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!

Requirements for Linearity

A system is called *linear* if it has two mathematical properties: homogeneity
(hōma-gen-ā-ity) and additivity. If you can show that a system has both
properties, then you have proven that the system is linear. Likewise, if you can
show that a system doesn't have one or both properties, you have proven that it
isn't linear. A third property, shift invariance, is not a strict requirement for
linearity, but it is a mandatory property for most DSP techniques. When you
see the term *linear system* used in DSP, you should assume it includes *shift
invariance* unless you have reason to believe otherwise. These three properties
form the mathematics of how linear system theory is defined and used. Later
in this chapter we will look at more intuitive ways of understanding linearity.
For now, let's go through these formal mathematical properties.

As illustrated in Fig. 5-2, homogeneity means that a change in the input signal's
amplitude results in a corresponding change in the output signal's amplitude.
In mathematical terms, if an input signal of *x[n]* results in an output signal of
*y[n]*, an input of *kx[n]* results in an output of *ky[n]*, for any input signal and constant, *k*.

A simple resistor provides a good example of both homogenous and non-homogeneous systems. If the input to the system is the voltage across the
resistor, *v(t)*, and the output from the system is the current through the resistor, *i(t)*
, the system is homogeneous. Ohm's law guarantees this; if the voltage is
increased or decreased, there will be a corresponding increase or decrease in the
current. Now, consider another system where the input signal is the voltage
across the resistor, *v(t)*, but the output signal is the power being dissipated in
the resistor, *p(t)*. Since power is proportional to the square of the voltage, if the
input signal is increased by a factor of *two*, the output signal is increase by a
factor of *four*. This system is not homogeneous and therefore cannot be linear.

The property of additivity is illustrated in Fig. 5-3. Consider a system where an
input of *x _{1}[n]* produces an output of

The important point is that *added* signals pass through the system without
interacting. As an example, think about a telephone conversation with your
Aunt Edna and Uncle Bernie. Aunt Edna begins a rather lengthy story about
how well her radishes are doing this year. In the background, Uncle Bernie is
yelling at the dog for having an accident in his favorite chair. The two voice
signals are added and electronically transmitted through the telephone network.
Since this system is additive, the sound you hear is the sum of the two voices as
they would sound if transmitted individually. You hear *Edna* and *Bernie*, not
the creature, *Ednabernie*.

A good example of a *nonadditive* circuit is the mixer stage in a radio transmitter.
Two signals are present: an audio signal that contains the voice or music, and
a carrier wave that can propagate through space when applied to an antenna.
The two signals are added and applied to a nonlinearity, such as a pn junction
diode. This results in the signals *merging* to form a third signal, a modulated
radio wave capable of carrying the information over great distances.

As shown in Fig. 5-4, shift invariance means that a shift in the input signal will
result in nothing more than an identical shift in the output signal. In more
formal terms, if an input signal of *x[n]* results in an output of *y[n]*, an input
signal of *x[n + s]* results in an output of *y[n + s]*, for any input signal and any
constant, *s*. Pay particular notice to how the mathematics of this shift is written,
it will be used in upcoming chapters. By adding a constant, *s*, to the
independent variable, *n*, the waveform can be advanced or retarded in the
horizontal direction. For example, when *s* = 2, the signal is shifted *left* by two
samples; when *s* = -2, the signal is shifted *right* by two samples.

Shift invariance is important because it means the characteristics of the system
do not change with time (or whatever the independent variable happens to be).
If a *blip* in the input causes a *blop* in the output, you can be assured that
another *blip* will cause an identical *blop*. Most of the systems you encounter
will be shift invariant. This is fortunate, because it is difficult to deal with
systems that change their characteristics while in operation. For example,
imagine that you have designed a digital filter to compensate for the degrading
effects of a telephone transmission line. Your filter makes the voices sound
more natural and easier to understand. Much to your surprise, along comes
winter and you find the characteristics of the telephone line have changed with
temperature. Your compensation filter is now mismatched and doesn't work
especially well. This situation may require a more sophisticated algorithm that
can *adapt* to changing conditions.

Why do homogeneity and additivity play a critical role in linearity, while shift
invariance is something on the side? This is because linearity is a very broad
concept, encompassing much more than just signals and systems. For example,
consider a farmer selling oranges for $2 per crate and apples for $5 per crate.
If the farmer sells only oranges, he will receive $20 for 10 crates, and $40 for
20 crates, making the exchange *homogenous*. If he sells 20 crates of oranges
and 10 crates of apples, the farmer will receive: . This
is the same amount as if the two had been sold individually, making the
transaction *additive*. Being both homogenous and additive, this sale of goods
is a linear process. However, since there are no signals involved, this is not a
*system*, and *shift invariance* has no meaning. Shift invariance can be thought of
as an additional aspect of linearity needed when signals and systems are
involved.