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!

Other Transform Pairs

Figure 11-5 (a) and (b) show the duality of the above: a rectangular pulse in the frequency domain corresponds to a sinc function (plus aliasing) in the time domain. Including the effects of aliasing, the time domain signal is given by:

To eliminate the effects of aliasing from this equation, imagine that the frequency domain is so finely sampled that it turns into a continuous curve. This makes the time domain infinitely long with no periodicity. The DTFT is the Fourier transform to use here, resulting in the time domain signal being given by the relation:

This equation is very important in DSP, because the rectangular pulse in the
frequency domain is the perfect *low-pass filter*. Therefore, the sinc function
described by this equation is the filter kernel for the perfect low-pass filter.
This is the basis for a very useful class of digital filters called the *windowed-sinc
filters*, described in Chapter 15.

Figures (c) & (d) show that a triangular pulse in the time domain coincides with
a sinc function *squared* (plus aliasing) in the frequency domain. This transform
pair isn't as important as the *reason* it is true. A 2*M* - 1 point triangle in the time domain can be formed by convolving an *M* point rectangular pulse with itself.
Since convolution in the time domain results in multiplication in the frequency
domain, convolving a waveform with itself will *square* the frequency spectrum.

Is there a waveform that is its own Fourier Transform? The answer is yes, and
there is *only* one: the Gaussian. Figure (e) shows a Gaussian curve, and (f)
shows the corresponding frequency spectrum, also a Gaussian curve. This
relationship is only true if you ignore aliasing. The relationship between the
standard deviation of the time domain and frequency domain is given by: 2πσ_{f} = 1/σ_{t}. While only one side of a Gaussian is shown in (f), the negative frequencies in the spectrum complete the full curve, with the center of symmetry at zero frequency.

Figure (g) shows what can be called a Gaussian burst. It is formed by
multiplying a sine wave by a Gaussian. For example, (g) is a sine wave
multiplied by the same Gaussian shown in (e). The corresponding frequency
domain is a Gaussian centered somewhere other than zero frequency. As before,
this transform pair is not as important as the *reason* it is true. Since the time
domain signal is the multiplication of two signals, the frequency domain will be
the convolution of the two frequency spectra. The frequency spectrum of the
sine wave is a delta function centered at the frequency of the sine wave. The
frequency spectrum of a Gaussian is a Gaussian centered at zero frequency.
Convolving the two produces a Gaussian centered at the frequency of the sine
wave. This should look familiar; it is identical to the procedure of *amplitude
modulation* described in the last chapter.