When discussing electronic circuits, I often lock onto the circuit's magnitude characteristics. For example, we refer to filters as high pass, low pass, or band pass, which is a shorthand way to describe the shape of the magnitude of the transfer function. Many times, the phase response is an afterthought, but this overlooked characteristic can be vitally important.

Distortionless transmission

In Linear does not mean no distortion, I used this definition of distortionless: A system or network is called distortionless if its output is an exact replica of its input, except for amplitude scaling and time delay [Ref 1]. Put mathematically,




where
y(t) = output signal
x(t) = input signal
k = amplitude scale factor
t0 = time delay in the system

As discussed in the previous post, k represents constant gain for all frequencies of interest. Now let’s look at the x(tt0) term, which represents a time delay of t0.

Applying the Fourier transform to the equation results in



The gain factor, k is still present and remains a constant. The t0 delay becomes a linear phase term in the frequency domain (the exponential term). A specific time delay in seconds maps to an ever-increasing phase shift that is linear with frequency. Higher frequencies must have larger phase shifts compared with lower frequencies. For a circuit to conform to the distortionless definition, the transfer function must have linear phase.

Fifth-harmonic square wave

In Find a signal's bandwidth from its harmonics, we constructed a square wave from its frequency components up to the fifth harmonic (Figure 1). The waveform includes the fundamental, third harmonic and fifth harmonic—the even harmonics are not present. It's not a perfect square wave, of course, but it does a reasonable job of approximating a square wave. This fifth-harmonic square wave is a handy tool for demonstrating the effects of phase shifts.

square waveFigure 1. This square wave consists of the fundamental frequency plus the third and fifth harmonics.

Figure 2 shows the individual frequency components of the fifth-harmonic square wave: the fundamental, the third harmonic and the fifth harmonic. Take a look at the phase of each of these sinusoids and you’ll see that they line up just right to create the waveform in Fig. 1. The largest sine wave is the fundamental and its maximum value is aligned with the desired square wave shape. The third and fifth harmonics are smaller in amplitude, flattening the top of the waveform and filling in the corners.

square wave componentsFigure 2. The individual components of the square wave: fundamental, third harmonic, and fifth harmonic.

Figure 3 shows what happens when we disturb this alignment by shifting the fundamental by 30 degrees. (The third and fifth harmonics are not changed.) Note that the resulting waveform has the same frequency components as Fig. 1 but the change in phase distorts the waveform shape.

square wave distortionFigure 3. Distortion introduced to the square wave by shifting the phase of the fundamental by 30 degrees.

As shown earlier, a linear phase shift will delay the waveform but will not introduce distortion. Figure 4 shows the same fifth-harmonic square wave with a linear phase shift applied. The fundamental is again shifted by 30 degrees, while the third harmonic has three times the phase shift (90 degrees) and the fifth harmonic has five times the phase shift (150 degrees). We see the resulting waveform, which is identical to that in Fig. 1, but delayed in time.

square wave linear phase shiftFigure 4. Square wave with linear phase shift applied to all frequency components.

Nonlinear phase causes distortion

We’ve shown that the phase characteristics of a system can introduce distortion into a waveform. The square wave is a convenient waveform to use due to its harmonic content but other waveforms will also be affected in a similar manner. The key point is that a linear phase response (over the frequency range of interest) will avoid introducing distortion.

This disclaimer still applies: “One person's distortion is another person's desired signal.”

Reference

  1. Witte, Robert A. Spectrum and Network Measurements, Scitech Publishing, 2014 review.