathematically defined by three
idealized characteristics: (1) the signal must be infinitesimally brief, (2) the
pulse must occur at time zero, and (3) the pulse must have an area of one.

Since the delta function is defined to be infinitesimally narrow *and* have a fixed
area, the amplitude is implied to be *infinite*. Don't let this bother you; it is
completely unimportant. Since the amplitude is part of the *shape* of the
impulse, you will never encounter a problem where the amplitude makes any
difference, infinite or not. The delta function is a mathematical construct, not
a real world signal. Signals in the real world that *act* as delta functions will
always have a finite duration and amplitude.

Just as in the discrete case, the continuous delta function is given the
mathematical symbol: δ( ). Likewise, the output of a continuous system in
response to a delta function is called the impulse response, and is often denoted by: *h*( ). Notice that parentheses, ( ), are used to denote continuous signals, as compared to brackets, [ ], for discrete signals. This notation is used in this
book and elsewhere in DSP, but isn't universal. Impulses are displayed in
graphs as vertical arrows (see Fig. 13-1d), with the *length* of the arrow
indicating the *area* of the impulse.

To better understand real world impulses, look into the night sky at a *planet* and
a *star*, for instance, Mars and Sirius. Both appear about the same brightness
and size to the unaided eye. The reason for this similarity is not

obvious, since the viewing geometry is drastically different. Mars is about 6000
kilometers in diameter and 60 million kilometers from earth. In comparison,
Sirius is about 300 times larger and over one-million times farther away. These
dimensions should make Mars appear more than *three-thousand* times larger
than Sirius. How is it possible that they look alike?

These objects look the same because they are small enough to be *impulses* to the
human visual system. The perceived shape is the impulse response of the eye,
not the actual image of the star or planet. This becomes obvious when the two
objects are viewed through a small telescope; Mars appears as a dim disk, while
Sirius still appears as a bright impulse. This is also the reason that stars twinkle
while planets do not. The image of a star is small enough that it can be briefly
blocked by particles or turbulence in the atmosphere, whereas the larger image
of the planet is much less affected.