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!

Harmonics

If a signal is periodic with frequency *f*, the only frequencies composing the
signal are integer multiples of *f*, i.e., *f*, 2*f*, 3*f*, 4*f*, etc. These frequencies are
called harmonics. The first harmonic is *f*, the second harmonic is 2*f*, the third
harmonic is 3*f*, and so forth. The first harmonic (i.e., *f*) is also given a special name, the fundamental frequency. Figure 11-7 shows an

example. Figure (a) is a pure sine wave, and (b) is its DFT, a single peak. In
(c), the sine wave has been distorted by poking in the tops of the peaks. Figure
(d) shows the result of this distortion in the frequency domain. Because the
distorted signal is periodic with the same frequency as the original sine wave,
the frequency domain is composed of the original peak plus harmonics.
Harmonics can be of any amplitude; however, they usually become smaller as
they increase in frequency. As with any signal, *sharp edges* result in *higher
frequencies*. For example, consider a common TTL logic gate generating a 1
kHz square wave. The edges rise in a few nanoseconds, resulting in harmonics
being generated to nearly 100 MHz, the *ten-thousandth* harmonic!

Figure (e) demonstrates a subtlety of harmonic analysis. If the signal is symmetrical around a horizontal axis, i.e., the top lobes are mirror images of the bottom lobes, all of the even harmonics will have a value of zero. As shown in (f), the only frequencies contained in the signal are the fundamental, the third harmonic, the fifth harmonic, etc.

All *continuous* periodic signals can be represented as a summation of
harmonics, just as described. *Discrete* periodic signals have a problem that
disrupts this simple relation. As you might have guessed, the problem is
*aliasing*. Figure 11-8a shows a sine wave distorted in the same manner as
before, by poking in the tops of the peaks. This waveform looks much less
regular and smooth than in the previous example because the sine wave is at a
much higher frequency, resulting in fewer samples per cycle. Figure (b) shows
the frequency spectrum of this signal. As you would expect, you can identify
the fundamental and harmonics. This example shows that harmonics can extend
to frequencies greater than 0.5 of the sampling frequency, and will be *aliased*
to frequencies somewhere between 0 and 0.5. You don't notice them in (b)
because their amplitudes are too low. Figure (c) shows the frequency spectrum
plotted on a logarithmic scale to reveal these low amplitude aliased peaks. At
first glance, this spectrum looks like random noise. It isn't; this is a result of the
many harmonics overlapping as they are aliased.

It is important to understand that this example involves distorting a signal *after*
it has been digitally represented. If this distortion occurred in an analog signal,
you would remove the offending harmonics with an antialias filter *before*
digitization. Harmonic aliasing is only a problem when nonlinear operations are
performed directly on a discrete signal. Even then, the amplitude of these
aliased harmonics is often low enough that they can be ignored.

The concept of harmonics is also useful for another reason: it explains why the
DFT views the time and frequency domains as *periodic*. In the frequency
domain, an *N* point DFT consists of *N*/2+1 equally spaced frequencies. You can
view the frequencies *between* these samples as (1) having a value of zero, or (2)
not existing. Either way they don't contribute to the synthesis of the time
domain signal. In other words, a *discrete* frequency spectrum consists of
*harmonics*, rather than a continuous range of frequencies. This requires the time
domain to be periodic with a frequency equal to the lowest sinusoid in the
frequency domain, i.e., the fundamental frequency. Neglecting the DC value,
the lowest frequency represented in the frequency domain makes one complete
cycle every *N* samples, resulting in the time domain being periodic with a period
of *N*. In other words, if one domain is *discrete*, the other domain must be
*periodic*, and vice versa. This holds for all four members of the Fourier
transform family. Since the DFT views both domains as discrete, it must also
view both domains as periodic. The samples in each domain represent
harmonics of the periodicity of the opposite domain.