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!

Analog Measurements on a DSP System

For just a few moments, forget that you are studying *digital* techniques.
Let's take a look at this from the standpoint of an engineer that
specializes in *analog* electronics. He doesn't care what is inside of the
EZ-KIT Lite, only that it has an analog input and an analog output. As
shown in Fig. 29-3, he would invoke the traditional analog method of
analyzing a "black box," attach a signal generator to the input, and look
at the output on an oscilloscope.

What does our analog guru find? First, the system is *linear* (as least as
far as this simple test can tell). If a sine wave is placed into the input,
a sine wave is observed on the output. If the amplitude or frequency of
the input is changed, a corresponding change is seen in the output.
When the input frequency is slowly increased, there comes a point where
the amplitude of the output sine wave decreases rapidly to zero. That
occurs just below one-half the sampling rate, due to the action of the
anti-alias filter on the ADC.

Now our engineer notices something unknown in the analog world: the
system has a perfect *linear phase*. In other words, there is a constant
delay between an event occurring in the input signal, and the result of
that event in the output signal. For instance, consider our example filter
kernel in Fig. 29-3. Since the center of symmetry is at sample 150, the
output signal will be delayed by 150 samples relative to the input signal.
If the system is sampling at 8 kHz, for example, this delay will be 18.75
milliseconds. In addition, the sigma-delta converter will also provide a
small additional fixed delay.

Our analog engineer will become very agitated when he sees this linear phase. The signals won't appear the way he thinks they should, and he will start twisting knobs at lightning speed. He will complain that the triggering isn't working right, and mumble such things as: "this doesn't make sense," what's going on here?", and "who's been playing with my oscilloscope?" The performance of DSP systems is so good, it will take him a few minutes before he understands what he is seeing.

To make him even more impressed, we ask our engineer to manually measure the frequency response of the system. To do this, he will step the signal generator through all the frequencies between 125 Hz and 10 kHz in increments of 125 Hz. At each frequency he measures the amplitude of the output signal and divides it by the amplitude of the input signal. (Of course, the easiest way to do this is to keep the input signal at a constant amplitude). We set the sampling rate of the EZ-KIT Lite at 22 kHz for this test. In other words, the 0 to 0.5 digital frequency of Fig. 29-2a is mapped to DC to 11 kHz in our real world measurement.

Figure 29-4 shows actual measurements taken on the EZ-KIT Lite; it
couldn't be better! The measured data points agree with the theoretical
curve within the limit of measurement error. This is something our
analog engineer has *never* seen with filters made from resistors,
capacitors, and inductors.

However, even this doesn't give the DSP the credit it deserves. Analog
measurements using oscilloscopes and digital-volt-meters have a typical
accuracy and precision of about 0.1% to 1%. In comparison, this DSP
system is limited only by the â ˜0.001% round-off error of the 16 bit
codec, since the internal calculations use floating point. In other words,
the device being evaluated is *one-hundred times *more precise than the
measurement tool being used. A proper evaluation of the frequency
response would require a specialized instrument, such as a computerized
data acquisition system with a 20 bit ADC. Given these facts, it is not
surprising that DSPs are often used in measurement instruments to
achieve high precision.

Now we can answer the question: Why does *Analog** Devices* sell *Digital*
*Signal Processors*? Only a decade ago, state-of-the-art signal processing
was carried out with precision op amps and similar transistor circuits.
Today, the highest quality *analog* processing is accomplished with *digital*
techniques. Analog Devices is a great role-model for individuals and
other companies; hold on to your vision and goals, but don't be afraid
to adapt with the changing technology!