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!

Special Properties of Linearity

Linearity is commutative, a property involving the combination of two or more
systems. Figure 5-10 shows the general idea. Imagine two systems combined
in a cascade, that is, the output of one system is the input to the next. If each
system is linear, then the overall combination will also be linear. The
commutative property states that the order of the systems in the cascade can be
rearranged without affecting the characteristics of the overall combination.
You probably have used this principle in electronic circuits. For example,
imagine a circuit composed of two stages, one for amplification, and one for
filtering. Which is best, amplify and then filter, or filter and then amplify? If
both stages are linear, the order doesn't make any difference and the overall
result is the same. Keep in mind that actual electronics has *nonlinear* effects
that may make the order important, for instance: interference, DC offsets,
internal noise, slew rate distortion, etc.

Figure 5-8 shows the next step in linear system theory: multiple inputs and
outputs. A system with multiple inputs and/or outputs will be linear if it is
composed of *linear subsystems* and *additions of signals*. The complexity does
not matter, only that nothing *nonlinear* is allowed inside of the system.

To understand what linearity means for systems with multiple inputs and/or outputs, consider the following thought experiment. Start by placing a signal on one input while the other inputs are held at zero. This will cause the multiple outputs to respond with some pattern of signals. Next, repeat the procedure by placing another signal on a different input. Just as before, keep all of the other inputs at zero. This second input signal will result in another pattern of signals appearing on the multiple outputs. To finish the experiment, place both signals on their respective inputs simultaneously. The signals appearing on the outputs will simply be the superposition (sum) of the output signals produced when the input signals were applied separately.

The use of multiplication in linear systems is frequently misunderstood. This
is because multiplication can be either linear or nonlinear, depending on what
the signal is multiplied by. Figure 5-9 illustrates the two cases. A system that
multiplies the input signal by a *constant*, is linear. This system is an amplifier
or an attenuator, depending if the constant is greater or less than one,
respectively. In contrast, multiplying a signal by *another signal* is nonlinear.
Imagine a sinusoid multiplied by another sinusoid; the resulting waveform is
clearly not sinusoidal.

Another commonly misunderstood situation relates to parasitic signals added
in electronics, such as DC offsets and thermal noise. Is the addition of these
extraneous signals linear or nonlinear? The answer depends on where the
contaminating signals are viewed as originating. If they are viewed as coming
from *within* the system, the process is nonlinear. This is because a sinusoidal
input does not produce a pure sinusoidal output. Conversely, the extraneous
signal can be viewed as *externally* entering the system on a separate input of a
multiple input system. This makes the process linear, since only a signal
addition is required within the system.