The Passband Inversion Trick

Recently I learned a clever trick from Ken Wetzel and Bob Smith, each of whom independently discovered it. The problem is this: how do you measure the distortion of an FM signal generator? The difficulty is that any FM detector you might use will have its own distortion. How can you separate the distortion of the generator from that of the detector?

Until recently, I never imagined that such a thing was possible. My approach was to build the most linear FM detector I could manage, usually a pulse counter, and assume that it was perfectly linear. It can't be, of course, but by careful design and attention to detail, you can minimize the major sources of error. However, measuring whatever residual distortion remains is a problem.

It is possible to characterize a detector if you have an essentially perfect generator. One way to make one is to digitally generate an FM signal by direct synthesis. You can put an upper bound on the distortion based on the D/A linearity, and you can bound S/N by word length and dither. This is practical, and the residual distortion can be extremely low and S/N very high. But I haven't yet built such a device.

Bob and Ken's trick lets you independently characterize even-order distortion products for any generator and detector. It relies on the asymmetry of even-order distortion. Any nonlinearity that follows a power law, where the power is an even integer, causes the shape of positive and negative waveform excursions to differ.

This shows a sine wave with 25% second-harmonic distortion. The waveform asymmetry is obvious. When a signal like this is frequency modulated, the instantaneous RF frequency does not mirror-image about the center frequency. It traces out a curve like that shown. If you could invert the signal spectrum, the phase of the detected fundamental would flip 180 degrees. The key insight is that the even-order distortion products also would invert their phase. Any product that added to the detector's own distortion would subtract after inversion. You measure different distortion values for the two cases, and with a little algebra, you can separate the two contributions. Moreover, if the generator distortion is adjustable, you can just tweak it until the distortion products for the normal and inverted signals are equal. The resulting distortion is the detector distortion, and you've nulled the generator distortion. Since the dominant distortion product for a varactor frequency modulator is the second, and the generator will use second-order compensation to cancel the distortion, in practice this means that you can null the generator's second harmonic. What's so surprising is that the detector's second harmonic need not be zero to accomplish this.

Let g be the generator distortion, d the detector distortion, and m1 and m2 the two measurements, all linear amplitudes. Then

d + g = m1
d – g = m2

d = (m1 + m2) / 2
g = (m1 – m2) / 2

m1 = m2 => d = m1 and g = 0

This trick relies on the distortion products being either in phase or out of phase so that scalar arithmetic works and cancellation can occur if you adjust the generator. For example, if d and g were 90 degrees out of phase, m1 could equal m2 without either d or g being zero. For the usual 1-kHz test signal, phase must not vary much between 1 and 2 kHz. This is almost certainly true for a varactor frequency modulator. It also very likely holds for a signal generator's audio and stereo-generator circuits, even though the composite lowpass filter may introduce a very slight phase shift. Phase may not be quite as constant in a tuner with narrow RF tuned circuits, narrow ceramic IF filters with substantial group delay, an aggressive postdetection filter, or a narrowband quadrature detector or discriminator. Still, I believe low-frequency phase shift is negligible for modern wide-IF designs, although this could stand verification. This won't be an issue for laboratory modulation analyzers.

To invert the signal spectrum, use a mixer and a second signal generator, which I'll call the local oscillator. In one case put the LO below the signal frequency, and in the other case, above. For example, to characterize a signal generator at 98 MHz, tune a low-distortion FM tuner to 108 MHz. Set the LO at 10 MHz. The mixer sum product at 108 MHz will be noninverted. Measure the distortion level and then set the LO at 206 MHz. The mixer difference product at 108 MHz will be inverted. The signal generator and tuner frequencies remain constant so that their distortion levels don't vary.

Instead of a mixer, I use my Marconi 2305 modulation meter. This instrument has a wideband, low-distortion, charge-pump detector (a specific kind of pulse counter). Because it has an untuned front-end, I can tune either to the signal frequency or 3 MHz below to obtain a signal at the 1.5-MHz IF. The passband inverts when I switch frequency.

The generator and detector distortions should not greatly differ for accurate results. For example, if g and d differ by 20 dB, m1 and m2 will differ by about 1.7 dB. This is a small number, but one that is pretty easy to measure, or null, to a fraction of a dB. However, accuracy may begin to degrade much beyond this. A 10-dB difference would be even better. The technique is most sensitive when the distortion levels are equal.

Ken and Bob's trick won't help with odd-order distortion. Such waveforms, like this sine wave with a 10% third harmonic, are always symmetrical. But the third harmonic is often smaller than the second, and it has another nice property. When the fundamental drops, the second harmonic generally drops more rapidly, and the third even more so. For stereo with a 9% pilot, the baseband fundamental drops nearly 7 dB. This often puts the second harmonic near the noise floor and the third below it. Things are more complex and less favorable for a tuner, where the baseband and subchannel signals sum in the stereo decoder, providing no drop in fundamental level past that point. In addition, nonuniform group delay in the ceramic IF filters distorts the stereo subchannel more than the baseband signal. These factors usually cause tuner distortion to be higher in stereo than in mono, sometimes with a significant third harmonic.

More is here.

Updated May 7, 2008