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!

Examples of Windowed-Sinc Filters

An electroencephalogram, or EEG, is a measurement of the electrical activity
of the brain. It can be detected as millivolt level signals appearing on electrodes
attached to the surface of the head. Each nerve cell in the brain generates small
electrical pulses. The EEG is the combined result of an enormous number of
these electrical pulses being generated in a (hopefully) coordinated manner.
Although the relationship between thought and this electrical coordination is
very poorly understood, different frequencies in the EEG can be identified with
specific mental states. If you close your eyes and relax, the predominant EEG
pattern will be a slow oscillation between about 7 and 12 hertz. This waveform
is called the *alpha rhythm*, and is associated with contentment and a decreased
level of attention. Opening your eyes and looking around causes the EEG to
change to the *beta rhythm*, occurring between about 17 and 20 hertz. Other
frequencies and waveforms are seen in children, different depths of sleep, and
various brain disorders such as epilepsy.

In this example, we will assume that the EEG signal has been amplified by analog electronics, and then digitized at a sampling rate of 100 samples per second. Acquiring data for 50 seconds produces a signal of 5,000 points. Our goal is to separate the alpha from the beta rhythms. To do this, we will design a digital low-pass filter with a cutoff frequency of 14 hertz, or 0.14

of the sampling rate. The transition bandwidth will be set at 4 hertz, or 0.04 of the sampling rate. From Eq. 16-3, the filter kernel needs to be about 101 points long, and we will arbitrarily choose to use a Hamming window. The program in Table 16-1 shows how the filter is carried out. The frequency response of the filter, obtained by taking the Fourier Transform of the filter kernel, is shown in Fig. 16-5.

In a second example, we will design a *band-pass filter* to isolate a *signaling tone*
in an audio signal, such as when a button on a telephone is pressed. We will
assume that the signal has been digitized at 10 kHz, and the goal is to isolate an
80 hertz band of frequencies centered on 2 kHz. In terms of the sampling rate,
we want to block all frequencies below 0.196 and above 0.204 (corresponding
to 1960 hertz and 2040 hertz, respectively). To achieve a transition bandwidth
of 50 hertz (0.005 of the sampling rate), we will make the filter kernel 801
points long, and use a Blackman window. Table 16-2 contains a program for
calculating the filter kernel, while Fig. 16-6 shows the frequency response. The
design involves several steps. First, *two* low-pass filters are designed, one with
a cutoff at 0.196, and the other with a cutoff at 0.204. This second filter is then
*spectrally inverted*, making it a high-pass filter (see Chapter 14, Fig. 14-6).
Next, the two filter kernels are added, resulting in a band-reject filter (see Fig.
14-8). Finally, another *spectral inversion* makes this into the desired band-pass
filter.