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 Chebyshev and Butterworth Responses

The Chebyshev response is a mathematical strategy for achieving a faster *roll-off* by allowing *ripple* in the frequency response. Analog and digital filters that
use this approach are called *Chebyshev filters*. For instance, analog Chebyshev
filters were used in Chapter 3 for analog-to-digital and digital-to-analog
conversion. These filters are named from their use of the *Chebyshev
polynomials*, developed by the Russian mathematician Pafnuti Chebyshev
(1821-1894). This name has been translated from Russian and appears in the
literature with different spellings, such as: Chebychev, Tschebyscheff,
Tchebysheff and Tchebichef.

Figure 20-1 shows the frequency response of low-pass Chebyshev filters with passband ripples of: 0%, 0.5% and 20%. As the ripple increases (bad), the roll-off becomes sharper (good). The Chebyshev response is an optimal trade-off between these two parameters. When the ripple is set to 0%, the filter is called a maximally flat or Butterworth filter (after S. Butterworth, a British engineer who described this response in 1930). A ripple of 0.5% is a often good choice for digital filters. This matches the typical precision and accuracy of the analog electronics that the signal has passed through.

The Chebyshev filters discussed in this chapter are called type 1 filters,
meaning that the ripple is only allowed in the *passband*. In comparison,

type 2 Chebyshev filters have ripple only in the *stopband*. Type 2 filters are
seldom used, and we won't discuss them. There is, however, an important
design called the elliptic filter, which has ripple in *both* the passband and the
stopband. Elliptic filters provide the fastest roll-off for a given number of
poles, but are much harder to design. We won't discuss the elliptic filter here,
but be aware that it is frequently the first choice of professional filter designers,
both in analog electronics and DSP. If you need this level of performance, buy
a software package for designing digital filters.