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!

Polar Notation

As it has been described so far, the frequency domain is a group of amplitudes
of cosine and sine waves (with slight scaling modifications). This is called
rectangular notation. Alternatively, the frequency domain can be expressed in
polar form. In this notation, *ReX*[ ] & *ImX*[ ] are replaced with two other arrays, called the Magnitude of *X*[ ] , written in equations as: *Mag X*[ ], and the
Phase of *X*[ ], written as: Phase *X*[ ]. The magnitude and phase are a pair-for-pair replacement for the real and imaginary parts. For example, *Mag X*[0] and *Phase X*[0]
are calculated using only *ReX*[0] and *ImX*[0]. Likewise, *Mag X*[14] and *Phase X*[14] are calculated using only *ReX*[14] and *ImX*[14], and so forth. To
understand the conversion, consider what happens when you add a cosine wave
and a sine wave of the same frequency. The result is a cosine wave of the same
frequency, but with a new amplitude and a phase shift. In equation form, the
two representations are related:

The important point is that no information is lost in this process; given one
representation you can calculate the other. In other words, the information
contained in the amplitudes *A* and *B*, is also contained in the variables *M* and θ. Although this equation involves sine and cosine waves, it follows the same
conversion equations as do simple vectors. Figure 8-9 shows the analogous
vector representation of how the two variables, *A* and *B*, can be viewed in a
rectangular coordinate system, while *M* and θ are parameters in polar
coordinates.

In polar notation, *Mag X*[ ] holds the amplitude of the cosine wave (*M* in Eq. 8-4 and Fig. 8-9), while *Phase X*[ ] holds the phase angle of the cosine wave (θ
in Eq. 8-4 and Fig. 8-9). The following equations convert the frequency
domain from rectangular to polar notation, and vice versa:

Rectangular and polar notation allow you to think of the DFT in two different
ways. With rectangular notation, the DFT decomposes an *N* point signal into *N*/2 + 1
cosine waves and *N*/2 + 1 sine waves, each with a specified *amplitude*. In polar
notation, the DFT decomposes an *N* point signal into *N*/2 + 1 cosine waves, each
with a specified *amplitude* (called the *magnitude*) and *phase shift*. Why does
polar notation use cosine waves instead of sine waves? Sine waves cannot
represent the DC component of a signal, since a sine wave of zero frequency is
composed of all zeros (see Figs. 8-5 a&b).

Even though the polar and rectangular representations contain exactly the same information, there are many instances where one is easier to use that the other. For example, Fig. 8-10 shows a frequency domain signal in both rectangular and polar form. Warning: Don't try to understand the shape of the real and imaginary parts; your head will explode! In comparison, the polar curves are straightforward: only frequencies below about 0.25 are present, and the phase shift is approximately proportional to the frequency. This is the frequency response of a low-pass filter.

When should you use rectangular notation and when should you use polar?
Rectangular notation is usually the best choice for calculations, such as in
equations and computer programs. In comparison, graphs are almost always in
polar form. As shown by the previous example, it is nearly impossible for
*humans* to understand the characteristics of a frequency domain signal by
looking at the real and imaginary parts. In a typical program, the frequency
domain signals are kept in rectangular notation until an observer needs to look
at them, at which time a rectangular-to-polar conversion is done.

Why is it easier to understand the frequency domain in polar notation? This
question goes to the heart of why decomposing a signal into sinusoids is *useful*.
Recall the property of *sinusoidal fidelity* from Chapter 5: if a sinusoid enters a
linear system, the output will also be a sinusoid, and at exactly the same
frequency as the input. Only the amplitude and phase can change. Polar
notation directly represents signals in terms of the amplitude and phase of the
component cosine waves. In turn, systems can be represented by how they
modify the amplitude and phase of each of these cosine waves.

Now consider what happens if rectangular notation is used with this scenario. A mixture of cosine and sine waves enter the linear system, resulting in a mixture of cosine and sine waves leaving the system. The problem is, a cosine wave on the input may result in both cosine and sine waves on the output. Likewise, a sine wave on the input can result in both cosine and sine waves on the output. While these cross-terms can be straightened out, the overall method doesn't match with why we wanted to use sinusoids in the first place.