There are many nuisances associated with using polar notation. None of these are overwhelming, just really annoying! Table 8-3 shows a computer program for converting between rectangular and polar notation, and provides solutions for some of these pests.
Nuisance 1: Radians vs. Degrees
It is possible to express the phase in either degrees or radians. When expressed in degrees, the values in the phase signal are between -180 and 180. Using radians, each of the values will be between -π and π, that is, -3.141592 to 3.141592. Most computer languages require the use radians for their trigonometric functions, such as cosine, sine, arctangent, etc. It can be irritating to work with these long decimal numbers, and difficult to interpret the data you receive. For example, if you want to introduce a 90 degree phase shift into a signal, you need to add 1.570796 to the phase. While it isn't going to kill you to type this into your program, it does become tiresome. The best way to handle this problem is to define the constant, PI = 3.141592, at the beginning of your program. A 90 degree phase shift can then be written as . Degrees and radians are both widely used in DSP and you need to become comfortable with both.
Nuisance 2: Divide by zero error
When converting from rectangular to polar notation, it is very common to find frequencies where the real part is zero and the imaginary part is some nonzero value. This simply means that the phase is exactly 90 or -90 degrees. Try to tell your computer this! When your program tries to calculate the phase from: Phase X[k] = arctan(ImX[k]/ReX[k]), a divide by zero error occurs. Even if the program execution doesn't halt, the phase you obtain for this frequency won't be correct. To avoid this problem, the real part must be tested for being zero before the division. If it is zero, the imaginary part must be tested for being positive or negative, to determine whether to set the phase to π/2 or -π/2, respectively. Lastly, the division needs to be bypassed. Nothing difficult in all these steps, just the potential for aggravation. An alternative way to handle this problem is shown in line 250 of Table 8-3. If the real part is zero, change it to a negligibly small number to keep the math processor happy during the division.
Nuisance 3: Incorrect arctan
Consider a frequency domain sample where ReX[k] = 1 and ImX[k] = 1. Equation 8-6 provides the corresponding polar values of Mag X[k] = 1.414 and Phase X[k] = 45°. Now consider another sample where ReX[k] = -1 and ImX[k] = -1. Again, Eq. 8-6 provides the values of Mag X[k] = 1.414 and Phase X[k] = 45?. The problem is, the phase is wrong! It should be -135°. This error occurs whenever the real part is negative. This problem can be corrected by testing the real and imaginary parts after the phase has been calculated. If both the real and imaginary parts are negative, subtract 180° (or π radians) from the calculated phase. If the real part is negative and the imaginary part is positive, add 180° (or π radians). Lines 340 and 350 of the program in Table 8-3 show how this is done. If you fail to catch this problem, the calculated value of the phase will only run between -π/2 and π/2, rather than between -π and π. Drill this into your mind. If you see the phase only extending to ±1.5708, you have forgotten to correct the ambiguity in the arctangent calculation.
Nuisance 4: Phase of very small magnitudes
Imagine the following scenario. You are grinding away at some DSP task, and suddenly notice that part of the phase doesn't look right. It might be noisy, jumping all over, or just plain wrong. After spending the next hour looking through hundreds of lines of computer code, you find the answer. The corresponding values in the magnitude are so small that they are buried in round-off noise. If the magnitude is negligibly small, the phase doesn't have any meaning, and can assume unusual values. An example of this is shown in Fig. 8-11. It is usually obvious when an amplitude signal is lost in noise; the values are so small that you are forced to suspect that the values are meaningless. The phase is different. When a polar signal is contaminated with noise, the values in the phase are random numbers between -π and π. Unfortunately, this often looks like a real signal, rather than the nonsense it really is.
Nuisance 5: 2π ambiguity of the phase
Look again at Fig. 8-10d, and notice the several discontinuities in the data. Every time a point looks as if it is going to dip below -3.14592, it snaps back to 3.141592. This is a result of the periodic nature of sinusoids.
For example, a phase shift of θ, is exactly the same as a phase shift of θ+2π, θ+4π, θ+6π, etc. Any sinusoid is unchanged when you add an integer multiple of 2π to the phase. The apparent discontinuities in the signal are a result of the computer algorithm picking its favorite choice from an infinite number of equivalent possibilities. The smallest possible value is always chosen, keeping the phase between -π and π.
It is often easier to understand the phase if it does not have these discontinuities, even if it means that the phase extends above π, or below -π. This is called unwrapping the phase, and an example is shown in Fig. 8-12. As shown by the program in Table 8-4, a multiple of 2π is added or subtracted from each value of the phase. The exact value is determined by an algorithm that minimizes the difference between adjacent samples.
Nuisance 6: The magnitude is always positive (π ambiguity of the phase)
Figure 8-13 shows a frequency domain signal in rectangular and polar form. The real part is smooth and quite easy to understand, while the imaginary part is entirely zero. In comparison, the polar signals contain abrupt
discontinuities and sharp corners. This is because the magnitude must always be positive, by definition. Whenever the real part dips below zero, the magnitude remains positive by changing the phase by π (or -π, which is the same thing). While this is not a problem for the mathematics, the irregular curves can be difficult to interpret.
One solution is to allow the magnitude to have negative values. In the example of Fig. 8-13, this would make the magnitude appear the same as the real part, while the phase would be entirely zero. There is nothing wrong with this if it helps your understanding. Just be careful not to call a signal with negative values the "magnitude" since this violates its formal definition. In this book we use the weasel words: unwrapped magnitude to indicate a "magnitude" that is allowed to have negative values.
Nuisance 7: Spikes between π and -π
Since π and -π represent exactly the same phase shift, round-off noise can cause adjacent points in the phase to rapidly switch between the two values. As shown in (d), this can produce sharp breaks and spikes in an otherwise smooth curve. Don't be fooled, the phase isn't really this discontinuous.