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 Fourier Transform

The Fourier Transform for continuous signals is divided into two categories,
one for signals that are *periodic*, and one for signals that are *aperiodic*. Periodic
signals use a version of the Fourier Transform called the Fourier Series, and are
discussed in the next section. The Fourier Transform used with aperiodic
signals is simply called the Fourier Transform. This chapter describes these
Fourier techniques using only *real* mathematics, just as the last several chapters
have done for discrete signals. The more powerful use of *complex* mathematics
will be reserved for Chapter 29.

Figure 13-8 shows an example of a continuous aperiodic signal and its frequency spectrum. The time domain signal extends from negative infinity to positive infinity, while each of the frequency domain signals extends from zero to positive infinity. This frequency spectrum is shown in rectangular form (real and imaginary parts); however, the polar form (magnitude and phase) is also used with continuous signals. Just as in the discrete case, the synthesis equation describes a recipe for constructing the time domain signal using the data in the frequency domain. In mathematical form:

In words, the time domain signal is formed by adding (with the use of an
integral) an infinite number of scaled sine and cosine waves. The real part of
the frequency domain consists of the scaling factors for the cosine waves, while
the imaginary part consists of the scaling factors for the sine waves. Just as
with discrete signals, the synthesis equation is usually written with *negative* sine
waves. Although the negative sign has no significance in this discussion, it is
necessary to make the notation compatible with the complex mathematics
described in Chapter 29. The key point to remember is that some authors put
this negative sign in the equation, while others do not. Also notice that
frequency is represented by the symbol, ω, a lower case

Greek omega. As you recall, this notation is called the natural frequency, and
has the units of radians per second. That is, ω = 2π*f*, where *f* is the frequency in cycles per second (hertz). The natural frequency notation is favored by
mathematicians and others doing signal processing by *solving equations*,
because there are usually fewer symbols to write.

The analysis equations for continuous signals follow the same strategy as the
discrete case: *correlation* with sine and cosine waves. The equations are:

As an example of using the analysis equations, we will find the frequency response of the RC low-pass filter. This is done by taking the Fourier transform of its impulse response, previously shown in Fig. 13-4, and described by:

The frequency response is found by plugging the impulse response into the analysis equations. First, the real part:

Using this same approach, the imaginary part of the frequency response is calculated to be:

Just as with discrete signals, the rectangular representation of the frequency
domain is great for mathematical manipulation, but difficult for human
understanding. The situation can be remedied by converting into polar notation
with the standard relations: *MagH*(ω) = [*ReH*(ω^{2}) + *ImH*(ω^{2})]^{1/2} and *Phase H*(ω) = arctan[*ReH*(ω)/*ImH*(ω)]. Working through the algebra provides
the frequency response of the RC low-pass filter as magnitude and phase (i.e.,
polar form):

Figure 13-9 shows graphs of these curves for a cutoff frequency of 1000 hertz (i.e., α = 2π1000).